Topics in algebraic geometry
and geometric modeling
Pål Hermunn Johansen
ii
Contents
1 Introduction
2 The
2.1
2.2
2.3
2.4
2.5
1
tangent developable
Introduction . . . . . . . . . . . . . . . . . . . .
Tangent developables . . . . . . . . . . . . . . .
Local properties of a real tangent developable .
Illustrations . . . . . . . . . . . . . . . . . . . .
The tangent developable of a complex algebraic
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3 Closest points, moving surfaces and algebraic geometry
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 The underlying idea . . . . . . . . . . . . . . . . . . . . .
3.3 Degrees of the moving surfaces . . . . . . . . . . . . . . .
3.4 Implementation of a test algorithm . . . . . . . . . . . . .
3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4 Solving a closest point problem by subdivision
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Definition of the problem . . . . . . . . . . . . . . . . . .
4.2.1 The basic method . . . . . . . . . . . . . . . . . .
4.2.2 Quality of the output and special cases . . . . . .
4.3 Improving the basic method . . . . . . . . . . . . . . . . .
4.3.1 Changing the subdivision . . . . . . . . . . . . . .
4.3.2 Changing the multiplication algorithm to allow an
exit . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.3 Introducing a box test and a plane test . . . . . .
4.3.4 Using the second order derivatives . . . . . . . . .
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iv
CONTENTS
4.4
4.5
4.3.5 The recursive algorithm explained . . . .
4.3.6 The basic method with the box and plane
4.3.7 Doing a preconditioned constant sign test
4.3.8 Speed measurements . . . . . . . . . . . .
Error analysis . . . . . . . . . . . . . . . . . . . .
Conclusion . . . . . . . . . . . . . . . . . . . . .
5 Monoid hypersurfaces
5.1 Introduction . . . . . . .
5.2 Basic properties . . . . .
5.3 Monoid surfaces . . . . .
5.4 Quartic monoid surfaces
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6.1
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tests
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strata of quartic monoids
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Definition of the strata . . . . . . . . . . . . . . . . . . . . . . . . 71
Types 1 to 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Type 9 - the tangent cone is smooth . . . . . . . . . . . . . . . . 95
Chapter 1
Introduction
This PhD thesis was started as a part of the European Community funded
project “Intersection algorithms for geometry based IT-applications using approximate algebraic methods”. The project was coordinated by Tor Dokken of
SINTEF in Oslo and included partners from many countries: The University of
Cantabria from Spain, INRIA and the University of Nice from France, think3
from Italy and France, the University of Linz from Austria, and SINTEF and
the University of Oslo from Norway.
The vision of the project was to bring algebraic geometry and approximation
theory together and apply it to problems in Computer Aided Geometric Design
(CAGD). The imperfect quality of intersection algorithms in CAGD systems
imposes high costs on the product creation process in industry, and it was
deemed necessary to find new and better methods for solving these problems.
A better understanding of the geometrical objects in CAGD is needed to fulfill
this goal.
The vision and goal of the project has certainly influenced my work and
focus as a PhD student. As a direct result of that, the topics in this thesis cover
different parts of the project plan. Chapters 2, 5 and 6 provide insight into
objects that are interesting in CAGD and geometric modelling.
Chapters 3 and 4, on the other hand, investigate an important problem in
CAGD and other industrial applications, the closest point problem. The investigation covers both theoretical aspects and pure optimalization. The closest
point problem is so common that having fast algorithms is of great importance.
The topics of the chapters may seem unrelated, but they all address central
problems in applied geometry.
1
2
CHAPTER 1. INTRODUCTION
Chapter 2 is a study of tangent developables and was published [15] in the
proceedings of the first COMPASS workshop. Developable surfaces are common
in CAGD. A good understanding of these objects is helpful for programmers that
wish to use them in CAGD application programs.
The tangent developable of a curve C ⊂ P3 is a singular surface with cuspidal
edges along C and the flex tangents of C. It also contains a multiple curve,
typically double. We express the degree of this curve in terms of the invariants
of C. In many cases we can describe the intersections of C with the multiple
curve, and pictures of these cases are provided.
Chapter 3 describes a method of computing closest points to a parametric
surface patch. The chapter is the result of collaboration with Jan B. Thomassen
and Tor Dokken and was published [41] in the Proceedings after the conference
“Spline curves and surfaces” in Tromsø. The article was a collaboration between
three people and my responsibility was mainly the degree formulas and text in
Section 3.3.
The method for computing closest points to a given parametric surface patch
is based on “moving surfaces”. For each parametric surface there are two natural
moving surfaces, one for each parameter direction. These two objects let us
reduce the closest point problem for a given point to solving two univariate
polynomial equations. We also describe an implementation of our algorithm
which – although not being fast – is very reliable.
Chapter 4 is a written and improved version of a talk given at the MEGA
2005 conference. This chapter deals with the same problem as the previous
chapter, but solves the problem by subdivision techniques. Different ways of
solving this problem through subdivision are explored, and different optimizations are timed. An error analysis for subdivision methods is carried out, and
this gives the user full control over the guaranteed accuracy of the subdivision
methods.
Chapter 5 is an article written with my advisor Ragni Piene and one of her
other students, Magnus Løberg. This article has been accepted for the proceedings of the COMPASS 2 workshop, and is a study of monoid hypersurfaces.
A monoid hypersurface is an irreducible hypersurface of degree d which has a
singular point of multiplicity d − 1. Any monoid hypersurface admits a rational
parameterization, and is hence of potential interest in computer aided geometric
design. We study properties of monoids in general and of monoid surfaces in
particular. The main results include a description of the possible real forms of
the singularities on a monoid surface other than the point of multiplicity d − 1.
These results are applied to the classification of singularities on quartic monoid
surfaces, complementing earlier work on the subject.
3
My contribution has been formulating and proving the lemmas and propositions in this chapter, building on the work started by the other authors. In
particular, Proposition 5.8 and its constructive proof has been my contribution,
and also extending work in [38] into a complete classification of singularities
away from the triple point.
In Chapter 6 the classification of monoids is continued by considering the
space of quartic monoids in P3 with only isolated singularities. This space has a
natural stratification based on geometric invariants related to the singularites of
the monoids. The strata of monoids are defined by first defining the invariants,
and then defining when two different monoids are considered to have the same
set of invariants.
The result is a very high number of strata of monoids. By using the classification in the previous chapter, we are able to calculate the dimension of each
stratum. Also, if a stratum is associated to a singular tangent cone, then the
stratum can be expressed as an image of a certain map, and this construction
let us recover the number of components of the stratum.
During the work on the thesis over the last four years I have met many
interesting people and made several new friends. Many of these have inspired
and helped me complete my work, and I am happy to mention some of them
here.
First of all I would like to thank my advisor, Ragni Piene, for always being
positive and supportive in my efforts. I will also thank her for answering lots of
questions, for asking me the right questions, and for providing small hints when
my research has been incomplete or temporarily stuck.
I would also like to thank my cand.scient. advisor Jan Christophersen for his
effort in turning me into a worthy PhD candidate. Many thanks to Tor Dokken
for leading the successful GAIA II project and providing insight into the world
of CAGD.
I would like to thank Mohammed Elkadi, Bernard Mourrain and André
Galligo for help and advice during my stay in Nice.
Finally, I would like to thank the many fellow students with whom I shared
an office, Torquil Macdonald Sørensen, Tore Halsne Flåtten, Le Thi Ha, An
Ta Thi Kieu, Guillaume Chèze, Ola Nilsson, my good friends George Harry
Hitching and Oliver Labs, and, most of all, my girlfriend Maria Samuelsen. You
have all made the work on this thesis a better experience.
4
CHAPTER 1. INTRODUCTION
Chapter 2
The tangent developable
2.1
Introduction
If we have a curve on which tangents can be defined, then the associated tangent
developable is the surface swiped out by the tangents. Tangent developables
have a cuspidal edge, and are easy to generate. Since most developable surfaces
are tangent developables, the Computer Aided Geometric Design community
should be interested in their properties. This article describes the local and
global geometry of tangent developables.
For the local study of tangent developables we consider analytic real curves.
Cleave showed in [5] that for most curves the tangent developable has a cuspidal
edge along most of the curve. This was extended by Mond in [23] and [24]
where he analyzed the tangent developable of more special curves. This work
was further extended by Ishikawa in [13], and results from that article are used
in section 2.3.
The following section contains figures illustrating the local behavior of tantgent developables, and one may want to have a brief look at these before reading
the rest of the text.
In section 2.5 the tangent developables of complex projective algebraic curves
are described. Algebraic geometrical invariants are introduced and relations
between these invariants are taken from [31]. We also show that tangent developables of rational curves of degree ≥ 4 have a double curve.
Many thanks goes to Ragni Piene for lots of good advice and considerable
help with this article.
5
6
2.2
CHAPTER 2. THE TANGENT DEVELOPABLE
Tangent developables
Given a curve in some space, its tangent developable is the union of the tangent
lines to the curve. The tangent line at a singular point is defined as the limit of
tangent lines at non-singular points. If the curve is algebraic, then its tangent
developable will be an algebraic surface.
Assume we have a parameterization of a curve with a non-vanishing derivative. Then we can make a map that parameterizes the corresponding tangent
developable. Let U ⊂ R, γ : U → R3 be a map with a non-vanishing derivative.
Define the map Γ : U × R → R3 by
Γ(t, u) = γ(t) + uγ 0 (t)
(2.1)
In this case the tangent developable of γ(U ) is the image of Γ. The following example uses this technique to calculate the implicit equation of a tangent
developable.
Example 2.1 (The tangent developable of the twisted cubic). Consider the
twisted cubic curve parameterized by γ : R → R3 where γ(t) = (t, t2 , t3 ).
The tangent developable is then the image of Γ : R2 → R3 where Γ(t, u) =
(t + u, t2 + 2ut, t3 + 3ut2 ). The algebra program Singular [10] can calculate the
implicit equation of the surface:
z 2 − 6xyz + 4x3 z + 4y 3 − 3x2 y 2 = 0.
In this case the implicit equation describe the same set of points as the
the image of Γ. However, when dealing with real parameterizations this is not
always true.
Calculating the Jacobian ideal shows us that the tangent developable is
singular exactly at γ(R). Moreover, if the surface is intersected with a general
plane, the resulting curve will have a cusp singularity at each intersection point
with γ(R).
Definition 2.2 (The type of a germ). Let γ be a smooth (C ∞ ) curve germ,
γ : (R, p) → (R3 , q). We say that the germ is of finite type if the vectors
γ 0 (p), γ 00 (p), γ 000 (p), γ (4) (p), . . .
span R3 . In this case, let ai = min{k | dimhγ 0 (p), γ 00 (p), . . . , γ (k) (p)i = i} and
define the type of the germ to be the triple (a1 , a2 , a3 ).
2.3. LOCAL PROPERTIES OF A REAL TANGENT DEVELOPABLE
7
In this article we will only look at parameterizations where all the germs are
of finite type.
What does a tangent developable look like? Along most of the curve, the
tangent developable has a cuspidal edge singularity, so it is never smooth.
2.3
Local properties of a real tangent developable
We now want to study the local properties of the tangent developable close to
the curve. Now we are no longer forced to use complex numbers, so we choose
to study only real tangent developables. Since this is a local study, we now look
at germs of curves γ : (R, 0) → (R3 , 0), as in definition 2.2.
Cleave shows in [5] that the tangent developable of most smooth curves γ
have a cuspidal edge along most of the curve. That is, the cuspidal edge exists
at intervals of points of type (1, 2, 3). We have already decided only to look at
curves where all the points are of finite type, and for all of these curves we will
have a cuspidal edge along most of the curve.
In the language of Cleave: Given a curve with nonzero curvature and torsion
at a point γ(t0 ). If the tangent developable is intersected with a general plane
through γ(t0 ), the resulting curve will have a cusp at that point. In [24] Mond
provides drawings of the tangent developable at points of type (1, 2, k) for 3 ≤
k ≤ 7. This is (in the language of differential geometry) when the torsion
vanishes to order ≤ 4.
This was extended by Goo Ishikawa in [13], where he proves the following:
The local diffeomorphism class of the tangent developable is determined by the
type of the point if and only if the type is one of the following: (1, 2, 2+r) where
r is a positive integer, (1, 3, 4), (1, 3, 5), (2, 3, 4) or (3, 4, 5).
In other words, for these types we can restrict our study to curves on the
form
x = tl1 +1 =: ta
y
= tl2 +2 =: tb
z
= tl3 +3 =: tc
at the origin. For other types we have to include more terms (of the power
series) in the local parameterizations to study the point. In these cases we can
get several different real pictures, but since points of other types are quite exotic,
they will not be analyzed here.
8
CHAPTER 2. THE TANGENT DEVELOPABLE
Knowing this we can calculate local self intersection curves at points of type
(1, 2, k) quite easily:
Example 2.3 ((a, b, c) = (1, 2, k) for k ≥ 3). To find local self intersection
curves we need to solve the equation Γ(t, u) = Γ(s, v) where Γ is defined as in
equation (2.1), Γ(t, u) = (t + u, t2 + 2tu, tk + ktk−1 u). Some straightforward
calculations leads us to solving
−(t2 − s2 ) + 2w(t − s)
=
0
(1 − k)(t − sk ) + kw(tk−1 − sk−1 )
=
0,
k
where w = t + u = s + v.
Assuming s 6= t we (eventually) get
0
=
2(1 − k)(tk − sk ) + k(t + s)(tk−1 − sk−1 )
=
(2 − k)(tk−1 + sk−1 ) + 2(tk−2 s + tk−3 s2 + . . . + tsk−2 ).
It is not hard to prove that s = −t is the only possible real self intersection by
analyzing the polynomial f (t) = (2 − k)(tk−1 + 1) + 2(tk−2 + tk−3 + . . . + t) and
its derivative. The real self intersection occurs exactly when k is even. This is
compatible with what Mond found in [23], but since Mond looked at C ∞ curves
he could only draw the conclusion for k ≤ 7. Note that we have complex self
intersections for all k ≥ 5.
Example 2.4 (Types (1, 3, 4), (1, 3, 5), (2, 3, 4) and (3, 4, 5)). Points of types
(1, 3, 4), (1, 3, 5) and (2, 3, 4) each have one local real self intersection curve,
while points of type (3, 4, 5) have no real self intersection curves. This was
calculated using Singular [10].
The following section contains pictures of all of these types.
2.4
Illustrations
This section contains figures of tangent developables of different curves, each
parameterized by a map t → (ta , tb , tc ) for some triple (a, b, c). For each of the
curves, the origin is of type (a, b, c) and all other points (close to the origin)
are of type (1, 2, 3). For all the figures, we have drawn the points that are at a
distance of ≤ 2 from the origin, so the figures illustrate the local properties of
the tangent developable.
The first five figures show points of type (1, 2, k). We can see that we have
self intersection curves exactly when k is even, as calculated in example 2.3.
In the first figure, all points are of type (1, 2, 3):
2.4. ILLUSTRATIONS
9
For most curves, the only types are (1, 2, 3) and (1, 2, 4). The following figure
shows a point of type (1, 2, 4):
The following figures show points of type (1, 2, k).
10
CHAPTER 2. THE TANGENT DEVELOPABLE
The tangent developable of the curve (t, t2 , t5 )
The tangent developable of the curves (t, t2 , t6 ) and (t, t2 , t7 )
The rest of the figures come from example 2.4. Note that for the points
where k1 (0) = 1 (types (1, 3, 4) and (1, 3, 5)) the line which is a cuspidal edge,
but not part of the curve, is an inflectional tangent line. This corresponds to
the Plücker formula mentioned in section 2.5, c = r0 + k1 , where c is the degree
of the cuspidal edge.
The tangent developable of the curve (t, t3 , t4 )
2.4. ILLUSTRATIONS
The tangent developable of the curve (t, t3 , t5 )
The tangent developable of the curve (t2 , t3 , t4 )
The tangent developable of the curve (t3 , t4 , t5 )
11
12
2.5
CHAPTER 2. THE TANGENT DEVELOPABLE
The tangent developable of a complex algebraic curve
To any projective algebraic curve, there are associated several invariants, most
importantly the degree and genus of the curve. Classical algebraic geometry
gives many relations between these values and the geometry of the curve. In
[31] Piene obtained results for the tangent developable, and the formulas have
been taken from that article.
In this section a curve will be a reduced algebraic curve C0 in the projective
complex 3-space, P3C . We also assume that the curve spans the space. Let
X ⊂ P3C denote the tangent developable of C0 .
Let h : C → C0 be the normalization map, so that C is the desingularization
of C0 . Let g denote the (geometric) genus of the curve and r0 the degree.
The rank r1 is defined as the number of tangents that intersect a general line.
Clearly this is the same as the degree of the tangent developable. The class
r2 is defined as the number of osculating planes to C0 that contain a general
point. The osculating plane at a point on the curve is the plane with the highest
order of contact with the curve at that point. Another point of view is that the
osculating plane at a point x0 is the limit of the planes containing x0 , x1 and
x2 as x1 , x2 → x0 .
For each point p ∈ C, we can choose affine coordinates around h(p) such
that the branch of C0 determined by p has a (formal) parameterization at h(p)
equal to
x = tl1 +1 + . . .
y = tl2 +2 + . . .
z
= tl3 +3 + . . .
with l0 := 0 ≤ l1 ≤ l2 ≤ l3 . This (formal) parameterization is also a curve germ
γ : (C, 0) → (C3 , 0). Because of this we extend the notion of the type to the
complex domain, and say that the type of the germ determined by p is equal to
(l1 + 1, l2 + 2, l3 + 3).
The coordinates are chosen such that p is the origin, the tangent is the line
y = z = 0, and the osculating plane is z = 0. We call ki (p) = li+1 − li the ith
stationary indexPof p. Since ki (p) 6= 0 only for a finite number of points p we
can define ki = p∈C ki (p).
If l1 = 0, then the germ is nonsingular. If l1 ≥ 1 we say that the germ has
a cusp, and if l1 = 1 the cusp is said to be ordinary. If l1 = 0 and l2 ≥ 1 we
2.5. THE TANGENT DEVELOPABLE OF A COMPLEX CURVE
13
call the point h(p) an inflection point or flex, and if l2 = 1 the flex is ordinary.
If l1 = l2 = 0 and l3 ≥ 1 we say that the curve has a stall or a point of
hyperosculation. For most curves we will have no cusps and no flexes.
Now it is time to state the relations between these values, all taken from
[31]:
r1
=
2r0 + 2g − 2 − k0
(2.2)
r2
=
3(r0 + 2g − 2) − 2k0 − k1
(2.3)
k2
=
4(r0 + 3g − 3) − 3k0 − 2k1
(2.4)
Note that r1 ≥ 3 since since r1 is the degree of the tangent developable, and no
quadric surface with a cuspidal edge exists. Furthermore, r2 ≥ 3 since r2 is the
degree of the dual curve, and the dual curve must span the space. From the
definition we get k2 ≥ 0.
The tangent developable X of C0 has degree µ0 = r1 , rank µ1 = r2 (defined
as the class of the intersection of the tangent developable with a general plane,
a plane curve) and class µ2 = 0 (defined as the number of tangent planes
containing a general line). Its cuspidal edge consists of C0 and the flex tangents
of C0 . The cuspidal edge has degree c = r0 + k1 .
Formulas involving algebraic invariants, as those above, are often called
Plücker formulas, and such formulas is central in enumerative algebraic geometry. There are lots of Plücker formulas, relating many different algebraic
invariants.
In addition to the cuspidal edge, X has a double (or higher order multiple)
curve, some times called the nodal curve of C0 . It consists of points that are on
more than one tangent of C0 . Eventual bitangents are part of the nodal curve.
Let b denote the degree of the nodal curve. If the nodal curve is double and
the flexes of C0 are ordinary, then [31] gives the following expressions for b:
2b
= µ0 (µ0 − 1) − µ1 − 3c = r1 (r1 − 1) − r2 − 3(r0 + k1 )
= r1 (r1 − 4) − k0 − 2k1
=
(2r0 + 2g − 2 − k0 )(2r0 + 2g − 6 − k0 ) − k0 − 2k1 .
For rational curves, g = 0, so then
2b = (2r0 − 2 − k0 )(2r0 − 6 − k0 ) − k0 − 2k1 .
In this case we see that
k2 = 4(r0 − 3) − 3k0 − 2k1 ≥ 0
14
CHAPTER 2. THE TANGENT DEVELOPABLE
implies
k0 =
4
3 r0
− 4 − 23 k1 − 13 k2 ≤
4
3 r0
−4
We can find a lower bound for b for rational curves of degree r0 ≥ 4 by first
eliminating k1 (using equation (2.4)) in the expression for b:
2b
=
(2r0 − 2 − k0 )(2r0 − 6 − k0 ) − k0 − 2k1
=
(2r0 − 2 − k0 )(2r0 − 6 − k0 ) + 2k0 + k2 − 4r0 + 12
≥ (2r0 − 2 − k0 )(2r0 − 6 − k0 ) + 2k0 − 4r0 + 12 (using k2 ≥ 0).
As a function in k0 the expression above is strictly decreasing (for k0 ≤ 34 r0 −4).
In other words, we can set k0 = 34 r0 − 4 and not break the inequality:
2b
≥ (2r0 − 2 − ( 34 r0 − 4))(2r0 − 6 − ( 43 r0 − 4)) + 2( 34 r0 − 4) − 4r0 + 12
=
4
9 r0 (r0
− 3).
We conclude that rational curves with b = 0 must have degree ≤ 3, and the
twisted cubic is the only one of these that is not planar. It follows that every
rational curve C0 of degree greater than 3 gives a tangent developable with a
nodal curve of positive degree.
We want to check if b = 1 is possible. If g = 0 and b = 1, then 2b ≥ 94 r0 (r0 −3)
implies r0 = 4. Also, k0 ≤ 43 r0 − 4 = 43 . This leads us to consider two cases,
k0 = 0 and k0 = 1. If k0 = 0 the formula for b implies k1 = 5 and equation (2.4)
gives k2 = −6. If k0 = 1 the formula for b implies k1 = 1 and equation (2.4)
give k2 = −1. The second stationary index k2 cannot be negative, so b = 1 is
impossible.
The following example shows that b = 2 actually can occur for g = 0 and
r0 ≥ 4:
Example 2.5 (A singular curve of degree 4). Let the curve γ0 : C → C3 be
given by γ0 (t) = (t, t2 , t3 +t4 ). This is an imbedding that is one to one on points,
so the degree is 4. Note that γ0 is nonsingular, but if we take the projective
completion γ : P1 → P3 given by
γ(s; t) = (s4 ; s3 t; s2 t2 ; st3 + t4 )
we get a singular curve. In fact, setting t = 1 yields the local parameterization
at (0; 1), s → (s4 ; s3 ; s2 ; s + 1). Let (w; x; y; z) be the projective coordinates for
P3C . Since 1/(1 + s) = 1 − s + s2 − s3 + . . . in a neighborhood of 0, setting z = 1
2.5. THE TANGENT DEVELOPABLE OF A COMPLEX CURVE
15
gives the local parameterization
w
= s2 − s3 + s4 − s5 + . . .
x = s3 − s4 + s5 − s6 + . . .
y
= s4 − s5 + s6 − s7 + . . . .
We see that the type of the local parameterization is (2, 3, 4), and thus k0 (γ(0; 1)) =
1 and k1 (γ(0; 1)) = k2 (γ(0; 1)) = 0. At any other point we see that the first and
second derivative are linearly independent, so each of them are of type (1, 2, n)
for some value of n. This means that we have k0 = 1 and k1 = 0. The degree
of the curve is r0 = 4, and the genus of the curve is g = 0 since the curve is
rational. Now we can calculate the rest of the invariants mentioned above.
From the formulas we get the rank of the curve, r1 = 5, the class of the
curve, r2 = 4, the second stationary index, k2 = 1, the degree of the surface
µ0 = r1 = 5, the rank of the surface µ1 = r2 = 4, and finally the degree of the
nodal curve, b = 2.
Using Singular [10], we can verify some of the results. A Gröbner bases
computation gives us the implicit equation of the surface:
F
=
3wx2 y 2 + 12x3 y 2 − 4w2 y 3 − 14wxy 3 + 8x2 y 3 − 9wy 4 − 4wx3 z
−16x4 z + 6w2 xyz + 24wx2 yz − 6w2 y 2 z − w3 z 2
This equation is, predictably, of degree µ0 = r1 = 5. We can find the singular
locus by setting the four partial derivatives equal to zero. The last one,
1 ∂F
·
= −2wx3 − 8x4 + 3w2 xy + 12wx2 y − 3w2 y 2 − w3 z,
2 ∂z
leads us to consider two cases, w = 0 and w 6= 0.
The first case implies x = 0 from ∂F/∂z = 0, and then ∂F/∂w = 0 gives
y = 0. This leaves us with one point, namely (0; 0; 0; 1) = γ(0; 1), the singular
point of the curve.
If w 6= 0 we can choose w = 1 and solve the system of equation quite easily.
This is because ∂F/∂z = 0 becomes
0 = −2x3 − 8x4 + 3xy + 12x2 y − 3y 2 − z,
(2.5)
so we can substitute z into the other equations. In other words, assuming
∂F/∂z = 0, the equation ∂F/∂y = 0 gives
0
=
16x6 + 8x5 − 32x4 y + x4 − 16x3 y + 16x2 y 2 − 2x2 y + 8xy 2 + y 2
=
(4x + 1)2 (x2 − y)2 .
16
CHAPTER 2. THE TANGENT DEVELOPABLE
If x2 − y = 0, then equation (2.5) gives z = x3 + x4 , as expected.
Setting x = −1/4 in the rest of the equations gives us a solution for every
y, so z is a polynomial of degree 2 in y given by (2.5). This is the degree of the
nodal curve that we calculated earlier.
Note that most curves will have k0 = k1 = 0, with a nodal curve of degree
b = 2(r0 + g − 1)(r0 + g − 3). Unless r0 = 3 and g = 0, the nodal curve will not
be empty.
The cuspidal edge and the nodal curve may both be singular, and they will
usually intersect. If the nodal curve is double and the flexes are ordinary, X
will have a finite number of points with multiplicity ≥ 3. These points can be
of different types.
If the nodal curve has a node at q outside the cuspidal edge, then q must
lie on at least 3 tangents, and therefore the nodal curve must have at least
multiplicity 3 at q since any selection of two out of three tangents will give a
branch in the nodal curve.
The total number T of triple points of the tangent developable X of C0 is
given in [31] and is
T = 61 (r1 − 4)((r1 − 3)(r1 − 2) − 6g).
(2.6)
The formula (2.6) is valid when the nodal curve is double. When the nodal
curve is more than double we have to use a generalized formula for the degree
of the multiple curves (also found in [31]). If the nodal curve consists of curves
Dj , where Dj is ordinary j-multiple, then the degrees bj of Dj satisfy
X
j(j − 1)bj = r1 (r1 − 1) − r2 − 3(r0 + k1 ),
(2.7)
j
still assuming the flexes to be ordinary. Note that this is a very special case,
and that producing interesting examples with high j may be hard.
An example where the nodal curve is triple can be found in [40, p. 65], and
we have calculated, using Singular [10], the details1 .
√
Example 2.6 (The equianharmonic rational quartic). Let α = 31 −3, let C0
be the rational curve defined by the map γ : P1C → P3C where
γ(s; t) = (αs4 − s2 t2 ; αs3 t; αst3 ; αt4 − s2 t2 ),
1 There
m=
is an error in [40], m is not supposed to be
1√
−3.
3
√
−3, but the same as α in the example,
2.5. THE TANGENT DEVELOPABLE OF A COMPLEX CURVE
17
and let X be its tangent developable. A Gröbner bases computation gives us
the implicit equation F = 0 of the surface X. Here F is a polynomial of degree
6 in the projective coordinates (w; x; y; z):
F
=
12w2 x3 y + 3w4 y 2 − 72αw2 x2 y 2 + 12w2 xy 3 − 256αx3 y 3
+18αw3 xyz + 24wx3 yz + 6w3 y 2 z + 48αwx2 y 2 z + 24wxy 3 z
+3w2 x2 z 2 − 12αw2 xyz 2 + 12x3 yz 2 + 3w2 y 2 z 2 − 72αx2 y 2 z 2
+12xy 3 z 2 + 4αw3 z 3 + 6wx2 z 3 + 18αwxyz 3 + 3x2 z 4
Taking a primary decomposition of the Jacobian ideal of F , we find that the
singular locus of X consists of two components, the curve C0 and the conic D
defined by z 2 + 4xy = 0 in the plane w + z = 0.
We want to show that D is a triple curve of X. The conic D can be parameterized by θ : P1C → P3C where
θ(u; v) = (−2uv; −v 2 ; u2 ; 2uv).
Using this parameterization we find the following: The point θ(u; v) lies on the
tangent to C0 at γ(s; t) if and only if
G(s, t, u, v) := s3 u − 3αst2 u + 3αs2 tv − t3 v = 0.
For a fixed (u; v) ∈ P1C , zeros of G(s, t, u, v) corresponds to points on C0 whose
tangent contain θ(u; v). For most (u; v) ∈ P1C we will get three distinct tangents.
In fact, let ∆(u, v) denote the discriminant of G with respect to (s; t). In this
case
∆(u, v) = (u2 + (3α + 1)uv − v 2 )(u2 − (3α + 1)uv − v 2 ).
If ∆(u, v) 6= 0, then the point θ(u; v) lies on three distinct tangents to C0 .
Let A denote the four points on D corresponding to ∆(u, v) = 0. We conclude that each point on D not in A lies on exactly three tangents of C0 . This
means that D is a triple curve of X.
Moreover, A is exactly the intersection of D and C0 , and these four points
are the only points on C0 whose local parameterization is not of type (1, 2, 3).
In fact, the local parameterizations in each point of A is of type (1, 2, 4). This
means that k0 = k1 = 0 and k2 = 4. Furthermore, the degree of C0 is r0 = 4 and
the formulas give the rank r1 = 6 and the class r2 = 6 of the curve. The multiple
curve only have one component, the triple curve D, and this corresponds to
b3 = 2 in equation (2.7).
18
CHAPTER 2. THE TANGENT DEVELOPABLE
The set A form an equianharmonic set on D, and that is why C0 is called
the equianharmonic rational quartic. Note that this example is very special
and arise from the thorough study [40] of the rational normal curve in P4C . The
curve C0 is constructed by projecting the rational normal curve in P4C from a
general point on a quadric called the nucleus of the polarity. The equation of
the nucleus of the polarity is x0 x4 − 4x1 x3 + 3x22 and the projection centre of
this example is (1, 0, α, 0, 1).
All the formulas in this section holds for curves in P3C . We can not make
similar equalities for real curves, but the projective invariants of the complex
curve give results for the real part in the form of inequalities. However, these
inequalities will not be made explicit in this article.
Chapter 3
Closest points, Moving
Surfaces and Algebraic
Geometry1
Jan B. Thomassen, Pål H. Johansen, and Tor Dokken
3.1
Introduction
In this paper, we present a new method for calculating closest points to a parametric surface. The method is based on algebraic techniques, in particular on
moving surfaces. Moving surfaces are objects that have previously been used
for implicitization [34], but the closest point problem now provides another application of these.
Recently, there has been renewed interest in exploring links between geometric modeling and algebraic geometry [9]. The work presented in this paper is
a part of this trend, and extends work from the European Commission project
GAIA II. Algebraic geometry has many uses in geometric modeling, including
such applications as point classification, implicitization, intersection and selfintersection problems, ray-tracing, etc. It was therefore natural to ask whether
1 This
chapter is the article [15]. My contribution is mainly the theory in Section 3.3.
19
20
CHAPTER 3. CLOSEST POINTS BY MOVING SURFACES
algebraic geometry also can be used in algorithms for computing closest points.
The closest point problem is a generic problem in CAGD. Applications include surface smoothing, surface fitting, and curve or surface selection. The
closest point problem can be described in the following way. We are given a
parametric surface p(u, v) and a point x0 in space. We want to find the point
pcl on the surface that is closest to x0 , or more precisely, we want to find the
parameters (ucl , vcl ) of pcl .
The conventional way to compute closest points involves iterative methods,
like Newton’s method, to minimize the distance function from x0 to a point on
the surface. This leads to solving a set of two polynomial equations in u and v,
(x0 − p(u, v)) · pu (u, v)
=
0,
(x0 − p(u, v)) · pv (u, v)
=
0,
(3.1)
for the footpoints to x0 . We recall that a footpoint p to x0 is a point on the
surface such that the vector (x0 − p) is orthogonal to the tangent plane at p.
Eqs. (3.1) express an orthogonality condition: The vector (x0 −pcl ) is orthogonal
to the tangent vectors pu (ucl , vcl ) and pv (ucl , vcl ) at the closest point. This is
illustrated in Fig. 3.1.
x0
pv (u,v )
pcl
pu(u,v )
Figure 3.1: Orthogonality conditions for the closest point.
One disadvantage of iterative methods is that we need an initial guess. It
is a problem to come up with a good initial guess [20]. A bad guess may give
a sequence of iterations that does not converge, or that converges to the wrong
solution. Furthermore, if a large number of closest points needs to be computed,
the method may be slow.
3.2. THE UNDERLYING IDEA
21
Another way to solve Eqs. (3.1) is to use subdivision techniques. An example
of this is Bézier clipping [27]. These methods are often robust and effective, but
may be unstable and use a long time to converge for some difficult surfaces, like
surfaces with singularities. Such methods are probably the methods of choice
in real applications, but we will not discuss them further here.
The method we propose in this paper for solving the closest point problem
uses moving surfaces, as already mentioned. A moving surface in our setting is a
one-parameter family of surfaces. We construct two such surfaces: one moving
in the u-direction and one moving in the v-direction. The two moving surfaces
give us two polynomial equations that are univariate. Univariate polynomial
equations can be solved fast with a recursive solver and all roots may be found on
the interval of interest within a predefined accuracy. This will give an algorithm
that does not need any initial guess, has no convergence problems, and is fast
when many closest points are to be calculated.
We may use elimination theory and Sylvester resultants to construct the
moving surfaces in our method. From this construction, we obtain formulas
for the algebraic degrees of the geometric objects involved when the surfaces
addressed are Bézier surfaces.
We also describe an implementation of an algorithm for computing closest
points based on the moving surface method. In this implementation, we construct the moving surfaces by solving a system of linear equations, rather than
by using resultants. The implementation produced accurate results when run
on test cases of biquadratic Bézier surfaces. Unfortunately, it couldn’t be applied to bicubic surfaces due to memory shortage when building certain matrices
necessary for the construction of the moving surfaces.
The organization of the paper is the following. In the following section, we
describe the way we use moving surfaces and the idea behind our method. In
Section 3.3, we analyze the method by using elimination theory and Sylvester’s
resultant, which gives us formulas for the algebraic degrees of the moving surfaces in the scheme. In Section 3.4, we present an algorithm for our method and
describe some results we have obtained from implementing it. Finally, Section
3.5 is a discussion of these results.
3.2
The underlying idea
Our method involves moving surfaces, which have been introduced by Sederberg
for implicitization [34]. In that context, a moving surface is an implicit surface
depending on two parameters, but in our setting a moving surface is a one-
22
CHAPTER 3. CLOSEST POINTS BY MOVING SURFACES
parameter family of implicit surfaces. Let us make the assumption that we
are dealing with parametric surfaces that are single rational patches. Thus, a
moving surface q(x; u), depending on the parameter u, is given by
q(x; u)
=
N
X
qi (x)Bi,N (u).
(3.2)
i=0
Here, Bi,N (u) are Bernstein basis polynomials of degree N , and qi (x) is a set
of N + 1 algebraic functions. In other words, q is given in terms of a Bernstein
polynomial in u of degree N , where the coefficients qi are implicit surfaces.
Furthermore, the moving surface q follows a surface p(u, v) (in the parameter
u) if
q(p(u, v); u)
=
0.
(3.3)
Moving surfaces may in this way follow a parametric surface in either the u or
the v direction.
How can we make use of such moving surfaces? Suppose we find a moving
surface q1 (x; u) with the following properties:
• q1 follows the given surface p(u, v) in u. This means that the surface
defined by q1 (x; u) = 0 intersects p in u-isoparameter curves.
• q1 is orthogonal to p for each u.
• q1 is ruled for each u. More precisely, it is swept out by lines spanned by
the normal n(u, v) along the u-isoparameter curves.
Then, for a given point x0 in space, the equation
q1 (x0 ; u)
=
0
(3.4)
is a univariate equation for the u-parameter of all footpoints to x0 . An example
of a moving surface with these properties is shown in Figure 3.2. Clearly, we
may have a similar moving surface q2 in the v direction. In the following, the
subscript 1 or 2 on q refers to either u or v.
A possible exception to this situation is that we are dealing with certain nongeneric surfaces, like surfaces of revolution. For these surfaces some points (like
those lying on the axis of revolution) may give, not footpoints, but footcurves.
I.e. the set of points with the same distance to x0 is a curve on the surface.
This is presumably a problem for most methods of computing closest points,
3.2. THE UNDERLYING IDEA
q1(x0;u)=0
23
n(u,v)
v
p(u,v)
u
Figure 3.2: A moving surface q1 that intersects p at u-isocurves, is orthogonal
to it, and is ruled.
and requires a separate discussion. For simplicity we assume that all the surfaces
we consider are sufficiently generic for this to happen.
Based on the considerations above, we propose a method for computing
closest points in two steps:
1. Preprocessing. Construct two moving surfaces: q1 for the u-direction, and
q2 for the v-direction. This is done once for each surface.
2. For each given point x0 , use the two moving surfaces to get two univariate
equations in u and v:
q1 (x0 ; u)
=
0,
q2 (x0 ; v)
=
0.
(3.5)
Check each pair of solutions (u, v) to these equations, along with the
closest point on the border, to find which one corresponds to the closest
point.
Finding the closest point on the border amounts to running a similar algorithm
for the four border curves.
24
CHAPTER 3. CLOSEST POINTS BY MOVING SURFACES
A sketch of a situation where we get a solution u0 and v0 from Step 2 is
shown in Figure 3.3.
q1(x0;u0)=0
q2(x0;v0)=0
x0
pv(u0;v0)
pcl
pu(u0;v0)
Figure 3.3: When the solutions u0 and v0 are found in Step 2, we can draw the
moving surfaces for these two parameter values. The point x0 , the closest point
pcl , and the straight line between them, lie on both of these surfaces.
Let us also make a remark about curves. A similar construction works for
curves, both in 2D and 3D. In 2D we have moving lines, while in 3D we have
moving planes. Since lines and planes are described implicitly by algebraic
functions that are linear, the algorithms become simpler. Furthermore, for
curves there is only one equation in Step 2. This equation is in fact equivalent
to the orthogonality condition (x0 − p(t)) · p0 (t) = 0.
3.3
Degrees of the moving surfaces
The two moving surfaces described in the previous section can be analyzed more
formally. In this section we will use elimination theory, in particular Sylvester’s
resultant, to perform this analysis [6]. We will assume that the surface p is a
single polynomial patch, i.e. a Bézier patch. In this case we obtain formulas
for the algebraic degrees involved in q1 and q2 given a parametric surface of
bidegree (nu , nv ). Referring back to the form (3.2) of a moving surface, the
3.3. DEGREES OF THE MOVING SURFACES
25
required degrees are:
d1
≡ degx (q1 ) = the degree of q1 (or q1,i ) in x
d2
≡ degx (q2 ) = the degree of q2 (or q2,i ) in x
N
≡ degu (q1 ) = the degree of q1 (or Bi,N ) in u
It turns out that degu (q1 ) is equal to degv (q2 ) so we need only one N . This is
connected with the fact that N counts the number of possible footpoints, and
this is given by the number of roots of q1 and q2 , respectively.
Thus we have a parameterized surface p : R2 → R3 , where p is given by
three polynomials p1 , p2 , p3 ∈ R[u, v] of degree (nu , nv ). We assume that this
description of the surface is sufficiently general, so that the degrees cannot be
reduced. Now let V be the set of points (u, v, x) ∈ R × R × R3 such that x is
on the normal of p given by the parameter values (u, v).
The set V is described by the two equations
F1 (u, v, x) := (x − p) · pu
=
0,
F2 (u, v, x) := (x − p) · pv
=
0.
(3.6)
The points satisfying these equations make a variety in R5 .
Using a resultant, we can eliminate one variable, and get one polynomial
defining a hypersurface in R4 . If we eliminate u, this set of points is exactly
the set V 0 = {(v, x) ∈ R × R3 | ∃u ∈ R s.t. F1 (u, v, x) = F2 (u, v, x) = 0}, which
corresponds to the moving surface q2 .
We want to determine the degrees in v and x of the equation defining V 0 .
First, we write
F1 (u, v, x)
=
2n
u −1
X
fi (v, x)ui ,
(3.7)
i=0
F2 (u, v, x)
=
2nu
X
gi (v, x)ui ,
i=0
and then use the Sylvester resultant to eliminate u. By examining the Sylvester
matrix, we can determine the degrees of the equation defining V 0 . The Sylvester
26
CHAPTER 3. CLOSEST POINTS BY MOVING SURFACES
matrix is a square matrix of size (4nu − 1) × (4nu − 1). It looks like this:


f0
0
···
0
g0
···
0
 f1
g1
···
0 
f0
···
0



..
..
.
.
.. 
.
.
.
.
.
.

.
.
.
.
.
.
. 


 f2nu −1 f2nu −2 · · ·
f0
g2nu −1 · · ·
g1 



g2nu
···
g2 
0
f2nu −1 · · ·
f1



..
..
..
..
.. 
..
..

.
.
.
.
.
.
. 
0
· · · g2nu
0
0
· · · f2nu −1
There are 2nu columns to the left, and the degree in v is 2nv for each of
these entries. There are 2nu − 1 columns to the right, each entry being of degree
2nv − 1. The total degree in v of the resultant is thus
N
=
4nu nv + (2nu − 1)(2nv − 1).
(3.8)
Note the symmetry of this expression with respect to nu and nv , which confirms
what we said previously about needing only one N .
The degree in x, that is, d2 , is a little trickier to work out. The polynomials
f0 , . . . , fnu −1 are of degree 1 in x, but the polynomials fnu , . . . , f2nu −1 are of
degree 0. Furthermore, the polynomials g0 , . . . , gnu are of degree 1 and the
polynomials gnu +1 , . . . , g2nu are of degree 0 in x. This means that the bottom
nu rows are of degree 0 in x and the rest of the 3nu − 1 rows are of degree 1.
The total degrees are therefore
d1
=
3nv − 1,
(3.9)
d2
=
3nu − 1.
(3.10)
As mentioned, the degree formulas for d1,2 and N are derived for general
parametrized surfaces, and as such are upper bounds. For some surfaces the
degrees could be effectively lower. This happens, for example, if the degree of
p is artificially high, so it can be obtained from a degree elevation of a lowerdegree parametrization. The degree can drop in other cases, but if the degree
in v drops, then the corresponding degree in u will typically drop in the same
way. For this reason, there will still be only one N .
A similar analysis can be carried out for rational surface patches. The degree
formulas are then:
N
=
9nu nv + (3nu − 2)(3nv − 2)
d1
=
4nv − 2
d2
=
4nu − 2
(3.11)
3.4. IMPLEMENTATION OF A TEST ALGORITHM
d1,2
N
(1, 1)
2
5
Bézier
(2, 2) (3, 3)
5
8
26
61
(4, 4)
11
113
(1, 1)
2
10
Rational
(2, 2) (3, 3)
6
10
52
130
27
(4, 4)
14
244
Table 3.1: Degrees for Bézier and rational surfaces of degrees of the form (n, n).
Since nu = nv we also have d1 = d2 .
Examples of the degrees for Bézier and rational surfaces of degrees (n, n)
with n ranging from 1 to 4 is shown in Table 3.1. As far as we know, these
results are new2 . The numbers d1 and d2 are the degrees of an algebraic surface
that is perpendicular to a parametric surface along an entire isocurve, and this
has not been noted before.
3.4
Implementation of a test algorithm
To test our ideas, we have implemented an algorithm for computing closest
points for tensor product Bézier surfaces. We have chosen not to use resultants
for this. Instead we rely on solving a system of linear equations, which will be
explained below. The reason is that this is a numerically very stable method,
which allows us to use the Bernstein form for all polynomials in a straightforward
way, which would not have been the case for resultant based methods. Besides,
we do not get into possible problems with base points.
A central object in our implementation is the “moving ruled surface”
r(u, v, w)
= p(u, v) + wn(u, v),
(3.12)
where p is the given surface, n is the normal vector, and w is an additional
parameter. This can be thought of as a trivariate tensor product Bézier object.
For fixed (u0 , v0 ), the line r(u0 , v0 , w), w ∈ R, is orthogonal to the surface at
p(u0 , v0 ). In other words, all points on this line has p(u0 , v0 ) as a footpoint.
Another property we have used in our implementation, is that evaluating an algebraic function q(x) on an n-variate Bernstein tensor polynomial
r(u1 , . . P
. , un ) yields a new n-variate Bernstein tensor polynomial. If we write
q(x) = j bj xj , where j is a multi-index, xj is a monomial in (x, y, z) in multi2 We
thank the referee for urging us to make this point.
28
CHAPTER 3. CLOSEST POINTS BY MOVING SURFACES
index form, and bj are the coefficients, we have a factorization
q(r(u1 , . . . , un ))
= bT DT B(u1 , . . . , un ).
(3.13)
Here, b is the coefficients bj organized in a vector, D is a matrix of numbers, and
B(u1 , . . . , un ) is a basis of n-variate Bernstein tensor polynomials, also organized
in a vector. If q is a degree d algebraic function and r is a degree (m1 , . . . , mn )
Bernstein polynomial, then q(r) is a degree (dm1 , . . . , dmn ) Bernstein polynomial. In our implementation, we use evaluation routines for algebraic functions
on Bernstein polynomials in order to find such matrix factorizations.
The moving surfaces q1 and q2 are defined by an array of coefficients.
For
P
q1 (x; u) we need to determine the coefficients b1,i;j of q1,i (x) = j b1,i;j xj , see
Eq. (3.2). This means that we can use numerical linear algebra to find the vector
b1 of coefficients in q1 . More precisely, we need to find a vector in the null-space
of D1 . We used a technique based on Gauss elimination and back-substitution
for this, which is faster than, say, SVD of D1 . This way of using numerical
linear algebra has previously been used in implicitization, see [7].
The algorithm follows the two-step structure described in Section 3.2.
Step 1. Preprocessing
Input: A parametric surface p(u, v).
1. Construct a “moving ruled surface” r(u, v, w)
2. Insert r into q1 to get the equation q1 (r(u, v, w), u)) = 0. This can
be factored into the linear equation
BT (u, v, w)D1 b1
=
0,
(3.14)
where b1 is the vector of coefficients in q1,i . Similarly for the vdirection.
3. Solve the matrix equation D1 b1 = 0 by e.g. Gauss elimination and
back-substitution. Similarly for the v-direction.
Output: The vectors b1 and b2 , or equivalently, the moving surfaces q1
and q2 .
Step 2. For each given point x0
Input: A point x0 in space.
1. Find the closest point on the boundary curves.
3.4. IMPLEMENTATION OF A TEST ALGORITHM
Average no. of hits
Running times
Accuracy
Moving surfaces
413
Full algo.
∼ 1 − 2 min
Just Step 2
∼1s
∼ 10−7
29
Newton’s method
374
∼5s
∼ 10−13
Table 3.2: Average results for running the two closest point algorithms on ten
random biquadratic surfaces. For details, see the text.
2. Insert x0 in q1 and q2 to get univariate equations in u and v:
q1 (x0 ; u)
=
0,
q2 (x0 ; v)
=
0.
(3.15)
3. Find all roots ui and vj .
4. Check each pair (ui , vj ) and the closest point on the boundary to find
the closest point.
Output: The parameters (ucl , vcl ) of the closest point pcl .
As an example, we tested the algorithm on a set of ten random biquadratic
Bézier surfaces. That is, the control points were random points in the unit cube.
We expect that within the family of biquadratic surfaces such surfaces will be
a challenge for any closest point algorithm. For each surface, 1000 points were
generated randomly in the bounding box, and their closest points on the surface
were computed. However, points whose closest points were found to lie on the
boundary were discarded. For comparison, we also implemented an algorithm
based on Newton’s method. We used a PC with two Intel Pentium 4 2.8GHz
processors to run these algorithms. The results are shown in Table 3.2.
Table 3.2 reports the average number of closest points found (hits) for each
surface. For both algorithms, less than half of the 1000 random points gave hits
because a majority of the closest points were on the boundaries of the surfaces.
(The surfaces had complicated geometries with lots of self-intersections.) The
moving surfaces algorithm was consistently better for getting hits – Newton’s
method produced a lot of messages for “No convergence”.
Running times were considerably longer for the full moving surfaces algorithm, with 1 − 2 minutes. Most of this time is spent in the preprocessing step
where the moving surfaces are constructed. When only Step 2 of the algorithm
is considered it is much faster.
30
CHAPTER 3. CLOSEST POINTS BY MOVING SURFACES
Finally, the accuracy given in Table 3.2 is an average of the errors for the
reported closest points. The averaging is over the order of magnitude of the
errors, i.e. it is an average of the log of the errors for each point. An error for
a single point was defined in terms of the angle θ between the vector (x0 − pcl )
and the normal n(ucl , vcl ) at the computed closest point, see Figure 3.4. As
x0
n(ucl ,v cl )
θ
pcl
Figure 3.4: The error can be measured by the angle θ between the normal n at
pcl and the vector to the point x.
we can see, Newton’s method produced much better accuracy than the moving
surfaces. Sources of error for the moving surfaces method are the building of
the matrix D, the Gauss elimination, the insertion of x0 to get the polynomial
equations, and the solving of these equations. The results in Table 3.2, however,
does not include iterative refinements.
Looking into the details for each surface – not shown in Table 3.2 – it turns
out that there is a complementary property for the two algorithms: Surfaces
that had a low accuracy for moving surfaces also had a high number of hits.
For example, one random surface had an accuracy of 10−5 vs. 10−13 for moving
surfaces and Newton’s method, respectively, while the hit numbers were 265 vs.
193.
It is necessary to make some remarks about problems with the memory usage
of our implementation of the moving surfaces algorithm. The amount of memory
needed for the matrix D1 (or D2 ) was about 350 Mbytes for the biquadratic
surface. This is a lot, but does not cause any problems. For a bicubic Bézier
surface, however, the corresponding memory requirement is about 4 Gbytes with
double precision! But even going to single precision was too much to handle for
3.5. DISCUSSION
31
our PCs.
3.5
Discussion
Moving surfaces provides a new method for computing closest points to a parametric surface. It is an alternative to the conventional algorithms based on
iterations and Newton’s method, or to subdivision methods.
Compared to a Newton based closest point algorithm, it takes a long time
to set up the system of moving surfaces for each parametric surface, but a short
time to compute the closest point once a point in space is given. This suggests
that the potential use of the moving surfaces method is for situations where a
large number of closest points are to be computed for each given surface, and
where long preprocessing times are acceptable.
Furthermore, the moving surfaces method is better than Newton’s method
for actually finding the closest points for surfaces with complex geometry. Thus,
if this kind of stability is desired, the moving surfaces method may also be a better choice, combined with an iterative refinement of the resulting closest points.
In other words, the closest points found from the moving surfaces method could
be used as the starting point of the Newton iterations.
However, we had problems with the implementation of the algorithm, due
to memory shortage, when applied on the realistic case of bicubic surfaces.
The large amount of memory is mainly used for building the matrices D1 and
D2 . In contrast, the amount of memory needed for storing the moving surfaces
themselves corresponds to only (N +1)(d1,2 +1)(d1,2 +2)(d1,2 +3)/6 doubles (i.e.
the dimensions of the vectors b1 and b2 , respectively. In the bicubic case this
number is 10230. This shows that we must find another way of implementing
the algorithm, or that we must find a way to use moving surfaces together with
approximations. But if we can afford the preprocessing of the coefficients of the
moving surfaces we can get fast and accurate calculations of closest points.
32
CHAPTER 3. CLOSEST POINTS BY MOVING SURFACES
Chapter 4
Solving a closest point
problem by subdivision
4.1
Introduction
In this paper, a closest point problem is solved by using subdivision techniques,
and it is shown that a considerable speed-up is possible if relatively simple ideas
are used.
Closest point problems are heavily used in CAGD applications. Applications
include surface smoothing, surface fitting, and curve and surface selection. It is
also important to note that in some applications, solving closest point problems
becomes the bottleneck of the algorithm. This fact alone makes it interesting
to be able to solve such problems fast.
The common way of solving closest point problems is by using iterative
methods (see [12] and references therein). These are generally fast, but the
need for an initial guess makes them error prone. Some times these kinds of
errors, even when highly uncommon, can ruin the result. In these cases, a
subdivision method can be the best way of ensuring high quality output.
The closest point problem can be solved by using a numerically stable algebraic polynomial solver, and this approach is explored in [19, Section 4.2]. This
method is robust, but it is much slower than what one can get from a good
iterative method or a subdivision algorithm.
Note that closest point problems for discrete sets are very different from the
problem of finding closest points on smooth surfaces. However, in a CAGD
33
34
CHAPTER 4. CLOSEST POINTS BY SUBDIVISION
system it may be interesting to find the closest point on a model consisting of
many smooth surfaces patches. Given such a problem, the techniques concerning
the discrete case found in [37] will be useful in addition to a good understanding
of the smooth case considered here.
Section 4.2 defines the problem and explains the basic way of solving it by
subdivision. Notes about how different implementations should be evaluated
and what happens in special cases are also included here.
Section 4.3 explores different changes to the basic algorithm and how these
changes affect the speed of implementations. Section 4.3.8 contains a table
summarizing run times for the different implementations in Section 4.3.
Section 4.4 contains an error analysis of subdivision methods. A formula
giving a bound on the error is produced, and this formula can be used when
determining if the output is accurate enough for a given application.
The chapter ends with Section 4.5, the conclusion.
4.2
Definition of the problem
In this chapter, a surface patch ϕ of (bi)degree (d, e) is a map ϕ : [0, 1]2 → R3
given by an array of control points cij ∈ R3 and the formula
ϕ(u, v) =
d X
e
X
i=0 j=0
cij
e
d
(1 − v)j v e−j .
(1 − u)i ud−i
j
i
This surface is called a tensor product Bézier surface with control points cij .
For a point x ∈ R3 , we want to find the parameter point (u0 , v0 ) ∈ [0, 1]2
that minimizes the distance function (u, v) → kϕ(u, v) − xk.
Note that if x is in the image of ϕ, then the closest point problem specializes
to the inverse problem. This problem has been studied in, for example, [21].
In a CAGD system one would like to solve the same problem in a more
general setting, for example where ϕ is a tensor product spline. However, since
we can reduce the spline case to the Bézier case by knot insertion, considering
only the Bézier case is not a big limitation.
4.2.1
The basic method
The idea was to start by implementing a simple method for finding closest
points, and then introduce different changes that might or might not speed
4.2. DEFINITION OF THE PROBLEM
35
up the implementation. The basic method described here will be improved in
Section 4.3.
Let F : [0, 1]2 → R be defined as the distance squared function, F (u, v) =
kϕ(u, v)−xk2 . The critical points of F are the zero set of two partial derivatives,
Fu (u, v) and Fv (u, v). If we evaluate one of these we get
Fu (u, v) = 2(ϕ(u, v) − x) · ϕu (u, v),
where ϕu = ∂ϕ
∂u . Note that this is a very natural orthogonality condition: when
Fu (u0 , v 0 ) and Fv (u0 , v 0 ) are both zero, the vector ϕ(u0 , v 0 ) − x is orthogonal
to the tangent plane of ϕ associated to the parameter value (u0 , v 0 ), or the
parameterization is degenerate in this point. Abusing language, we say that
(u0 , v 0 ) is a critical point for x if Fu (u0 , v 0 ) = Fv (u0 , v 0 ) = 0. It is clear that
the closest point (u0 , v0 ) will be either a critical point or on the border of the
parameter domain.
The basic method is straightforward. Find the closest point on the border,
and then find all the critical points by applying a subdivision solver to the tensor
product functions Fu and Fv , and compare the distances. The shortest distance
will give the closest point.
One note on the closest point on the border may be useful: To find the
closest point on one side, for example (0, 1) × {0}, it is sufficient to evaluate
F (u, 0) on zeros of the univariate polynomial Fu (u, 0). Every implementation
in this chapter will start by finding the closest point on the border.
4.2.2
Quality of the output and special cases
Even though the algorithms described in this chapter have been tested thoroughly, it is the application of the output data that should determine if any
algorithm is “accurate enough”.
When doing subdivision, one may be very unlucky, and the subdivision
will take a lot of time. This happens when the set of critical points is onedimensional. The possibility of a one-dimensional set of solutions is by far the
biggest problem with subdivision methods, and usually must be handled with
great care.
When this happens in our case, all points on each connected component
of the set of critical points will give the same distance to x, so any point on
one such component will do as a solution to the problem. For this reason
special cases are not such a big problem, at least when we can detect it. We
solve this by counting the number of calls to the recursive function, and, if
36
CHAPTER 4. CLOSEST POINTS BY SUBDIVISION
it is called too many times, we abort and start over with a lower maximum
recursion depth. Together with iterative refinement of best points, this should
give us good answers in practically all cases.
When the processing of one point is completed, we know to which depth the
recursion has been completed successfully. Then the error analysis in Section
4.4 can be used to determine if the guaranteed accuracy is sufficient.
4.3
Improving the basic method
All implementations follow this pattern:
• Initialize (u0 , v0 , D) = (0, 0, F (0, 0)). This set of variables represents the
best parameter point so far with the distance squared.
• For each (u, v) ∈ {(0, 1), (1, 0), (1, 1)}, evaluate F (u, v) and, if F (u, v) <
D, update (u0 , v0 , D).
• For each side of the unit square, solve a univariate polynomial by subdivision and update (u0 , v0 , D) if a new closest point is found.
• Recursively find candidate points in the interior of the unit square and
update (u0 , v0 , D).
It is the last item that will change when improving the basic method. But before
we discuss these changes, we need some simple notation.
Let ϕc : [0, 1]2 → Rn be given by (d + 1)(e + 1) control points c = (cij ) and
the formula
ϕc (u, v) =
d X
e
X
i=0 j=0
cij
d
e
(1 − u)i ud−i
(1 − v)j v e−j .
i
j
We say that c = (cij ) represents ϕc on the square [0, 1]2 ⊂ R2 . Furthermore,
we say that b = (bij ) represents ϕc on the rectangle [a1 , b1 ] × [a2 , b2 ] if
ϕb (u, v) = ϕc ((1 − u)a1 + ub1 , (1 − v)a2 + vb2 ).
Such representations can be calculated easily by using de Casteljau’s algorithm,
and if 0 ≤ ai ≤ bi ≤ 1 for i = 1, 2, the calculations are stable.
The input to the basic recursive solver is a rectangle [a1 , b1 ] × [a2 , b2 ] and
two sets of control points (scalars) representing Fu and Fv . If either Fu or Fv
4.3. IMPROVING THE BASIC METHOD
37
has control points of constant signs, then the solver concludes that there are no
base points in the rectangle and returns. Else it subdivides the rectangle and
calls itself with new control points representing Fu and Fv in the subdivided
rectangles, once for each sub-rectangle. The algorithm also keeps track of the
recursion level, and when it reaches the bottom it evaluates the midpoint and
updates (u0 , v0 , D) if a new closest point has been found.
Now we will focus on different changes to the implementations.
4.3.1
Changing the subdivision
The first change is to subdivide ϕ(u, v) − x, ϕu (u, v) and ϕv (u, v) instead of
subdividing Fu and Fv . Since the cost of subdividing is roughly proportional to
the cube of the degree, subdividing these three vector valued functions is faster
than subdividing the two scalar functions Fu and Fv .
However, this change means that the dot product must be carried out when
checking whether the control points of Fu or Fv have constant sign. The cost of
the multiplication of tensor product Bézier functions is roughly proportional to
the fourth power of the degree, and that indicates that changing the subdivision
is a bad idea. However, even though changing the subdivision turned out to
make the algorithm slower, it allowed us to do other optimizations (see Sections
4.3.2 and 4.3.3) that let us do the full multiplication less often.
Experiments showed that only subdividing ϕ(u, v) − x and calculating the
subdivided ϕu (u, v) and ϕv (u, v) from ϕ(u, v) − x is a bad idea. The operation
is unstable, and this instability slowed down the implementations considerably.
On the other hand, it is conceivable that this instability can become negligible
in some cases, when the recursive level is much smaller than the precision of the
computer arithmetic used.
4.3.2
Changing the multiplication algorithm to allow an
early exit
The conventional way to do the multiplication of two tensor product polynomials is to initialize the result control points to zero, then loop through the control
coefficients of one factor and, for each such coefficient, loop through the coefficients of the other factor, while accumulating the result. The multiplication in
the dot products (ϕ(u, v) − x) · ϕu and (ϕ(u, v) − x) · ϕv was changed so that
each coefficient of the result was calculated before starting on the next. The
loops became more complicated, but the change allows the constant sign test to
stop early if both signs are encountered.
38
CHAPTER 4. CLOSEST POINTS BY SUBDIVISION
The implementation was further improved by first calculating the corner
coefficients. These coefficients only depend on one control point from each
factor, so they can be calculated very fast. Also, when checking for constant
sign in Fu and Fv , the corner coefficients are most likely to differ.
This change resulted in a considerable improvement in speed, but not enough
to offset the speed lost by the changes in Section 4.3.1.
4.3.3
Introducing a box test and a plane test
When we enter the recursive function, the variable D holds the best distance
squared. Let {bij } represent ϕ(u, v) − x in the rectangle [a1 , b1 ] × [a2 , b2 ], so
ϕ([a1 , b1 ], [a2 , b2 ]) − x = ϕb ([0, 1], [0, 1]) is contained in the convex hull of the
control points {bij }. We can estimate the distance from x to the surface patch
ϕ([a1 , b1 ], [a2 , b2 ]) and return if it is too big. This is done by calculating the
ij
distance
√ from 0 to the smallest box containing {b }. If this distance is bigger
than D, we can return before doing the multiplication. This is called the box
test.
While calculating the smallest box containing the control points {bij }, we
can also find the control point that is closest to 0. Let this point be bαβ , and
calculate bαβ · bij for each
point bij . If all of these dot products are
√ control
αβ
bigger than the constant Dkb k, we can return before doing the multiplication. This is called the plane test, since it checks if all control
points bij are on
√
αβ
αβ
the other side (than the origin) of the plane X · b = Dkb k.
For the box test and the plane test to work as well as possible we also
calculate distances at the middle point and the midpoints of the sides before
subdividing. This way, when entering the recursive function, we have already
tested the corner points. This increases the chance of the box test and the plane
test helping us.
The box test and the plane test resulted in a huge speed-up, making the
algorithm faster than the basic method.
4.3.4
Using the second order derivatives
For some rectangles [a1 , b1 ] × [a2 , b2 ] the function F may be convex, and therefore the rectangle can contain at most one base-point. This property can be
determined from the signs of the eigenvalues of the Hessian of F . If the product
of the eigenvalues is positive in the entire rectangle and the Newton refinement
converges to a point inside the rectangle, then this point is the only base-point
in the rectangle.
4.3. IMPROVING THE BASIC METHOD
39
A method for calculating the control coefficients of the product of the eigenvalues was introduced, but the high degree made this too costly, and it only
made the implementations slower.
4.3.5
The recursive algorithm explained
This section explains the algorithm we got after applying the changes in Sections 4.3.1, 4.3.2 and 4.3.3. The algorithm follows the pattern specified in the
beginning of Section 4.3. When we enter the recursive function, it is assumed
that the corners of the square to be considered has been evaluated, and that the
variables (u0 , v0 ) and D have been updated accordingly. We keep track of the
total number of calls to be able to abort in special cases. The recursive function
works as follows:
Input: The square [u1 , u2 ] × [v1 , v2 ], control points bij representing ϕ − x and
ij
vectors bij
u and bv representing ϕu and ϕv in this square.
Output: Void, the variables (u0 , v0 ) and D will be updated if necessary.
• If the number of calls to the recursive function is bigger than a constant,
in our case 16384, abort. If else, increase the variable holding the number
of calls to the recursive function.
• Calculate the distance from the origin to the
√ smallest box containing the
control points bij . If this is bigger than D, return. Also perform the
plane test described in Section 4.3.3.
• Calculate the control points aij ∈ R representing 21 Fu , one at a time,
2d−1,0
starting with the corner points a0,0 = b0,0 · b0,0
= bd,0 · bd−1,0
,
u , a
u
0,2e
0,e
0,e
2d,2e
d,e
d−1,e
a
= b · bu and a
= b · bu
. If both signs are encountered,
stop calculating coefficients and skip to the next bullet point. If all signs
are the same, return.
• Do the same as above for Fv .
• Let u0 = 21 (u1 + u2 ) and v 0 = 21 (v1 + v2 ). Evaluate F (u1 , v 0 ), F (u2 , v 0 ),
F (u0 , v1 ), F (u0 , v2 ) and F (u0 , v 0 ) and update (u0 , v0 ) and D accordingly.
• If the square is sufficiently small, return.
• Calculate representations of ϕ−x, ϕu and ϕv on the four squares [u1 , u0 ]×
[v1 , v 0 ], [u1 , u0 ] × [v 0 , v2 ], [u0 , u2 ] × [v1 , v 0 ] and [u0 , u2 ] × [v 0 , v2 ] using de
ij
Casteljau’s algorithm on bij , bij
u and bv . Then call the recursive function
40
CHAPTER 4. CLOSEST POINTS BY SUBDIVISION
with these values. In most cases it pays to sort the order of these calls
based on the values F (u1 , v 0 ), F (u2 , v 0 ), F (u0 , v1 ) and F (u0 , v2 ). To be
precise, if F (u1 , v 0 ) < F (u2 , v 0 ), then call the the recursive function for
the square [u1 , u0 ] × [v1 , v 0 ] before the square [u0 , u2 ] × [v1 , v 0 ] and the
square [u1 , u0 ] × [v 0 , v2 ] before [u0 , u2 ] × [v 0 , v2 ]. Similarly, if F (u0 , v1 ) <
F (u0 , v2 ), handle the square [u1 , u0 ] × [v1 , v 0 ] before [u1 , u0 ] × [v 0 , v2 ] and
[u0 , u2 ] × [v1 , v 0 ] before [u0 , u2 ] × [v 0 , v2 ].
4.3.6
The basic method with the box and plane tests
After seeing the huge improvement from the tests in Section 4.3.3, it was natural
to try to improve the basic method using the same tests. The resulting algorithm
calculates representations of Fu , Fv and ϕ(u, v)−x in the unit square. Then the
recursive function does essentially the same as the algorithm in Section 4.3.5,
except that no multiplication needs to be carried out.
The result is a method that is a little slower than the algorithm in Section 4.3.5. However, the difference is so small that a strong conclusion cannot
be drawn - the result can be very different on different hardware, and, most
importantly, on other test cases.
4.3.7
Doing a preconditioned constant sign test
General subdivision algorithms can be sped up considerably by doing a preconditioned constant sign test [26]. The idea is to do a special linear transformation
of the equations to be solved, and then to check if any of the resulting equations
has control points of constant sign. The linear transformation is determined
by the cofactor matrix of the Jacobian matrix evaluated in the midpoint of the
square.
Preconditioning requires the equations to be of the same degree, so we elevate
the degree of Fu and Fv to (2d, 2e), and transform the system by using the
cofactor matrix of the Hessian of F at the midpoint:
G1
Fvv (u0 , v 0 ) −Fuv (u0 , v 0 )
Fu
=
−Fuv (u0 , v 0 ) Fuu (u0 , v 0 )
G2
Fv
If the control points of either G1 or G2 have constant sign (different from zero)
we can conclude that the rectangle contains no critical point.
For our datasets, doing preconditioning helped a lot when the box and plane
tests where not present, cutting processing time in half. When the box and
plane tests were used, the effect ranged from a slowdown of a few percent to a
4.3. IMPROVING THE BASIC METHOD
41
speedup of a few percent. It is highly likely that this will be different on other
datasets.
4.3.8
Speed measurements
The implementations were tested on many surfaces and on many points per
surface. To be exact, for each bi-degree, 100 surfaces with control points placed
randomly in the unit cube were tested, and for each surface 100 random points
in the unit cube were selected. Table 4.1 shows the times for seven algorithms:
• the algorithm described in 4.3.5
• the algorithm from Section 4.3.5 without the plane test from Section 4.3.3
• the algorithm from Section 4.3.5 without the plane test and the box test
from Section 4.3.3
• the algorithm from Section 4.3.5 without the improved calculation of the
product 12 Fu (u, v) = (ϕ(u, v) − x) · ϕu (u, v) from Section 4.3.2
• the basic algorithm
• the basic algorithm with a box test
• the basic algorithm with a box test and a plane test
Each of these algorithms were tested with and without preconditioning, and
the table shows the best time for each algorithm.
Degree
Algorithm from Section 4.3.5
no plane test
no plane or box test
no multiplication optimization
Basic algorithm
with box test
with box and plane test
(2, 2)
2.31
3.89
12.2
3.65?
5.80
2.73
2.40
(3, 3)
4.78
9.59
52.8
9.14?
21.1
6.21
5.10?
(4, 4)
8.37
20.1
170
19.3?
52.9?
11.5
8.79?
(3, 9)
18.2?
43.9
495
46.2?
138
22.3
17.9
(7, 7)
32.0?
98.6
1671
110?
390?
45.9
32.0?
(20, 20)
652?
2823
204009
4348?
23586
861?
561?
Table 4.1: Time in seconds spent to solve the closest point problem for 100
surfaces and 100 point per surface on a 2.80 GHz Pentium 4 CPU. A ? indicates
that it did not pay to do a preconditioned constant sign test.
42
CHAPTER 4. CLOSEST POINTS BY SUBDIVISION
The maximum recursion level was set to 32, but numbers should be representative. Experiments showed that the time was roughly proportional to the
level of recursion, at least for reasonable values.
The usual cautions apply: It is unlikely that these test cases give a very good
indication of what is fastest for less random data sets on different hardware.
Because of this, it is recommended that anyone who needs to calculate closest
points fast do their own timings.
Remark: I also experimented with different compiler optimization flags, and
the optimal set of flags was not constant for different degrees. To be specific,
it was the -funroll-loops options that helped in some cases, but made things
worse in others. If this problem is to be solved in production code, the best
compiler available should be used. The table shows times for the best set of
compiler settings for each case.
4.4
Error analysis
If infinite precision in the calculations is assumed, then we can develop a lower
bound L of kF ([0, 1], [0, 1])k in terms of the distance squared D returned by
the algorithm, the bi-degree (d, e) of ϕ, the depth n of the recursion, and the
diameter
R = max {kcij − ckl k}
i,j,k,l
of the control points. We also assume that the number of recursive calls is not
constrained in any way except in terms of depth. This means that the actual
constrained algorithm will give worse results in some rare cases, but it will be
able to report what level was reached successfully, giving us a worse guaranteed
accuracy. We say that the depth of recursion is n if any square bigger than
2−n × 2−n that may contain the closest point is subdivided.
We know that a square of size 2−n × 2−n containing the closest point (u0 , v0 )
will be considered by the algorithm. Let this square be denoted [u1 , u2 ] × [v1 , v2 ]
with u1 ≤ u0 ≤ u2 , v1 ≤ v0 ≤ v2 and u2 − u1 = v2 − v1 ≤ 2−n . The corners
of this small square have been evaluated as candidates for the return value of
the algorithm, so D ≤ F (ui , vj ) for i = 1, 2 and j = 1, 2. We now assume
u0 − u1 ≤ 2−n−1 and v0 − v1 ≤ 2−n−1 , so that (u1 , v1 ) is the sample point
closest to (u0 , v0 ).
The derivatives ϕu and ϕv have limited range:
kϕu (u, v)k
≤ dR for all (u, v) ∈ [0, 1]2
(4.1)
kϕv (u, v)k
2
(4.2)
≤ eR for all (u, v) ∈ [0, 1]
4.4. ERROR ANALYSIS
43
This limits the difference ϕ(u0 , v0 ) − ϕ(u1 , v1 ):
kϕ(u0 , v0 ) − ϕ(u1 , v1 )k ≤ 2−n−1 (d + e)R
(4.3)
Thus we have a lower bound for the shortest distance,
√
kϕ(u0 , v0 ) − xk ≥ D − 2−n−1 (d + e)R.
This bound is not very impressive, but we can improve this bound by using the
fact that Fu (u0 , v0 ) = 0 and that the second derivatives are bounded as follows:
kϕuu (u, v)k
≤ 2d(d − 1)R for all (u, v) ∈ [0, 1]2
2
(4.4)
kϕuv (u, v)k
≤ 2deR for all (u, v) ∈ [0, 1]
(4.5)
kϕvv (u, v)k
≤ 2e(e − 1)R for all (u, v) ∈ [0, 1]2
(4.6)
We can now make a distance preserving coordinate change such that ϕ(u0 , v0 ) =
(0, 0, 0), x = (−kϕ(u0 , v0 ) − xk, 0, 0) and ϕ(u1 , v1 ) =: (a, b, c). From equations
(4.1) and (4.2) we get
b2 + c2 ≤ (2−n−1 (d + e)R)2 =: B.
Furthermore, equations (4.4), (4.5) and (4.6) gives
|ϕu (u, v) · (1, 0, 0)|
≤ 2−n d(e + d − 1)
|ϕv (u, v) · (1, 0, 0)|
≤ 2−n e(d + e − 1)
on the rectangle [u1 , u0 ] × [v1 , v0 ]. This gives
|a| ≤ 2−2n−1 (d + e)(d + e − 1).
We can refine this a little bit: Equations (4.4), (4.5) and (4.6) give
|ϕu (u, v) · (1, 0, 0)|
≤ 2−n d((d − 1)|u0 − u| + e|v0 − v|) and
|ϕv (u, v) · (1, 0, 0)|
≤ 2−n e(d|u0 − u| + (e − 1)|v0 − v|)
Setting u0 − u = v0 − v = t and integrating the bound on the derivatives, we
get
Z
|a| ≤
2−n−1
−n
2
0
(d + e)(d + e − 1)t dt = 2−2n−2 (d + e)(d + e − 1) =: A.
44
CHAPTER 4. CLOSEST POINTS BY SUBDIVISION
From Pythagoras we get D ≤ (kϕ(u0 , v0 ) − xk + A)2 + B. From this we get
the lower bound L of the shortest distance
√
L := D − B − A ≤ kϕ(u0 , v0 ) − xk.
√
The corresponding upper bound of the error D − kϕ(u0 , v0 ) − xk can be calculated in a stable way:
√
√
√
√
D − kϕ(u0 , v0 ) − xk ≤ D − L =
D− D−B+A
B
= √
+A
√
D+ D−B
This is better than the error bound in equation (4.3) in most cases, when D
is much bigger than B. A few examples can illustrate this pretty well. If the
bi-degree
is (3, 3), the diameter of the control points is 1, the distance returned
√
is D = 0.01, the depth of the recursion is 26, then the error is at most
1.016 · 10−13 . If the recursion level is increased to 32, then the error is at most
2.48 · 10−17 .
4.5
Conclusion
The closest point problem treated in this chapter is quite common in geometric
applications, and often needs to be solved by a computer algorithm in the fastest
possible way. Subdivision methods has the advantage that they can be made
very fast in almost all cases, and the guaranteed accuracy of the algorithm is
known. For special points, when the subdivision method takes too long, the
application must decide the proper action. In some cases it is natural to fall
back to a more accurate method. In other cases it is natural to simply discard
the point and move on to the next.
The result is a flexible set of algorithms that should be usable for most
applications.
Chapter 5
Monoid hypersurfaces1
Pål Hermunn Johansen, Magnus Løberg, Ragni Piene
5.1
Introduction
A monoid hypersurface is an (affine or projective) irreducible algebraic hypersurface which has a singularity of multiplicity one less than the degree of the
hypersurface. The presence of such a singular point forces the hypersurface to
be rational: there is a rational parameterization given by (the inverse of) the
linear projection of the hypersurface from the singular point.
The existence of an explicit rational parameterization makes such hypersurfaces potentially interesting objects in computer aided design. Moreover, since
the “space” of monoids of a given degree is much smaller than the space of all
hypersurfaces of that degree, one can hope to use monoids efficiently in (approximate or exact) implicitization problems. These were the reasons for considering
monoids in the paper [35]. In [28] monoid curves are used to approximate other
curves that are close to a monoid curve, and in [29] the same is done for monoid
surfaces. In both articles the error of such approximations are analyzed – for
each approximation, a bound on the distance from the monoid to the original
curve or surface can be computed.
1 This
chapter has been submitted as an article for the proceedings of the conference COMPASS II, and has been accepted by the editors of the book.
45
46
CHAPTER 5. MONOID HYPERSURFACES
In this article we shall study properties of monoid hypersurfaces and the
classification of monoid surfaces with respect to their singularities. Section 5.2
explores properties of monoid hypersurfaces in arbritrary dimension and over
an arbitrary base field. Section 5.3 contains results on monoid surfaces, both
over arbritrary fields and over R. The last section deals with the classification of
monoid surfaces of degree four. Real and complex quartic monoid surfaces were
first studied by Rohn [32], who gave a fairly complete description of all possible
cases. He also remarked [32, p. 56] that some of his results on quartic monoids
hold for monoids of arbitrary degree; in particular, we believe he was aware
of many of the results in Section 5.3. Takahashi, Watanabe, and Higuchi [38]
classify complex quartic monoid surfaces, but do not refer to Rohn. (They cite
Jessop [14]; Jessop, however, only treats quartic surfaces with double points and
refers to Rohn for the monoid case.) Here we aim at giving a short description
of the possible singularities that can occur on quartic monoids, with special
emphasis on the real case.
5.2
Basic properties
Let k be a field, let k̄ denote its algebraic closure and Pn := Pnk̄ the projective
n-space over k̄. Furthermore we define the set of k-rational points Pn (k) as the
set of points that admit representatives (a0 : · · · : an ) with each ai ∈ k.
For any homogeneous polynomial F ∈ k̄[x0 , . . . , xn ] of degree d and point
p = (p0 : p1 : · · · : pn ) ∈ Pn we can define the multiplicity of Z(F ) at p. We
know that pr 6= 0 for some r, so we can assume p0 = 1 and write
F =
d
X
xd−i
0 fi (x1 − p1 x0 , x2 − p2 x0 , . . . , xn − pn x0 )
i=0
where fi is homogeneous of degree i. Then the multiplicity of Z(F ) at p is
defined to be the smallest i such that fi 6= 0.
Let F ∈ k̄[x0 , . . . , xn ] be of degree d ≥ 3. We say that the hypersurface
X = Z(F ) ⊂ Pn is a monoid hypersurface if X is irreducible and has a singular
point of multiplicity d − 1.
In this article we shall only consider monoids X = Z(F ) where the singular
point is k-rational. Modulo a projective transformation of Pn over k we may –
and shall – therefore assume that the singular point is the point O = (1 : 0 :
· · · : 0).
5.2. BASIC PROPERTIES
47
Hence, we shall from now on assume that X = Z(F ), and
F = x0 fd−1 + fd ,
where fi ∈ k[x1 , . . . , xn ] ⊂ k[x0 , . . . , xn ] is homogeneous of degree i and fd−1 6=
0. Since F is irreducible, fd is not identically 0, and fd−1 and fd have no
common (non-constant) factors.
The natural rational parameterization of the monoid X = Z(F ) is the map
θF : Pn−1 → Pn
given by
θF (a) = (fd (a) : −fd−1 (a)a1 : . . . : −fd−1 (a)an ),
for a = (a1 : · · · : an ) such that fd−1 (a) 6= 0 or fd (a) 6= 0.
The set of lines through O form a Pn−1 . For every a = (a1 : · · · : an ) ∈ Pn−1 ,
the line
La := {(s : ta1 : . . . : tan )|(s : t) ∈ P1 }
(5.1)
intersects X = Z(F ) with multiplicity at least d − 1 in O. If fd−1 (a) 6= 0 or
fd (a) 6= 0, then the line La also intersects X in the point
θF (a) = (fd (a) : −fd−1 (a)a1 : . . . : −fd−1 (a)an ).
Hence the natural parameterization is the “inverse” of the projection of X from
the point O. Note that θF maps Z(fd−1 ) \ Z(fd ) to O. The points where the
parameterization map is not defined are called base points, and these points are
precisely the common zeros of fd−1 and fd . Each such point b corresponds to
the line Lb contained in the monoid hypersurface. Additionally, every line of
type Lb contained in the monoid hypersurface corresponds to a base point.
Note that Z(fd−1 ) ⊂ Pn−1 is the projective tangent cone to X at O, and
that Z(fd ) is the intersection of X with the hyperplane “at infinity” Z(x0 ).
Assume P ∈ X is another singular point on the monoid X. Then the line L
through P and O has intersection multiplicity at least d − 1 + 2 = d + 1 with
X. Hence, according to Bezout’s theorem, L must be contained in X, so that
this is only possible if dim X ≥ 2.
By taking the partial derivatives of F we can characterize the singular points
of X in terms of fd and fd−1 :
∂
Lemma 5.1. Let ∇ = ( ∂x
, . . . , ∂x∂n ) be the gradient operator.
1
(i) A point P = (p0 : p1 : · · · : pn ) ∈ Pn is singular on Z(F ) if and only if
fd−1 (p1 , . . . , pn ) = 0 and p0 ∇fd−1 (p1 , . . . , pn ) + ∇fd (p1 , . . . , pn ) = 0.
48
CHAPTER 5. MONOID HYPERSURFACES
(ii) All singular points of Z(F ) are on lines La where a is a base point.
(iii) Both Z(fd−1 ) and Z(fd ) are singular in a point a ∈ Pn−1 if and only if all
points on La are singular on X.
(iv) If not all points on La are singular, then at most one point other than O
on La is singular.
Proof. (i) follows directly from taking the derivatives of F = x0 fd−1 + fd , and
(ii) follows from (i) and the fact that F (P ) = 0 for any singular point P .
Furthermore, a point (s : ta1 : . . . : tan ) on La is, by (i), singular if and only if
s∇fd−1 (ta) + ∇fd (ta) = td−1 (s∇fd−1 (a) + t∇fd (a)) = 0.
This holds for all (s : t) ∈ P1 if and only if ∇fd−1 (a) = ∇fd (a) = 0. This proves
(iii). If either ∇fd−1 (a) or ∇fd−1 (a) are nonzero, the equation above has at
most one solution (s0 : t0 ) ∈ P1 in addition to t = 0, and (iv) follows.
Note that it is possible to construct monoids where F ∈ k[x0 , . . . , xn ], but
where no points of multiplicity d − 1 are k-rational. In that case there must be
(at least) two such points, and the line connecting these will be of multiplicity
d − 2. Furthermore, the natural parameterization will typically not induce a
parameterization of the k-rational points from Pn−1 (k).
5.3
Monoid surfaces
In the case of a monoid surface, the parameterization has a finite number of
base points. From Lemma 5.1 (ii) we know that all singularities of the monoid
other than O, are on lines La corresponding to these points. In what follows
we will develop the theory for singularities on monoid surfaces — most of these
results were probably known to Rohn [32, p. 56].
We start by giving a precise definition of what we shall mean by a monoid
surface.
Definition 5.2. For an integer d ≥ 3 and a field k of characteristic 0 the polynomials fd−1 ∈ k[x1 , x2 , x3 ]d−1 and fd ∈ k[x1 , x2 , x3 ]d define a normalized nondegenerate monoid surface Z(F ) ⊂ P3 , where F = x0 fd−1 +fd ∈ k[x0 , x1 , x2 , x3 ]
if the following hold:
(i) fd−1 , fd 6= 0
5.3. MONOID SURFACES
49
(ii) gcd(fd−1 , fd ) = 1
(iii) The curves Z(fd−1 ) ⊂ P2 and Z(fd ) ⊂ P2 have no common singular point.
The curves Z(fd−1 ) ⊂ P2 and Z(fd ) ⊂ P2 are called respectively the tangent
cone and the intersection with infinity.
Unless otherwise stated, a surface that satisfies the conditions of Definition
5.2 shall be referred to simply as a monoid surface.
Since we have finitely many base points b and each line Lb contains at most
one singular point in addition to O, monoid surfaces will have only finitely many
singularities, so all singularities will be isolated. (Note that Rohn included
surfaces with nonisolated singularities in his study [32].) We will show that the
singularities other than O can be classified by local intersection numbers.
Definition 5.3. Let f, g ∈ k[x1 , x2 , x3 ] be nonzero and homogeneous. Assume
p = (p1 : p2 : p3 ) ∈ Z(f, g) ⊂ P2 , and define the local intersection number
Ip (f, g) = lg
k̄[x1 , x2 , x3 ]mp
,
(f, g)
where k̄ is the algebraic closure of k, mp = (p2 x1 −p1 x2 , p3 x1 −p1 x3 , p3 x2 −p2 x3 )
is the homogeneous ideal of p, and lg denotes the length of the local ring as a
module over itself.
Note that Ip (f, g) ≥ 1 if and only if f (p) = g(p) = 0. When Ip (f, g) = 1 we
say that f and g intersect transversally at p. The terminology is justified by
the following lemma:
Lemma 5.4. Let f, g ∈ k[x1 , x2 , x3 ] be nonzero and homogeneous and p ∈
Z(f, g). Then the following are equivalent:
(i) Ip (f, g) > 1
(ii) f is singular at p, g is singular at p, or ∇f (p) and ∇g(p) are nonzero and
parallel.
(iii) s∇f (p) + t∇g(p) = 0 for some (s, t) 6= (0, 0)
Proof. (ii) is equivalent to (iii) by a simple case study: f is singular at p if and
only if (iii) holds for (s, t) = (1, 0), g is singular at p if and only if (iii) holds for
(s, t) = (0, 1), and ∇f (p) and ∇g(p) are nonzero and parallel if and only if (iii)
holds for some s, t 6= 0.
50
CHAPTER 5. MONOID HYPERSURFACES
We can assume that p = (0 : 0 : 1), so Ip (f, g) = lg S where
S=
k̄[x1 , x2 , x3 ](x1 ,x2 )
.
(f, g)
Furthermore, let d = deg f , e = deg g and write
f=
d
X
i=1
fi xd−i
and g =
3
e
X
gi xe−i
3
i=1
where fi , gi are homogeneous of degree i.
If f is singular at p, then f1 = 0. Choose ` = ax1 + bx2 such that ` is not
a multiple of g1 . Then ` will be a nonzero non-invertible element of S, so the
length of S is greater than 1.
We have ∇f (p) = (∇f1 (p), 0) and ∇g(p) = (∇g1 (p), 0). If they are parallel,
choose ` = ax0 + bx1 such that ` is not a multiple of f1 (or g1 ), and argue as
above.
Finally assume that f and g intersect transversally at p. We may assume
that f1 = x1 and g1 = x2 . Then (f, g) = (x1 , x2 ) as ideals in the local ring
k̄[x1 , x2 , x3 ](x1 ,x2 ) . This means that S is isomorphic to the field k̄(x3 ). The
length of any field is 1, so Ip (f, g) = lg S = 1.
Now we can say which are the lines Lb , with b ∈ Z(fd−1 , fd ), that contain a
singularity other than O:
Lemma 5.5. Let fd−1 and fd be as in Definition 5.2. The line Lb contains a
singular point other than O if and only if Z(fd−1 ) is nonsingular at b and the
intersection multiplicity Ib (fd−1 , fd ) > 1.
Proof. Let b = (b1 : b2 : b3 ) and assume that (b0 : b1 : b2 : b3 ) is a singular
point of Z(F ). Then, by Lemma 5.1, fd−1 (b1 , b2 , b3 ) = fd (b1 , b2 , b3 ) = 0 and
b0 ∇fd−1 (b1 , b2 , b3 ) + ∇fd (b1 , b2 , b3 ) = 0, which implies Ib (fd−1 , fd ) > 1. Furthermore, if fd−1 is singular at b, then the gradient ∇fd−1 (b1 , b2 , b3 ) = 0, so fd ,
too, is singular at b, contrary to our assumptions.
Now assume that Z(fd−1 ) is nonsingular at b = (b1 : b2 : b3 ) and the intersection multiplicity Ib (fd−1 , fd ) > 1. The second assumption implies fd−1 (b1 , b2 , b3 ) =
fd (b1 , b2 , b3 ) = 0 and s∇fd−1 (b1 , b2 , b3 ) = t∇fd (b1 , b2 , b3 ) for some (s, t) 6= (0, 0).
Since Z(fd−1 ) is nonsingular at b, we know that ∇fd−1 (b1 , b2 , b3 ) 6= 0, so t 6= 0.
Now (−s/t : b1 : b2 : b3 ) 6= (1 : 0 : 0 : 0) is a singular point of Z(F ) on the line
Lb .
5.3. MONOID SURFACES
51
Recall that an An singularity is a singularity with normal form x21 +x22 +xn+1
,
3
see [3, p. 184].
Proposition 5.6. Let fd−1 and fd be as in Definition 5.2, and assume P = (p0 :
p1 : p2 : p3 ) 6= (1 : 0 : 0 : 0) is a singular point of Z(F ) with I(p1 :p2 :p3 ) (fd−1 , fd ) =
m. Then P is an Am−1 singularity.
Proof. We may assume that P = (0 : 0 : 0 : 1) and write the local equation
g := F (x0 , x1 , x2 , 1) = x0 fd−1 (x1 , x2 , 1) + fd (x1 , x2 , 1) =
d
X
gi
(5.2)
i=2
with gi ∈ k̄[x0 , x1 , x2 ] homogeneous of degree i. Since Z(fd−1 ) is nonsingular at
0 := (0 : 0 : 1), we can assume that the linear term of fd−1 (x1 , x2 , 1) is equal to
x1 . The quadratic term g2 of g is then g2 = x0 x1 + ax21 + bx1 x2 + cx22 for some
a, b, c ∈ k. The Hessian matrix of g evaluated at P is


0 1 0
H(g)(0, 0, 0) = H(g2 )(0, 0, 0) = 1 2a b 
0 b 2c
which has corank 0 when c 6= 0 and corank 1 when c = 0. By [3, p. 188], P is
an A1 singularity when c 6= 0 and an An singularity for some n when c = 0.
The index n of the singularity is equal to the Milnor number
µ = dimk̄
k̄[x0 , x1 , x2 ](x0 ,x1 ,x2 )
k̄[x0 , x1 , x2 ](x0 ,x1 ,x2 )
.
= dimk̄ ∂g
Jg
, ∂g , ∂g
∂x0
∂x1
∂x2
We need to show that µ = I0 (fd−1 , fd )−1. From the definition of the intersection
multiplicity, it is not hard to see that
I0 (fd−1 , fd ) = dimk̄
k̄[x1 , x2 ](x1 ,x2 )
.
(fd−1 (x1 , x2 , 1), fd (x1 , x2 , 1))
The singularity at p is isolated, so the Milnor number is finite. Furthermore,
since gcd(fd−1 , fd ) = 1, the intersection multiplicity is finite. Therefore both
dimensions can be calculated in the completion rings. For the rest of the proof we
view fd−1 and fd as elements of the power series rings k̄[[x1 , x2 ]] ⊂ k̄[[x0 , x1 , x2 ]],
and all calculations are done in these rings.
52
CHAPTER 5. MONOID HYPERSURFACES
Since Z(fd−1 ) is smooth at O, we can write
fd−1 (x1 , x2 , 1) = (x1 − φ(x2 )) u(x1 , x2 )
for some power series φ(x2 ) and invertible power series u(x1 , x2 ). To simplify
notation we write u = u(x1 , x2 ) ∈ k̄[[x1 , x2 ]].
The Jacobian ideal Jg is generated by the three partial derivatives:
∂g
∂x0
∂g
∂x1
∂g
∂x2
(x1 − φ(x2 )) u
∂u
∂fd
= x0 u + (x1 − φ(x2 ))
+
(x1 , x2 )
∂x1
∂x1
∂u
∂fd
= x0 −φ0 (x2 )u + (x1 − φ(x2 ))
(x1 , x2 )
+
∂x2
∂x2
∂g
By using the fact that x1 − φ(x2 ) ∈ ∂x
we can write Jg without the symbols
0
∂u
∂x1
and
=
∂u
∂x2 :
Jg = x1 − φ(x2 ), x0 u +
∂fd
0
∂x1 (x1 , x2 ), −x0 uφ (x2 )
+
∂fd
∂x2 (x1 , x2 )
To make theP
following calculations clear, define the polynomials hi by writing
d
fd (x1 , x2 , 1) = i=0 xi1 hi (x2 ). Now
Pd
Pd
0
i 0
(x
)
,
h
Jg = x1 − φ(x2 ), x0 u + i=1 ixi−1
h
(x
),
−x
uφ
(x
)
+
x
i 2
0
2
1
i=0 1 i 2
so
k̄[[x2 ]]
k̄[[x0 , x1 , x2 ]]
=
Jg
(A(x2 ))
where
A(x2 ) = φ0 (x2 )
P
d
i=1
P
d
i 0
iφ(x2 )i−1 hi (x2 ) +
φ(x
)
h
(x
)
.
2
2
i
i=0
For the intersection multiplicity we have
k̄[[x2 ]]
k̄[[x1 , x2 ]]
=
=
Pd
i
fd−1 (x1 , x2 , 1), fd (x1 , x2 , 1)
x1 − φ(x2 ), i=0 x1 hi (x2 )
B(x2 )
k̄[[x1 , x2 ]]
Pd
where B(x2 ) = i=0 φ(x2 )i hi (x2 ). Observing that B 0 (x2 ) = A(x2 ) gives the
result µ = I0 (fd−1 , fd ) − 1.
5.3. MONOID SURFACES
53
Corollary 5.7. A monoid surface of degree d can have at most 12 d(d − 1)
singularities in addition to O. If this number of singularities is obtained, then
all of them will be of type A1 .
Proof. The sum of all local intersection numbers Ia (fd−1 , fd ) is given by Bézout’s
theorem:
X
Ia (fd−1 , fd ) = d(d − 1).
a∈Z(fd−1 ,fd )
The line La will contain a singularity other than O only if Ia (fd−1 , fd ) ≥ 2,
giving a maximum of 12 d(d − 1) singularities in addition to O. Also, if this
number is obtained, all local intersection numbers must be exactly 2, so all
singularities other than O will be of type A1 .
Both Proposition 5.6 and Corollary 5.7 were known to Rohn, who stated
these results only in the case d = 4, but said they could be generalized to
arbitrary d [32, p. 60].
For the rest of the section we will assume k = R. It turns out that we can find
a real normal form for the singularities other than O. The complex singularities
of type An come in several real types, with normal forms x21 ±x22 ±xn+1
. Varying
3
the ± gives two types for n = 1 and n even, and three types for n ≥ 3 odd.
The real type with normal form x21 − x22 + xn+1
is called an A−
n singularity, or
3
−
of type A , and is what we find on real monoids:
Proposition 5.8. On a real monoid, all singularities other than O are of type
A− .
Proof. Assume p = (0 : 0 : 1) is a singular point on Z(F ) and set g =
F (x0 , x1 , x2 , 1) as in the proof of Proposition 5.6.
First note that u−1 g = x0 (x1 − φ(x2 )) + fd (x1 , x2 )u−1 is an equation for the
singularity. We will now prove that u−1 g is right equivalent to ±(x20 − x21 + xn2 ),
for some n, by constructing right equivalent functions u−1 g =: g(0) ∼ g(1) ∼
g(2) ∼ g(3) ∼ ±(x20 − x21 + xn2 ). Let
g(1) (x0 , x1 , x2 )
= g(0) (x0 , x1 + φ(x2 ), x2 )
= x0 x1 + fd (x1 + φ(x2 ), x2 )u−1 (x1 + φ(x2 ), x2 )
= x0 x1 + ψ(x1 , x2 )
where ψ(x1 , x2 ) ∈ R[[x1 , x2 ]]. Write ψ(x1 , x2 ) = x1 ψ1 (x1 , x2 ) + ψ2 (x2 ) and
define
g(2) (x0 , x1 , x2 ) = g(1) (x0 − ψ1 (x1 , x2 ), x1 , x2 ) = x0 x1 + ψ2 (x2 ).
54
CHAPTER 5. MONOID HYPERSURFACES
The power series ψ2 (x2 ) can be written on the form
ψ2 (x2 ) = sxn2 (a0 + a1 x2 + a2 x22 + . . . )
where s = ±1 and a0 > 0. We see that g(2) is right equivalent to g(3) =
x0 x1 + sxn2 since
q
n
2
g(2) (x0 , x1 , x2 ) = g(3) x0 , x1 , x2 a0 + a1 x2 + a2 x2 + . . . .
Finally we see that
g(4) (x0 , x1 , x2 ) := g(3) (sx0 − sx1 , x0 + x1 , x2 ) = s(x20 − x21 + xn2 )
proves that u−1 g is right equivalent to s(x20 − x21 + xn2 ) which is an equation for
an An−1 singularity with normal form x20 − x21 + xn2 .
Note that for d = 3, the singularity at O can be an A+
1 singularity. This
happens for example when f2 = x20 + x21 + x22 .
For a real monoid, Corollary 5.7 implies that we can have at most 12 d(d − 1)
real singularities in addition to O. We can show that the bound is sharp by a
simple construction:
Example. To construct a monoid with the maximal number of real singularities, it is sufficient to construct two affine real curves in the xy-plane defined
by equations fd−1 and fd of degrees d − 1 and d such that the curves intersect
in d(d − 1)/2 points with multiplicity 2. Let m ∈ {d − 1, d} be odd and set
m Y
2iπ
2iπ
x sin
fm = ε −
+ y cos
+1 .
m
m
i=1
For ε > 0 sufficiently small there exist at least m+1
radii r > 0, one for each
2
root of the univariate polynomial fm |x=0 , such that the circle x2 + y 2 − r2
intersects fm in m points with multiplicity 2. Let f2d−1−m be a product of such
circles. Now the homogenizations of fd−1 and fd define a monoid surface with
1 + 12 d(d − 1) singularities. See Figure 5.1.
Proposition 5.6 and Bezout’s theorem imply that the maximal Milnor number of a singularity other than O is d(d − 1) − 1. The following example shows
that this bound can be achieved on a real monoid:
Example. The surface X ⊂ P3 defined by F = x0 (x1 x2d−2 + x3d−1 ) + xd1 has exactly two singular points. The point (1 : 0 : 0 : 0) is a singularity of multiplicity
5.3. MONOID SURFACES
55
Figure 5.1: The curves fm for m = 3, 5 and corresponding circles
3 with Milnor number µ = (d2 − 3d + 1)(d − 2), while the point (0 : 0 : 1 : 0) is
an Ad(d−1)−1 singularity. A picture of this surfaces for d = 4 is shown in Figure
5.2.
Figure 5.2: The surface defined by F = x0 (x1 x2d−2 + xd−1
) + xd1 for d = 4.
3
56
CHAPTER 5. MONOID HYPERSURFACES
5.4
Quartic monoid surfaces
Every cubic surface with isolated singularities is a monoid. Both smooth and
singular cubic surfaces have been studied extensively, most notably in [33], where
real cubic surfaces and their singularities were classifed, and more recently in
[36], [4], and [16]. The site [17] contains additional pictures and references.
In this section we shall consider the case d = 4. The classification of real
and complex quartic monoid surfaces was started by Rohn [32]. (In addition
to considering the singularities, Rohn studied the existence of lines not passing
through the triple point, and that of other special curves on the monoid.) In [38],
Takahashi, Watanabe, and Higuchi described the singularities of such complex
surfaces. The monoid singularity of a quartic monoid is minimally elliptic [42],
and minimally elliptic singularities have the same complex topological type if
and only if their dual graphs are isomorphic [18]. In [18] all possible dual graphs
for minimally elliptic singularities are listed, along with example equations.
Using Arnold’s notation for the singularities, we use and extend the approach
of Takahashi, Watanabe, and Higuchi in [38].
Consider a quartic monoid surface, X = Z(F ), with F = x0 f3 + f4 . The
tangent cone, Z(f3 ), can be of one of nine (complex) types, each needing a
separate analysis.
For each type we fix f3 , but any other tangent cone of the same type will be
projectively equivalent (over the complex numbers) to this fixed f3 . The nine
different types are:
1. Nodal irreducible curve, f3 = x1 x2 x3 + x32 + x33 .
2. Cuspidal curve, f3 = x31 − x22 x3 .
3. Conic and a chord, f3 = x3 (x1 x2 + x23 )
4. Conic and a tangent line, f3 = x3 (x1 x3 + x22 ).
5. Three general lines, f3 = x1 x2 x3 .
6. Three lines meeting in a point, f3 = x32 − x2 x23
7. A double line and another line, f3 = x2 x23
8. A triple line f3 = x33
9. A smooth curve, f3 = x31 + x32 + x33 + 3ax0 x1 x3 where a3 6= −1
5.4. QUARTIC MONOID SURFACES
57
To each quartic monoid we can associate, in addition to the type, several
integer invariants, all given as intersection numbers. From [38] we know that,
for the types 1–3, 5, and 9, these invariants will determine the singularity type
of O up to right equivalence. In the other cases the singularity series, as defined
by Arnol’d in [1] and [2], is determined by the type of f3 . We shall use, without
proof, the results on the singularity type of O due to [38]; however, we shall use
the notations of [1] and [2].
We complete the classification begun in [38] by supplying a complete list
of the possible singularities occurring on a quartic monoid. In addition, we
extend the results to the case of real monoids. Our results are summarized in
the following theorem.
Theorem 5.9. On a quartic monoid surface, singularities other than the monoid
point can occur as given in Table 5.1. Moreover, all possibilities are realizable on
real quartic monoids with a real monoid point, and with the other singularities
being real and of type A− .
Proof. The invariants listed in the “Invariants and constraints” column are all
nonnegative integers, and any set of integer values satisfying the equations represents one possible set of invariants, as described above. Then, for each set
of invariants, (positive) intersection multiplicities, denoted mi , m0i and m00i , will
determine the singularities other than O. The column “Other singularities” give
these and the equations they must satisfy. Here we use the notation A0 for a
line La on Z(F ) where O is the only singular point.
The analyses of the nine cases share many similarities, and we have chosen
not to go into great detail when one aspect of a case differs little from the
previous one. We end the section with a discussion on the possible real forms
of the tangent cone and how this affects the classification of the real quartic
monoids.
In all cases, we shall write
f4
= a1 x41 + a2 x31 x2 + a3 x31 x3 + a4 x21 x22 + a5 x21 x2 x3
+ a6 x21 x23 + a7 x1 x32 + a8 x1 x22 x3 + a9 x1 x2 x23 + a10 x1 x33
+ a11 x42 + a12 x32 x3 + a13 x22 x23 + a14 x2 x33 + a15 x43
and we shall investigate how the coefficients a1 , . . . , a15 are related to the geometry of the monoid.
58
Case
1
2
3
CHAPTER 5. MONOID HYPERSURFACES
Triple point
T3,3,4
T3,3,3+m
Q10
T9+m
T3,4+r0 ,4+r1
4
S series
5
T4+jk ,4+jl ,4+jm
6
U series
7
V series
8
9
V 0 series
P8 = T3,3,3
Invariants and constraints
m = 2, . . . , 12
m = 2, 3
r0 = max(j0 , k0 ), r1 = max(j1 , k1 ),
j0 > 0 ↔ k0 > 0, min(j0 , k0 ) ≤ 1,
j1 > 0 ↔ k1 > 0, min(j1 , k1 ) ≤ 1
j0 ≤ 8, k0 ≤ 4, min(j0 , k0 ) ≤ 2,
j0 > 0 ↔ k0 > 0, j1 > 0 ↔ k0 > 1
m1 + l1 ≤ 4, k2 + m2 ≤ 4,
k3 + l3 ≤ 4, k2 > 0 ↔ k3 > 0,
l1 > 0 ↔ l3 > 0, m1 > 0 ↔ m2 > 0,
min(k2 , k3 ) ≤ 1, min(l1 , l3 ) ≤ 1,
min(m1 , m2 ) ≤ 1, jk = max(k2 , k3 ),
jl = max(l1 , l3 ), jm = max(m1 , m2 )
j1 > 0 ↔ j2 > 0 ↔ j3 > 0,
at most one of j1 , j2 , j3 > 1,
j1 , j2 , j3 ≤ 4
j0 > 0 ↔ k0 > 0, min(j0 , k0 ) ≤ 1,
j0 ≤ 4, k0 ≤ 4
Other singularities
P
Ami −1 , P mi = 12
Ami −1 , P mi = 12 − m
Ami −1 , P mi = 12
Ami −1 , P mi = 12 − m
Ami −1 , P mi = 4 − k0 − k1 ,
Am0 −1 ,
m0i = 8 − j0 − j1
i
P
Ami −1 , P mi = 4 − k0 ,
Am0 −1 ,
m0 = 8 − j0
i
P i
Ami −1 , P mi = 4 − m1 − l1 ,
Am0 −1 ,
m0 = 4 − k2 − m2 ,
i
P i00
mi = 4 − k3 − l3
Am00 −1 ,
i
P
Ami −1 , P mi = 4 − j1 ,
m0 = 4 − j2 ,
Am0 −1 ,
i
P i00
Am00 −1 ,
m = 4 − j3
i
P i
Ami −1 ,
mi = 4 − j0 ,
None
P
Ami −1 ,
mi = 12
Table 5.1: Possible configurations of singularities for each case
5.4. QUARTIC MONOID SURFACES
59
Case 1. The tangent cone is a nodal irreducible curve, and we can assume
f3 (x1 , x2 , x3 ) = x1 x2 x3 + x32 + x33 . The nodal curve is singular at (1 : 0 : 0).
If f4 (1, 0, 0) 6= 0, then O is a T3,3,4 singularity [38]. We recall that (1 : 0 : 0)
cannot be a singular point on Z(f4 ) as this would imply a singular line on the
monoid, so we assume that either (1 : 0 : 0) 6∈ Z(f4 ) or (1 : 0 : 0) is a smooth
point on Z(f4 ). Let m denote the intersection number I(1:0:0) (f3 , f4 ). Since
Z(f3 ) is singular at (1 : 0 : 0) we have m 6= 1. From [38] we know that O
is a T3,3,3+m singularity for m = 2, . . . , 12. Note that some of these complex
singularities have two real forms, as illustrated in Figure 5.3.
Figure 5.3: The monoids Z(x3 + y 3 + 5xyz − z 3 (x + y)) and Z(x3 + y 3 + 5xyz −
z 3 (x − y)) both have a T3,3,5 singularity, but the singularities are not right
equivalent over R. (The pictures are generated by the program [8].)
Bézout’s theorem and Proposition 5.6 limit the possible configurations of
singularities on the monoid for each m. Let θ(s, t) = (−s3 − t3 , s2 t, st2 ). Then
the tangent cone Z(f3 ) is parameterized by θ as a map from P1 to P2 . When
we need to compute the intersection numbers between the rational curve Z(f3 )
and the curve Z(f4 ), we can do that by studying the roots of the polynomial
60
CHAPTER 5. MONOID HYPERSURFACES
f4 (θ). Expanding the polynomial gives
f4 (θ)(s, t)
= a1 s12 − a2 s11 t + (−a3 + a4 )s10 t2 + (4a1 + a5 − a7 )s9 t3
+
(−3a2 + a6 − a8 + a11 )s8 t4 + (−3a3 + 2a4 − a9 + a12 )s7 t5
+
(6a1 + 2a5 − a7 − a10 + a13 )s6 t6
+
(−3a2 + 2a6 − a8 + a14 )s5 t7 + (−3a3 + a4 − a9 + a15 )s4 t8
+
(4a1 + a5 − a10 )s3 t9 + (−a2 + a6 )s2 t10 − a3 st11 + a1 t12 .
This polynomial will have roots at (0 : 1) and (1 : 0) if and only if f4 (1, 0, 0) =
a1 = 0. When a1 = 0 we may (by symmetry) assume a3 6= 0, so that (0 : 1) is
a simple root and (1 : 0) is a root of multiplicity m − 1. Other roots of f4 (θ)
correspond to intersections of Z(f3 ) and Z(f4 ) away from (1 : 0 : 0). The multiplicity mi of each root is equal to the corresponding intersection multiplicity,
giving rise to an Ami −1 singularity if mi > 0, as described by Proposition 5.6,
or a line La ⊂ Z(F ) with O as the only singular point if mi = 1.
The polynomial f4 (θ) defines a linear map from the coefficient space k 15 of f4
to the space of homogeneous polynomials of degree 12 in s and t. By elementary
linear algebra, we see that the image of this map is the set of polynomials of
the form
b0 s12 + b1 s11 t + b2 s10 t2 + · · · + b12 t12
where b0 = b12 . The kernel of the map corresponds to the set of polynomials of
the form `f3 where ` is a linear form. This means that f4 (θ) ≡ 0 if and only if
f3 is a factor in f4 , making Z(F ) reducible and not a monoid.
For every m = 0, 2, 3, 4, . . . , 12 we can select r parameter points
p1 , . . . , pr ∈ P1 \ {(0 : 1), (1 : 0)}
and positive multiplicities m1 , . . . , mr with m1 + · · · + mr = 12 − m and try to
describe the polynomials f4 such that f4 (θ) has a root of multiplicity mi at pi
for each i = 1, . . . , r.
Still assuming a3 6= 0 whenever a1 = 0, any such choice of parameter points
p1 , . . . , pr and multiplicities m1 , . . . , mr corresponds to a polynomial q = b0 s12 +
b1 s11 t + · · · + b12 t12 that is, up to a nonzero constant, uniquely determined.
Now, q is equal to f4 (θ) for some f4 if and only if b0 = b12 . If m ≥ 2, then q
contains a factor stm−1 , so b0 = b12 = 0, giving q = f4 (θ) for some f4 . In fact,
when m ≥ 2 any choice of p1 , . . . , pr and m1 , . . . , mr with m1 +· · ·+mr = 12−m
corresponds to a four dimensional space of equations f4 that gives this set of
roots and multiplicities in f4 (θ). If f40 is one such f4 , then any other is of the
5.4. QUARTIC MONOID SURFACES
61
form λf40 + `f3 for some constant λ 6= 0 and linear form `. All of these give
monoids that are projectively equivalent.
When m = 0, we write pi = (αi : βi ) for i = 1, . . . , r. The condition b0 = b12
on the coefficients of q translates to
α1m1 · · · αrmr = β1m1 · · · βrmr .
(5.3)
This means that any choice of parameter points (α1 : β1 ), . . . , (αr : βr ) and
multiplicities m1 , . . . , mr with m1 + · · · + mr = 12 that satisfy condition (5.3)
corresponds to a four dimensional family λf40 + `f3 , giving a unique monoid up
to projective equivalence.
For example, we can have an A11 singularity only if f4 (θ) is of the form
(αs − βt)12 . Condition (5.3) implies that this can only happen for 12 parameter
points, all of the form (1 : ω), where ω 12 = 1. Each such parameter point (1 :
ω) corresponds to a monoid uniquely determined up to projective equivalence.
However, since there are six projective transformations of the plane that maps
Z(f3 ) onto itself, this correspondence is not one to one. If ω112 = ω212 = 1,
then ω1 and ω2 will correspond to projectively equivalent monoids if and only if
ω13 = ω23 or ω13 ω23 = 1. This means that there are three different quartic monoids
with one T3,3,4 singularity and one A11 singularity. One corresponds to those
ω where ω 3 = 1, one to those ω where ω 3 = −1, and one to those ω where
ω 6 = −1. The first two of these have real representatives, ω = ±1.
It easy to see that for any set of multiplicities m1 + · · · + mr = 12, we can
find real points p1 , . . . , pr such that condition (5.3) is satisfied. This completely
classifies the possible configurations of singularities when f3 is an irreducible
nodal curve.
Case 2. The tangent cone is a cuspidal curve, and we can assume
f3 (x1 , x2 , x3 ) = x31 − x22 x3 . The cuspidal curve is singular at (0 : 0 : 1) and can
be parameterized by θ as a map from P1 to P2 where θ(s, t) = (s2 t, s3 , t3 ). The
intersection numbers are determined by the degree 12 polynomial f4 (θ). As in
the previous case, f4 (θ) ≡ 0 if and only if f3 is a factor of f4 , and we will assume
this is not the case. The multiplicity m of the factor s in f4 (θ) determines the
type of singularity at O. If m = 0 (no factor s), then O is a Q10 singularity. If
m = 2 or m = 3, then O is of type Q9+m . If m > 3, then (0 : 0 : 1) is a singular
point on Z(f4 ), so the monoid has a singular line and is not considered in this
article. Also, m = 1 is not possible, since f4 (θ(s, t)) = f4 (s2 t, s3 , t3 ) cannot
contain st11 as a factor.
For each m = 0, 2, 3 we can analyze the possible configurations of other
singularities on the monoid. Similarly to the previous case, any choice of parameter points p1 , . . . , pr ∈ P1 \ {(0 : 1)} and positive multiplicities m1 , . . . , mr
62
CHAPTER 5. MONOID HYPERSURFACES
P
with
mi = 12 − m corresponds, up to a nonzero constant, to a unique degree
12 polynomial q.
When m = 2 or m = 3, for any choice of parameter values and associated
multiplicities, we can find a four dimensional family f4 = λf40 + `f3 with the
prescribed roots in f4 (θ). As before, the family gives projectively equivalent
monoids.
When m = 0, one condition must be satisfied for q to be of the form f4 (θ),
namely b11 = 0, where b11 is the coefficient of st11 in q.
For example, we can have an A11 singularity only if q is of the form (αs −
βt)12 . The condition b11 = 0 implies that either q = λs12 or q = λt12 . The
first case gives a surface with a singular line, while the other gives a monoid
with an A11 singularity (see Figure 5.2). The line from O to the A11 singularity
corresponds to the inflection point of Z(f3 ).
For any set of multiplicities m1 , . . . , mr with m1 + · · · + mr = 12, it is not
hard to see that there exist real points p1 , . . . , pr such
Pthat the condition b11 = 0
is satisfied. It suffices to take pi = (αi : 1), with
mi αi = 0 (the condition
corresponding to b11 = 0). This completely classifies the possible configurations
of singularities when f3 is a cuspidal curve.
Case 3. The tangent cone is the product of a conic and a line that is not
tangent to the conic, and we can assume f3 = x3 (x1 x2 + x23 ). Then Z(f3 ) is
singular at (1 : 0 : 0) and (0 : 1 : 0), the intersections of the conic Z(x1 x2 + x23 )
and the line Z(x3 ). For each f4 we can associate four integers:
j0 := I(1:0:0) (x1 x2 + x23 , f4 ),
j1 := I(0:1:0) (x1 x2 + x23 , f4 ),
k0 := I(1:0:0) (x3 , f4 ),
k1 := I(0:1:0) (x3 , f4 ).
We see that k0 > 0 ⇔ f4 (1 : 0 : 0) = 0 ⇔ j0 > 0, and that Z(f4 ) is singular
at (1 : 0 : 0) if and only if k0 and j0 both are bigger than one. These cases
imply a singular line on the monoid, and are not considered in this article. The
same holds for k1 , j1 and the point (0 : 1 : 0).
Define ri = max(ji , ki ) for i = 1, 2. Then, by [38], O will be a singularity of
type T3,4+r0 ,4+r1 if r0 ≤ r1 , or of type T3,4+r1 ,4+r0 if r0 ≥ r1 .
We can parameterize the line Z(x3 ) by θ1 where θ1 (s, t) = (s, t, 0), and the
conic Z(x1 x2 + x23 ) by θ2 where θ2 (s, t) = (s2 , −t2 , st). Similarly to the previous
cases, roots of f4 (θ1 ) correspond to intersections between Z(f4 ) and the line
Z(x3 ), while roots of f4 (θ2 ) correspond to intersections between Z(f4 ) and the
conic Z(x1 x3 + x23 ).
For any legal values of of j0 , j1 , k0 and k1 , parameter points
(α1 : β1 ), . . . , (αmr : βmr ) ∈ P1 \ {(0 : 1), (1 : 0)},
5.4. QUARTIC MONOID SURFACES
63
with multiplicities m1 , . . . , mr such that m1 + · · · + mr = 4 − k0 − k1 , and
parameter points
0
0
1
(α10 : β10 ), . . . , (αm
0 : βm0 ) ∈ P \ {(0 : 1), (1 : 0)},
r
r
with multiplicities m01 , . . . , m0r0 such that m01 + · · · + m0r0 = 8 − j0 − j1 , we can
fix polynomials q1 and q2 such that
• q1 is nonzero, of degree 4, and has factors sk1 , tk0 and (βi s − αi t)mi for
i = 1, . . . , r,
0
• q2 is nonzero, of degree 8, and has factors sj1 , tj0 and (βi0 s − αi0 t)mi for
i = 1, . . . , r0 .
Now q1 and q2 are determined up to multiplication by nonzero constants. Write
q1 = b0 s4 + · · · + b4 t4 and q2 = c0 s8 + · · · + c8 t8 .
The classification of singularities on the monoid consists of describing the
conditions on the parameter points and nonzero constants λ1 and λ2 for the
pair (λ1 q1 , λ2 q2 ) to be on the form (f4 (θ1 ), f4 (θ2 )) for some f4 .
Similarly to the previous cases, f4 (θ1 ) ≡ 0 if and only if x3 is a factor in f4
and f4 (θ2 ) ≡ 0 if and only if x1 x2 +x23 is a factor in f4 . Since f3 = x3 (x1 x2 +x23 ),
both cases will make the monoid reducible, so we only consider λ1 , λ2 6= 0.
We use linear algebra to study the relationship between the coefficients
a1 . . . a15 of f4 and the polynomials q1 and q2 . We find (λ1 q1 , λ2 q2 ) to be of
the form (f4 (θ1 ), f4 (θ2 )) if and only if λ1 b0 = λ2 c0 and λ1 b4 = λ2 c8 . Furthermore, the pair (λ1 q1 , λ2 q2 ) will fix f4 modulo f3 . Since f4 and λf4 correspond
to projectively equivalent monoids for any λ 6= 0, it is the ratio λ1 /λ2 , and not
λ1 and λ2 , that is important.
Recall that k0 > 0 ⇔ j0 > 0 and k1 > 0 ⇔ j1 > 0. If k0 > 0 and k1 > 0,
then b0 = c0 = b4 = c8 = 0, so for any λ1 , λ2 6= 0 we have (λ1 q1 , λ2 q2 ) =
(f4 (θ1 ), f4 (θ2 )) for some f4 . Varying λ1 /λ2 will give a one-parameter family of
monoids for each choice of multiplicities and parameter points.
If k0 = 0 and k1 > 0, then b0 = c0 = 0. The condition λ1 b4 = λ2 c8 implies
λ1 /λ2 = c8 /b4 . This means that any choice of multiplicities and parameter
points will give a unique monoid up to projective equivalence. The same goes
for the case where k0 > 0 and k1 = 0.
Finally, consider the case where k0 = k1 = 0. For (λ1 q1 , λ2 q2 ) to be of the
form (f4 (θ1 ), f4 (θ2 )) we must have λ1 /λ2 = c8 /b4 = c0 /b0 . This translates into
a condition on the parameter points, namely
0
0
0
0
0 mr0
(α0 )m1 · · · (αr0 0 )mr0
(β10 )m1 · · · (βr0
)
= 1 m1
.
m1
mr
β1 · · · βr
α1 · · · αrmr
(5.4)
64
CHAPTER 5. MONOID HYPERSURFACES
In other words, if condition (5.4) holds, we have a unique monoid up to projective
equivalence.
It is easy to see that for any choice of multiplicities, it is possible to find
real parameter points such that condition (5.4) is satisfied. This completes the
classification of possible singularities when the tangent cone is a conic plus a
chordal line.
Case 4. The tangent cone is the product of a conic and a line tangent to
the conic, and we can assume f3 = x3 (x1 x3 + x22 ). Now Z(f3 ) is singular at
(1 : 0 : 0). For each f4 we can associate two integers
j0 := I(1:0:0) (x1 x3 + x22 , f4 )
and
k0 := I(1:0:0) (x3 , f4 ).
We have j0 > 0 ⇔ k0 > 0, j0 > 1 ⇔ k0 > 1. Furthermore, j0 and k0 are
both greater than 2 if and only if Z(f4 ) is singular at (1 : 0 : 0), a case we have
excluded. The singularity at O will be of the S series, from [1], [2].
We can parameterize the conic Z(x1 x3 + x22 ) by θ2 and the line Z(x3 ) by
θ1 where θ2 (s, t) = (s2 , st, −t2 ) and θ1 (s, t) = (s, t, 0). As in the previous case,
the monoid is reducible if and only if f4 (θ1 ) ≡ 0 or f4 (θ2 ) ≡ 0. Consider two
nonzero polynomials
q1
= b0 s4 + b1 s3 t + b2 s2 t2 + b3 st3 + b4 t4
q2
= c0 s8 + c1 s7 t + · · · + c7 st7 + c8 t8 .
Now (λ1 q1 , λ2 q2 ) = (f4 (θ1 ), f4 (θ2 )) for some f4 if and only if λ1 b0 = λ2 c0 and
λ1 b1 = λ2 c1 . As before, only the cases where λ1 , λ2 6= 0 are interesting. We
see that (λ1 q1 , λ2 q2 ) = (f4 (θ1 ), f4 (θ2 )) for some λ1 , λ2 6= 0 if and only if the
following hold:
• b0 = 0 ↔ c0 = 0 and b1 = 0 ↔ c1 = 0
• b0 c1 = b1 c0 .
The classification of other singularities (than O) is very similar to the previous case. Roots of f4 (θ1 ) and f4 (θ2 ) away from (1 : 0) correspond to intersections
of Z(f3 ) and Z(f4 ) away from the singular point of Z(f3 ), and when one such
intersection is multiple, there is a corresponding singularity on the monoid.
Now assume (λ1 q1 , λ2 q2 ) = (f4 (θ1 ), f4 (θ2 )) for some λ1 , λ2 6= 0 and some
f4 . If b0 6= 0 (equivalent to c0 6= 0) then j0 = k0 = 0 and λ1 /λ2 = c0 /b0 .
If b0 = c0 = 0 and b1 6= 0 (equivalent to c1 6= 0), then j0 = k0 = 1, and
λ1 /λ2 = c1 /b1 . If b0 = b1 = c0 = c1 = 0, then j0 , k0 > 1 and any value of λ1 /λ2
5.4. QUARTIC MONOID SURFACES
65
will give (λ1 q1 , λ2 q2 ) of the form (f4 (θ1 ), f4 (θ2 )) for some f4 . Thus we get a
one-dimensional family of monoids for this choice of q1 and q2 .
Now consider the possible configurations of other singularities on the monoid.
Assume that j00 ≤ 8 and k00 ≤ 4 are nonnegative integers such that j0 > 0 ↔
k0 > 0 and j0 > 1 ↔ k0 > 1. For any set of multiplicities m1 , . . . , mr with
m1 + · · · + mr = 4 − k00 and m01 , . . . , m0r0 with m01 + · · · + m0r0 = 8 − j00 , there
exists a polynomial f4 with real coefficients such that f4 (θ1 ) has real roots away
from (1 : 0) with multiplicities m1 , . . . , mr , and f4 (θ2 ) has real roots away from
(1 : 0) with multiplicities m01 , . . . , m0r0 . Furthermore, for this f4 we have k0 = k00
and j0 = j00 . Proposition 5.6 will give the singularities that occur in addition to
O.
This completes the classification of the singularities on a quartic monoid
(other than O) when the tangent cone is a conic plus a tangent.
Case 5. The tangent cone is three general lines, and we assume f3 = x1 x2 x3 .
For each f4 we associate six integers,
k2 := I(1,0,0) (f4 , x2 ),
l1 := I(0,1,0) (f4 , x1 ),
m1 := I(0,0,1) (f4 , x1 ),
k3 := I(1,0,0) (f4 , x3 ),
l3 := I(0,1,0) (f4 , x3 ),
m2 := I(0,0,1) (f4 , x2 ).
Now k2 > 0 ⇔ k3 > 0, l1 > 0 ⇔ l3 > 0, and m1 > 0 ⇔ m2 > 0. If both k2
and k3 are greater than 1, then the monoid has a singular line, a case we have
excluded. The same goes for the pairs (l1 , l3 ) and (m1 , m2 ).
When the monoid does not have a singular line, we define jk = max(k2 , k3 ),
jl = max(l1 , l3 ) and jm = max(m1 , m2 ). If jk ≤ jl ≤ jm , then [38] gives that O
is a T4+jk ,4+jl ,4+jm singularity.
The three lines Z(x1 ), Z(x2 ) and Z(x3 ) are parameterized by θ1 , θ2 and θ3
where θ1 (s, t) = (0, s, t), θ2 (s, t) = (s, 0, t) and θ3 (s, t) = (s, t, 0). Roots of the
polynomial f4 (θi ) away from (1 : 0) and (0 : 1) correspond to intersections
between Z(f4 ) and Z(xi ) away from the singular points of Z(f3 ).
As before, we are only interested in the cases where none of f4 (θi ) ≡ 0 for
i = 1, 2, 3, as this would make the monoid reducible.
For the study of other singularities on the monoid we consider nonzero polynomials
q1
= b0 s4 + b1 s3 t + b2 s2 t2 + b3 st3 + b4 t4 ,
q2
= c0 s4 + c1 s3 t + c2 s2 t2 + c3 st3 + c4 t4 ,
q3
= d0 s4 + d1 s3 t + d2 s2 t2 + d3 st3 + d4 t4 .
Linear algebra shows that (λ1 q1 , λ2 q2 , λ3 q3 ) = (f4 (θ1 ), f4 (θ2 ), f4 (θ3 )) for some
f4 if and only if λ1 b0 = λ3 d4 , λ1 b4 = λ2 c4 , and λ2 c0 = λ3 d0 . A simple analysis
66
CHAPTER 5. MONOID HYPERSURFACES
shows the following: There exist λ1 , λ2 , λ3 6= 0 such that
(λ1 q1 , λ2 q2 , λ3 q3 ) = (f4 (θ1 ), f4 (θ2 ), f4 (θ3 ))
for some f4 , and such that Z(f4 ) and Z(f3 ) have no common singular point if
and only if all of the following hold:
• b0 = 0 ↔ d4 = 0 and b0 = d4 = 0 → (b1 6= 0 or d3 6= 0),
• b4 = 0 ↔ c4 = 0 and b4 = c4 = 0 → (b3 6= 0 or c3 6= 0),
• c0 = 4 ↔ d0 = 0 and c0 = d0 = 0 → (c1 6= 0 or d1 6= 0),
• b0 c4 d0 = b4 c0 d4 .
Similarly to the previous cases we can classify the possible configurations of
other singularities by varying the multiplicities of the roots of the polynomials
q1 , q2 and q3 . Only the multiplicities of the roots (0 : 1) and (1 : 0) affect
the first three bullet points above. Then, for any set of multiplicities of the
rest of the roots, we can find q1 , q2 and q3 such that the last bullet point is
satisfied. This completes the classification when Z(f3 ) is the product of three
general lines.
Case 6. The tangent cone is three lines meeting in a point, and we can
assume that f3 = x32 − x2 x23 . We write f3 = `1 `2 `3 where `1 = x2 , `2 = x2 − x3
and `3 = x2 + x3 , representing the three lines going through the singular point
(1 : 0 : 0). For each f4 we associate three integers j1 , j2 and j3 defined as the
intersection numbers ji = I(1:0:0) (f4 , `i ). We see that j1 = 0 ⇔ j2 = 0 ⇔ j3 = 0,
and that Z(f4 ) is singular at (1 : 0 : 0) if and only if two of the integers j1 ,
j2 , j3 are greater then one. (Then all of them will be greater than one.) The
singularity will be of the U series [1], [2].
The three lines Z(`1 ), Z(`2 ) and Z(`3 ) can be parameterized by θ1 , θ2 , and
θ3 where θ1 (s, t) = (s, 0, t), θ2 (s, t) = (s, t, t) and θ2 (s, t) = (s, t, −t).
For the study of other singularities on the monoid we consider nonzero polynomials
q1
= b0 s4 + b1 s3 t + b2 s2 t2 + b3 st3 + b4 t4 ,
q2
= c0 s4 + c1 s3 t + c2 s2 t2 + c3 st3 + c4 t4 ,
q3
= d0 s4 + d1 s3 t + d2 s2 t2 + d3 st3 + d4 t4 .
Linear algebra shows that (λ1 q1 , λ2 q2 , λ3 q3 ) = (f4 (θ1 ), f4 (θ2 ), f4 (θ3 )) for some
f4 if and only if λ1 b0 = λ2 c4 = λ3 d0 , and 2λ1 b1 = λ2 c1 + λ3 d1 . There exist
5.4. QUARTIC MONOID SURFACES
67
λ1 , λ2 , λ3 6= 0 such that (λ1 q1 , λ2 q2 , λ3 q3 ) = (f4 (θ1 ), f4 (θ2 ), f4 (θ3 )) for some f4
and such that Z(f4 ) and Z(f3 ) have no common singular point if and only if all
of the following hold:
• b0 = 0 ↔ c0 = 0 ↔ d0 = 0,
• if b0 = c0 = d0 = 0, then at least two of b1 , c1 , and d1 are different from
zero,
• 2b1 c0 d0 = b0 c1 d0 + b0 c0 d1 .
As in all the previous cases we can classify the possible configurations of
other singularities for all possible j1 , j2 , j3 . As before, the first bullet point
only affect the multiplicity of the factor t in q1 , q2 and q3 . For any set of
multiplicities for the rest of the roots, we can find q1 , q2 , q3 with real roots of
the given multiplicities such that the last bullet point is satisfied. This completes
the classification of the singularities (other than O) when Z(f3 ) is three lines
meeting in a point.
Case 7. The tangent cone is a double line plus a line, and we can assume
f3 = x2 x23 . The tangent cone is singular along the line Z(x3 ). The line Z(x2 ) is
parameterized by θ1 and the line Z(x3 ) is parameterized by θ2 where θ1 (s, t) =
(s, 0, t) and θ2 (s, t) = (s, t, 0). The monoid is reducible if and only if f4 (θ1 )
or f4 (θ2 ) is identically zero, so we assume that neither is identically zero. For
each f4 we associate two integers, j0 := I(1:0:0) (f4 , x2 ) and k0 := I(1:0:0) (f4 , x3 ).
Furthermore, we write f4 (θ2 ) as a product of linear factors
f4 (θ2 ) = λsk0
r
Y
(αi s − t)mi .
i=0
Now the singularity at O will be of the V series and depends on j0 , k0 and
m1 , . . . , m r .
Other singularities on the monoid correspond to intersections of Z(f4 ) and
the line Z(x2 ) away from (1 : 0 : 0). Each such intersection corresponds to a
root in the polynomial f4 (θ1 ) different from (1 : 0). Let j00 ≤ 4 and k00 ≤ 4 be
integers such that j0 > 0 ↔ k0 > 0. Then, for any homogeneous polynomials
q1 , q2 in s, t of degree 4 such that s is a factor of multiplicity j00 in q1 and of
multiplicity k00 in q2 , there is a polynomial f4 and nonzero constants λ1 and λ2
such that k0 = k00 , j0 = j00 and (λ1 q1 , λ2 q2 ) = (f4 (θ1 ), f4 (θ2 )). Furthermore,
if q1 and q2 have real coefficients, then f4 can be selected with real coeficients.
This follows from an analysis similar to case 5 and completes the classification
of singularities when the tangent cone is a product of a line and a double line.
68
CHAPTER 5. MONOID HYPERSURFACES
Case 8. The tangent cone is a triple line, and we assume that f3 = x33 .
The line Z(x3 ) is parameterized by θ where θ(s, t) = (s, t, 0). Assume that
the polynomial f4 (θ) has r distinct roots with multiplicities m1 , . . . , mr . (As
before f4 (θ) ≡ 0 if and only if the monoid is reducible.) Then the type of the
singularity at O will be of the V 0 series [3, p. 267]. The integers m1 , . . . , mr are
constant under right equivalence over C. Note that one can construct examples
of monoids that are right equivalent over C, but not over R (see Figure 5.4).
Figure 5.4: The monoids Z(z 3 + xy 3 + x3 y) and Z(z 3 + xy 3 − x3 y) are right
equivalent over C but not over R.
The tangent cone is singular everywhere, so there can be no other singularities on the monoid.
Case 9. The tangent cone is a smooth cubic curve, and we write f3 =
x31 + x32 + x33 + 3ax1 x2 x3 where a3 6= −1. This is a one-parameter family of
elliptic curves, so we cannot use the parameterization technique of the other
cases. The singularity at O will be a P8 singularity (cf. [3, p. 185]), and other
singularities correspond to intersections between Z(f3 ) and Z(f4 ), as described
by Proposition 5.6.
To classify the possible configurations of singularities on a monoid with a
nonsingular (projective) tangent cone, we need to answer
the following quesPr
tion: For any positive integers m1 , . . . , mr such that i=1 mi = 12, does there,
for some a ∈ R \ {−1}, exist a polynomial f4 with real coefficients such that
Z(f3 , f4 ) = {p1 , . . . , pr } ∈ P2 (R) and Ipi (f3 , f4 ) = mi for i = 1, . . . , r? Rohn
5.4. QUARTIC MONOID SURFACES
69
[32, p. 63] says that one can always find curves Z(f3 ), Z(f4 ) with this property.
Here we shall show that for any a ∈ R \ {−1} we can find a suitable f4 .
In fact, in almost all cases f4 can be constructed as a product of linear and
quadratic terms in a simple way. The difficult cases are (m1 , m2 ) = (11, 1),
(m1 , m2 , m3 ) = (8, 3, 1), and (m1 , m2 ) = (5, 7). For example, the case where
(m1 , m2 , m3 ) = (3, 4, 5) can be constructed as follows: Let f4 = `1 `2 `23 where `1
and `2 define tangent lines at inflection points p1 and p3 of Z(f3 ). Let `3 define
a line that intersects Z(f3 ) once at p3 and twice at another point p2 . Note that
the points p1 , p2 and p3 can be found for any a ∈ R \ {−1}.
The case (m1 , m2 ) = (11, 1) is also possible for every a ∈ R \ {−1}. For any
point p on Z(f3 ) there exists an f4 such that Ip (f3 , f4 ) ≥ 11. For all except a
finite number of points, we have equality [25], so the case (m1 , m2 ) = (11, 1) is
possible for any a ∈ R \ {−1}. The case (m1 , m2 , m3 ) = (8, 3, 1) is similar, but
we need to let f4 be a product of the tangent at an inflection point with another
cubic.
The case (m1 , m2 ) = (5, 7) is harder. Let a = 0. Then we can construct a
conic C that intersects Z(f3 ) with multiplicity five in one point and multiplicity
one in an inflection point, and choosing Z(f4 ) as the union of C and twice the
tangent line through the inflection point will give the desired example. The same
can be done for a = −4/3. By using the computer algebra system Singular
[11] we can show that these constructions can be continuously extended to any
a ∈ R \ {−1}. This completes the classification of singularities on a monoid
when the tangent cone is smooth.
In the Cases 3, 5, and 6, not all real equations of a given type can be
transformed to the chosen forms by a real transformation.
In Case 3 the conic may not intersect the line in two real points, but rather
in two complex conjugate points. Then we can assume f3 = x3 (x1 x3 + x21 + x22 ),
and the singular points are (1 : ±i : 0). For any real f4 , we must have
I(1:i:0) (x1 x3 + x21 + x22 , f4 ) = I(1:−i:0) (x1 x3 + x21 + x22 , f4 )
and
I(1:i:0) (x3 , f4 ) = I(1:−i:0) (x3 , f4 ),
so only the cases where j0 = j1 and k0 = k1 are possible. Apart from that, no
other restrictions apply.
In Case 5, two of the lines can be complex conjugate, and we assume f3 =
x3 (x21 + x22 ). A configuration from the previous analysis is possible for real
coefficients of f4 if and only if m1 = m2 , k2 = l1 , and k3 = l3 . Furthermore,
only the singularities that correspond to the line Z(x3 ) will be real.
70
CHAPTER 5. MONOID HYPERSURFACES
In Case 6, two of the lines can be complex conjugate, and then we may
assume f3 = x32 + x33 . Now, if j3 denotes the intersection number of Z(f4 ) with
the real line Z(x2 + x3 ), precisely the cases where j1 = j2 are possible, and only
intersections with the line Z(x2 + x3 ) may contribute to real singularities.
This concludes the classification of real and complex singularities on real
monoids of degree 4.
Remark. In order to describe the various monoid singularities, Rohn [32]
computes the “class reduction” due to the presence of the singularity, in (almost)
all cases. (The class is the degree of the dual surface [30, p. 262].) The class
reduction is equal to the local intersection multiplicity of the surface with two
general polar surfaces. This intersection multiplicity is equal to the sum of the
Milnor number and the Milnor number of a general plane section through the
singular point [39, Cor. 1.5, p. 320]. It is not hard to see that a general plane
section has either a D4 (Cases 1–6, 9), D5 (Case 7), or E6 (Case 8) singularity.
Therefore one can retrieve the Milnor number of each monoid singularity from
Rohn’s work.
Chapter 6
The strata of quartic
monoids
In this chapter we will define the strata of a quartic monoid surfaces in P3 with
a fixed triple point, O = (1 : 0 : 0 : 0). Then we will calculate the dimension
of all strata and the number of components of each stratum where the tangent
cone is not of generic type. We use Definition 5.2 as the definition of a monoid
- monoids have only isolated singularities.
6.1
Definition of the strata
Let X be a quartic monoid surface (with only isolated singular points) where
O ∈ X is the point of multiplicity 3. Furthermore, let F be an equation of X,
and write F = x0 f3 + f4 where f3 , f4 ∈ k[x1 , x2 , x3 ], as before.
The plane curves Z(f3 ) and Z(f4 ) will be named as in Definition 5.2, Z(f3 )
is the tangent cone and Z(f4 ) is the intersection with infinity.
We will now define the set of invariants for a given monoid X = Z(F ). For
each quartic monoid surface X = Z(F ) in P3 there is an associated type j of the
tangent cone, as given by the list in Section 5.4 on page 56. Depending on the
type j, the tangent cone has a number of irreducible components with associated
multiplicities. Also, the tangent cone has a set of special points, denoted σX .
The number and types of these points depend only on the type j. For the types
where the tangent cone has only isolated singularities, σX is defined as the set
of singular points. For type 7, the set σX is defined as the intersection of the
71
72
CHAPTER 6. THE STRATA OF QUARTIC MONOIDS
two irreducible components of the tangent cone. Furthermore, for type 8 we set
σX = ∅, the triple line is considered to have no special points.
In addition to the type j we associate to X one invariant for every pair (p, C)
where p ∈ σX is a special point and C = Z(h) is an irreducible component of
Z(f3 ) containing p. This invariant is defined as ip,C := Ip (h, f4 ). Note that the
invariants correspond to integers in Table 5.1.
Furthermore, for each component C = Z(h) of the tangent cone, the monoid
X has an associated partition m1 , . . . , mr of
X
4 deg(h) −
Ip (f4 , h)
(6.1)
p∈σX
determined by the intersection multiplicities between C and Z(f4 ) away from
σX . For all partitions we assume m1 ≥ m2 ≥ · · · ≥ mr ≥ 1, and we allow r = 0
(then m1 , . . . , mr is empty and a partition of 0). Thus every partition has a
unique representation m1 , . . . , mr .
The stratum of X is defined as the set of quartic monoid hypersurfaces that
have the same type, the same invariants ip,C and the same partition(s) as X.
To be precise, X and Y are in the same stratum if and only if the tangent cones
are of the same type and there is a bijection Φ that associates components of
X to components of Y of the same type and multiplicity and special points of
X to special points of Y , such that
• the point p ∈ σX is on a component C ⊆ X if and only if the point Φ(p)
is on the component Φ(C)
• ip,C = iΦ(p),Φ(C) for all points p ∈ σX and all components C ⊆ X
• for each component C ⊆ X the associated partition of C is equal to the
associated partition of Φ(C)
The set of monoid quartic surfaces is a subset of the set of quartic surfaces,
usually identified with the coefficient space P34 . It is not hard to see that the
dimension of the space of monoids where the triple point is fixed at O is 24, and
the space of monoids where the triple point can vary is of dimension 27.
We define the space S as the space of quartic monoids with a triple point at
O and only isolated singularities. The set S is viewed as a subset of the space
(A10 \ {0}) × (A15 \ {0})/ ∼ ⊂ P24 , where the set A10 \ {0} corresponds to the
coefficients of f3 , the set A15 \ {0} corresponds to the coefficients of f4 and the
equivalence relation ∼ is defined in the usual way – F is equivalent to F 0 if and
6.2. TYPES 1 TO 8
73
only if F = λF 0 for some λ 6= 0. We will write [F ] for the monoid in S defined
by the equation F = x0 f3 + f4 throughout this chapter.
Furthermore, S is an open subset of P24 . Indeed, the complement of S is the
union of the following closed sets:
• the set of monoids Z(F ) where Z(fi ) for i = 3, 4 has a common singular
point. (Let X ⊂ P2 ×P24 be the set of pairs (x̄, F ) such that x̄ is a singular
point of both Z(f3 ) and Z(f4 ). This set is defined by equations on the form
∂fi
∂xj for i = 3, 4 and j = 1, 2, 3, and is thus closed. The projection to the
second factor is also closed.)
• the set corresponding to equations F where f3 and f4 has a common linear
factor (equal to the image of a closed set by a projective morphism, as
above)
• the set corresponding to equations F where f3 and f4 has a common
quadratic factor (equal to the image of a closed set by a projective morphism, as above)
• the set corresponding to equations F where f4 = `f3 where ` is a linear
form or zero
• the set corresponding to f3 ≡ 0
6.2
Types 1 to 8
For the types 1 to 8, each stratum S can be characterized as the image of a
certain morphism γS . In this section the morphisms γS will be constructed
and used to calculate the dimension and the number of components of S for
each stratum. Before doing a systematic analysis of the strata, we will describe
the general strategy and carry out some preliminary calculations. The purpose
of these calculations is to obtain a formula for the dimension of each S. This
formula will be correct for a stratum S if some properties of γS are proved.
For every S we will define γS as the composition γS = ψ ◦ (ϕS × id):
(ϕS ×id)
ψ
γS : BS × G −−−−−→ S × G −−−−→ S
Each BS will be defined to be equal to, or a hypersurface in, a product of
one or more of the following:
S
• Ar∆ := Ar \ i6=j Z(ui − uj )
74
CHAPTER 6. THE STRATA OF QUARTIC MONOIDS
S
S
• Ar∆,0 := Ar \ i6=j Z(ui − uj ) \ i Z(ui )
S
• Ar0 := Ar \ i Z(ui )
• k r := Ar
where u1 , . . . , ur are the coordinate functions on Ar . Thus BS will be a locally
closed, not necessarily irreducible variety in some affine space. Furthermore,
since BS can be defined by not more than one equation, we see the that BS is
of pure dimension.
We define G to be the group of projective transformations of P3 fixing O,
and let ψ be the action of G on S. We consider G as a subgroup of PGL(4) and
see that G is the set of elements in PGL(4) that can be written as a matrix on
the form


1 a1,1 a1,2 a1,3
0 a2,1 a2,2 a2,3 


0 a3,1 a3,2 a3,3  .
0 a4,1 a4,2 a4,3
Since such a matrix representative is unique as an element of PGL(4) we see
that G is of dimension 12.
One possible way of constructing the variety BS and the map γS is to let BS
parameterize, through ϕS , the monoids in S up to projective equivalence, and
let G correspond to projective equivalence. The construction of BS and ϕS will
indeed be such that all monoids of a stratum S will be projectively equivalent
to a monoid in ϕS (BS ), but it will typically not be true that all the monoids in
ϕS (BS ) will be projectively different. The construction of BS and ϕS will be
explained more thoroughly later. With this in mind, we will state some basic
properties that will allow us to write down a formula for the dimension of S.
The morphisms ϕS will be defined such that BS and ϕS (BS ) have the same
number of components. Furthermore, for types 1 to 7 the fibers of ϕS will all
be finite, so the dimension of ϕS (BS ) will be equal to the dimension of BS . For
strata of type 8 the dimension of the fibers of ϕS will vary.
(i)
Consider the case that BS is reducible and has c components. Let BS for
i = 1, . . . , c denote the components of BS . These are all hypersurfaces, one
for each factor of the equation defining BS , and the intersection of two such
hypersurfaces will always be empty. This will be easy to see from the equations
(i)
(i)
of the components. We define γS as the restriction of γS to BS for i = 1, . . . , c.
The dimension of S will then be the maximum of the dimension of the images
(i)
of γS .
6.2. TYPES 1 TO 8
75
(i)
(i)
In fact, each image γS (BS ) will be pure dimensional of the same dimension. To see this, it is sufficient (and slightly stronger) to check that the fiber
γS−1 (γS (b̃, g̃)) is pure dimensional and of the same dimension for any b̃ ∈ BS
(i)
(i)
(i)
(i)
and g ∈ G: If b̃ ∈ BS , then (γS )−1 (γS (b̃, g̃)) = γS −1 (γS (b̃, g̃)) ∩ BS . Since
the components of BS are disjoint, each of the components of γS−1 (γS (b̃, g̃))
(i)
will either not intersect, or be contained in, BS . Now, if γS−1 (γS (b̃, g̃)) is pure
(i)
(i)
dimensional, it follows that the fiber (γS )−1 (γS (b̃, g̃)) is pure dimensional and
of the same dimension.
(i)
For any irreducible component BS ×G ⊂ BS ×G we know that the dimension
of the image is equal to the dimension of BS × G minus the dimension of the
(i)
fibers of γS . From this we can give a formula for the dimension of the images
(i)
of γS and therefore also for S:
(i)
(i)
dim γS (BS ) = dim S = dim BS + 12 − dim γS−1 γS (b̃, g̃)
(6.2)
where b̃ ∈ BS and g̃ ∈ G.
(i)
Note that γS may identify some of the components BS . So, to find the
(j)
(i)
number of components of S, we need to check when γS (BS ) = γS (BS ) for
i 6= j.
When studying the map γS to find the dimension and components of S, we
will need to calculate the fiber γS−1 (γS (b̃, g̃)) for fixed (b̃, g̃) ∈ BS × G. This is
equivalent to solving the equation
γS (b, g) = γS (b̃, g̃)
(6.3)
for fixed (b̃, g̃) ∈ BS × G.
We will now describe the strategy for constructing the sets BS and morphisms ϕS (BS ), and then carry out some of the calculations needed to solve
(6.3).
In the classification of the quartic monoids we fix the tangent cone for each
type j = 1, . . . , 8. This can be done up to projective transformation. Thus,
we can, and will, for each type j = 1, . . . , 8, let the tangent cone of every
monoid in the image ϕS (BS ) be the same, and be described by one polynomial
(j)
f3 . This polynomial will be equal to f3 in the corresponding case in Chapter 5.
Furthermore, for each type j we will select one monomial xe11 xe22 xe33 with nonzero
(j)
coefficient in f3 and, for each stratum S of type j, define ϕS such that the image
ϕS (BS ) contains only monoids that can be described by a polynomial F whose
76
CHAPTER 6. THE STRATA OF QUARTIC MONOIDS
coefficients of xe11 +1 xe22 xe33 , xe11 x2e2 +1 x3e3 and xe11 xe22 xe33 +1 are all zero. Note that
these choices can be made, as any monoid is projectively equivalent to a monoid
[F ] where the coefficients of x1e1 +1 xe22 xe33 , xe11 x2e2 +1 xe33 and xe11 xe22 xe33 +1 are all
zero. Indeed, assuming that the coefficient of xe11 xe22 xe33 in f3 is 1, the monoid
x0 f3 + f˜4 where the coefficients of xe11 +1 xe22 xe33 , xe11 x2e2 +1 xe33 and xe11 xe22 x3e3 +1
in f˜4 are c1 , c2 and c3 , respectively, is projectively equivalent to the monoid
(x0 − c1 x1 − c2 x2 − c3 x3 )f3 + f˜4 having the special coefficients equal to zero.
For types 1 to 7 we also define ϕS such that if the (projectively equivalent)
monoids defined by x0 f3 + f4 and x0 f3 + λf4 are both in the image ϕS (BS ),
then λ = 1 (so they are the same). For type 8 we define ϕS such that [x0 f3 + f4 ]
is in the image of ϕS if and only if [x0 f3 + λf4 ] is in the image for all λ ∈ k ∗ .
We will now explain the idea of the construction of BS and ϕS . Assume that
(j)
the tangent cone Z(f3 ) decomposes into r ≥ 1 components C1 , . . . , Cr and let
θi for n = 1, . . . , r be as in the case-by-case proof of Theorem 5.9. Recall that
θn , viewed as a map from P1 to P2 , parameterizes Cn .
A stratum S of type j is defined as the set of monoids having a specific set of
invariants ip,Cn (equal to intersection numbers) and partitions associated to the
components C1 , . . . , Cr . Each invariant ip,Cn corresponds to to the multiplicity
of a specific root (depending on p) of f4 (θn (s, t)). The partition associated to
Cn corresponds to other roots of f4 (θn (s, t)).
(j)
(j)
If two monoids [x0 f3 + f4 ] and [x0 f3 + f˜4 ] are such that the polynomials
f4 (θn (s, t)) = f˜4 (θn (s, t)) for each n, then f4 − f˜4 is a multiple of g (j) , where
(j)
g (j) = f3 for j = 1, . . . , 6, g (7) = x2 x3 and g (8) = x3 .
The classification in Chapter 5 (Theorem 5.9) tells us which set of polynomials {qn } are on the form {f4 (θi (s, t))}. This enables us to define BS and ϕS
such that the image of γS is exactly S.
(j)
When f3 = g (j) the maps ϕS assign, to an element b ∈ BS , a unique monoid
(j)
[x0 f3 + f4 ] satisfying the properties above (specific coefficients equal to zero)
such that f4 (θn ) = qn (b) for each component Cn . The set BS and morphism
ϕS is constructed such that {qi (b)}b∈BS runs through all possible polynomials
on the form {f4 (θi (s, t))} for the given stratum S, up to multiplication by a
nonzero constant and up to symmetry in the variables x1 , x2 and x3 . The role
of the symmetries will be clear later, in the cases where {qi (b)}b∈BS does not
run through all possible polynomials on the form {f4 (θi (s, t))}.
(j)
(j)
(j)
Still assuming f3 = g (j) , any two monoids [x0 f3 + f4 ] and [x0 f3 + f˜4 ]
˜
such that f4 (θi (s, t)) = λf4 (θi (s, t)) for i = 1, . . . , r are projectively equivalent:
(j)
f4 − λf˜4 is a multiple of g (j) = f3 . Thus the image of γS is equal to S.
6.2. TYPES 1 TO 8
77
(j)
When f3 6= g (j) we use a similar construction, where we can easily check
that every monoid with any possible set of {f4 (θn (s, t))} is projectively equivalent to a monoid in ϕS (BS ).
As we will se in the systematic analysis of the strata, some of coordinates
in BS will correspond to roots of the polynomials {qn (b)}. The conditions on
{qn (b)} to be on the form {f4 (θn (s, t))} are linear conditions, but the pullback
of these will typically not be linear conditions on BS . Still, the sets BS can and
will be constructed to be (at most) hypersurfaces, and the equations of these
can be easily factorized, giving the components of BS .
For each type j, define Mj as the group of invertible 3 × 3 matrices M
such that f3 (x̄) = f3 (M x̄). Here we let x̄ be the column vector (x1 , x2 , x3 )T
and abuse the notation of f3 a little. Note that Mj can be considered as a
not necessarily irreducible locally closed variety in A9 , where f3 (x̄) = f3 (M x̄)
represents the closed condition, while det M 6= 0 represents the open condition
on Mj .
Now, (b, g) ∈ γS−1 (γS (b̃, g̃)) is equivalent to γS (b, g) = γS (b̃, g̃), which is
equivalent to ϕS (b) = ψ(ϕ(b̃), g −1 g̃). We write ϕ(b) = [x0 f3 (x̄) + f4 (x̄)] ∈ S
and ϕ(b̃) = [x0 f3 (x̄) + f˜4 (x̄)] ∈ S for the rest of this section.
Define h = g −1 g̃ and write h on the form


1 a1,1 a1,2 a1,3

0


(6.4)

0
A
0
By definition we have
ψ(ϕ(b̃), h) = [(x0 + a1,1 x1 + a1,2 x2 + a1,3 x3 )f3 (Ax̄) + f˜4 (Ax̄)]
= [x0 f3 (Ax̄) + (a1,1 x1 + a1,2 x2 + a1,3 x3 )f3 (Ax̄) + f˜4 (Ax̄)].
If ϕS (b̃) = ψ(ϕ(b̃), h), then it is clear that A = λM for some λ ∈ k ∗ and
some M ∈ M. We write ` = a1,1 x1 + a1,2 x2 + a1,3 x3 and see that
ψ(ϕ(b̃), h) = [x0 f3 (λM x̄) + `f3 (λM x̄) + f˜4 (λM x̄)]
= [λ3 x0 f3 (M x̄) + λ3 `f3 (M x̄) + λ4 f˜4 (M x̄)]
= [x0 f3 (M x̄) + `f3 (M x̄) + λf˜4 (M x̄)]
= [x0 f3 (x) + `f3 (M x̄) + λf˜4 (M x̄)].
78
CHAPTER 6. THE STRATA OF QUARTIC MONOIDS
Now assume that ψ(ϕ(b̃), h) = ϕ(b) for some b ∈ BS . From the condition
that the coefficients of xe11 +1 xe22 xe33 , xe11 x2e2 +1 xe33 and xe11 xe22 xe33 +1 in ϕ(b̃) are
all zero we see that ` is uniquely determined by λ and M , and can be the
written on the form λ`1 (M ) where `1 (M ) only depends on f˜4 and M . This gives
ψ(ϕ(b̃), h) = [x0 f3 (x) + λ(`1 (M )f3 (M x̄) + f˜4 (M x̄))]. Uniqueness of elements
on the form [x0 f3 + λf4 ] for strata of type 1 to 7 implies that only one λ is
possible for each M for types 1 to 7.
We define MS,b̃,g̃ as the set of matrices M ∈ M such that there exists
(b, g) ∈ BS × G such that γS (b̃, g̃) = γS (b, g) and h := g −1 g can be written on
the form (6.4) where A = λM for some λ ∈ k ∗ . This set is clearly independent
of g̃, so we define MS,b̃ := MS,b̃,g̃ for any g̃ ∈ G. A priori, the set MS,b̃
is not necessarily a subgroup of Mj . However, we will check that for any
stratum S and any b̃ ∈ BS , MS,b̃ is both a subgroup and a subvariety of Mj
of codimension zero.
(j)
Note that if M ∈ Mj and ηM ∈ Mj for some η ∈ k, then f3 (x) =
(j)
(j)
(j)
f3 (ηM x) = η 3 f3 (M x) = η 3 f3 (x), so η 3 = 1. Furthermore, we see that
M ∈ Mj,b̃ if and only if ω3 M ∈ Mj,b̃ , where ω3 is a primitive third root of
unity.
This means that when S is a stratum of type 1 to 7, then for each (b, g) ∈
γS−1 (γ(b̃, g̃)) there are exactly 3 corresponding elements in MS,b̃ . If M is one
such element, then ω3 M and ω32 M are the other two. It follows that the dimension of the fiber γS−1 (γS (b̃, g̃)) is equal to the dimension of MS,b̃ . This is again
equal to the dimension of Mj , so
dim γS−1 (γS (b̃, g̃)) = dim Mj
for every stratum S of type 1, . . . , 7.
Given this dimension formula we can write down a formula for the dimension
of S (of type 1 to 7):
dim S = dim(BS ) + dim(G) − dim γS−1 γ = dim BS + 12 − dim Mj
(6.5)
We will now consider the strata of S. For each type j we will calculate Mj
by using the computer algebra system Singular [11]. (In fact, we will just list
the elements of Mj or it generators, but the calculations are carried out by a
simple Singular script.) Then, for each stratum S of type j, we will define BS
and ϕS , and calculate the components of BS . We will calculate the set MS,b̃
6.2. TYPES 1 TO 8
79
for each b̃ ∈ BS (and check that MS,b̃ is in fact both a subgroup and subvariety
of Mj of codimension zero) and check if γS identifies any of the components of
BS . This will give the number of components of S, and we can safely calculate
the dimension of S using (6.5).
Primitive roots of unity will be used and we will write ωi for a primitive ith
root of unity. This means that ωij = 1 if and only if j is a multiple of i.
(j)
In cases j = 1, . . . , 6 the polynomial f3 is square free, and we have parame(j)
terizations (s, t) → θi (s, t) for i = 1, . . . , rj of the components of Z(f3 ). Then,
(j)
as we have seen in the classification of quartic monoids, f˜4 = f4 + `f3 for some
linear form ` if and only if f˜4 (θi (s, t)) = f4 (θi (s, t)) for all i = 1, . . . , rj . This
will be used throughout the case-by-case analysis of the strata.
Before studying the different strata, we summarize the results in the following theorem/table:
Theorem 6.1. In the space S of quartic monoid hypersurfaces with a triple
point at O = (1 : 0 : 0 : 0) and only isolated singularities, each stratum with
type 1 to 8 has dimension and number of components as given by Table 6.1.
(1)
Case 1. The tangent cone is a nodal cubic, and we set f3 := f3 = x1 x2 x3 +
x32 + x33 . The group M1 is isomorphic to the group A3 × Z3 of 18 elements,
where A3 denotes the permutation group of 3 elements. M1 is generated by the
following three matrices:






1 0 0
1 0
0
ω3 0
0
M 1 = 0 0 1  , M 2 = 0 ω 3 0  , M 3 =  0 ω 3 0  .
0 1 0
0 0 ω32
0
0 ω3
For the study of the different strata we have to distinguish between m = 0
and m > 0 where m is defined as in Chapter 5. Recall that we define θ(s, t) =
(−s3 − t3 , s2 t, st2 ) in this case.
Subcase 1a. Assume m = 0 and let m1 , . . . , mr be a partition of 12. This
m1
mr
defines a stratum S. We define BS ⊂ A12
− 1,
∆,0 by the equation u1 · · · ur
12
where we use u1 , . . . , ur as the coordinate functions on A .
For a point
Qrb = (b1 , . . . , br ) ∈ BS there is a unique polynomial f4 such that
f4 (θ(s, t)) = i=1 (s + bi t)mi and the coefficients of x1 x33 , x2 x33 and x43 in f4 are
all zero. The map ϕS is then defined as ϕS (b) = x0 f3 + f4 . The map ϕS is
clearly a morphism by the analysis in Chapter 5, and the image of γS is S ⊂ S.
Let c := gcd(m1 , . . . , mr ). Now BS is reducible if and only if c > 1, and then
m /c
m /c
BS has c components, each given by u1 1 · · · ur r − ωci . This follows from the
80
CHAPTER 6. THE STRATA OF QUARTIC MONOIDS
Type
1
2
3
4
5
6
7
8
Invariants
m = 0, m1 + · · · + mr = 12
2e1 3e2 := gcd(m1 , . . . , mr )
m = 2, . . . , 12, m1 + · · · + mr = 12 − m
m = 0, m1 + · · · + mr = 12
m = 2, 3, m1 + · · · + mr = 12 − m
j0 = j1 = k0 = k1 = 0
m1 + · · · + mr = 4, m01 + · · · + m0r0 = 8
2e := gcd(m1 , . . . , mr , m01 , . . . , m0r0 )
j0 = k0 = 0, j1 , k1 > 0
m1 + · · · + mr = 4 − k1
m01 + · · · + m0r0 = 8 − j1
j0 , k0 , j1 , k1 > 0
m1 + · · · + mr = 4 − k0 − k1
m01 + · · · + m0r0 = 8 − j0 − j1
j0 = k0 = 0, m1 + · · · + mr = 4
m01 + · · · + m0r0 = 8
j0 = k0 = 1, m1 + · · · + mr = 4 − k0
m01 + · · · + m0r0 = 8 − j0
j0 , k0 ≥ 2, m1 + · · · + mr = 4 − k0
m01 + · · · + m0r0 = 8 − j0
k1,2 = k2,1 = k1,3 = k3,1 = k2,3 = k3,2 = 0
m1 + · · · + mr = m01 + · · · + m0r0 = 4
00
m00
1 + · · · + mr 00 = 4
e
0
2 := gcd(m1,...,r , m1,...,r0 , m00
)
1,...,r 00
k1,2 , k1,3 > 0, k2,1 = k3,1 = k2,3 = k3,2 = 0
m1 + · · · + mr = 4
m01 + · · · + m0r0 = 4 − k1,2
00
m00
1 + · · · + mr 00 = 4 − k1,3
k1,2 , k1,3 , k2,1 , k2,3 > 0, k3,1 = k3,2 = 0
m1 + · · · + mr = 4 − k2,1
m01 + · · · + m0r0 = 4 − k1,2
00
m00
1 + · · · + mr 00 = 4 − k1,3 − k2,3
k1,2 , k1,3 , k2,1 , k2,3 , k3,1 , k3,2 > 0
m1 + · · · + mr = 4 − k2,1 − k3,1
m01 + · · · + m0r0 = 4 − k1,2 − k3,2
00
m00
1 + · · · + mr 00 = 4 − k1,3 − k2,3
j1 = j2 = j3 = 0, m1 + · · · + mr = 4
00
m01 + · · · + m0r0 = m00
1 + · · · + mr 00 = 4
j1 = j2 = j3 = 1, m1 + · · · + mr = 3
00
m01 + · · · + m0r0 = m00
1 + · · · + mr 00 = 3
j1 ≥ 2, j2 = j3 = 1,
m1 + · · · + mr = 4 − j1
00
m01 + · · · + m0r0 = m00
1 + · · · + mr 00 = 3
j0 = k0 = 0
m1 + · · · + mr = m01 + · · · + m0r0 = 4
j0 , k0 > 0, m1 + · · · + mr = 4 − j0
m01 + · · · + m0r0 = 4 − k0
m1 + · · · + mr = 4
dim S
11 + r
Components
1 + e1
12 + r
10 + r
11 + r
r + r0 + 10
1
1
1
1+e
r + r0 + 11
1
r + r0 + 12
1
r + r0 + 9
1
r + r0 + 10
1
r + r0 + 11
1
r + r0 + r00 + 9
1+e
r + r0 + r00 + 10
1
r + r0 + r00 + 11
1
r + r0 + r00 + 12
1
r + r0 + r00 + 8
1
r + r0 + r00 + 9
1
r + r0 + r00 + 10
1
r + r0 + 11
1
r + r0 + 12
1
r + 13
1
Table 6.1: The strata of S of type 1 to 8
6.2. TYPES 1 TO 8
81
m /c
m /c
fact that the polynomial u1 1 · · · ur r − ωci has no monomial factors and a
Newton polytope which has only two lattice points.
We will now study the group MS,b̃ through the equation γS (b, g) = γS (b̃, g̃).
Write ` = a1,1 x1 +a1,2 x2 +a1,3 x3 and h = g −1 g̃ on the form (6.4) where A = λM .
Then the equation γS (b, g) = γS (b̃, g̃) is equivalent to f4 (x̄) = `f3 + λf˜4 (M x̄).
Now f4 (x̄) = `f3 + λf˜4 (M x̄) for some ` if and only if
f4 (θ(s, t)) = λf˜4 (M θ(s, t))
where θ(s, t) is considered
a column vector. The degree 12 polynomial f4 (θ(s, t))
Qr
is by definition i=1 (s + bi t)mi , while λf˜4 (M θ(s, t)) needs to be considered
further.
Note that Mi θ(s, t) for i = 1, 2, 3 are all parameterizations of Z(f3 ). By
analyzing these parameterizations we can find which components of ϕS (BS ) are
identified by ψ and also calculate the group Ms,b̃ . First note that M1 θ(s, t) =
θ(t, s), M2 (θ(s, t)) = θ(s, ω3 t) and M3 (θ(s, t)) = ω3 θ(s, t). Now we have
f˜4 (M1 θ(s, t)) = f˜4 (θ(t, s)) =
r
Y
i=1
(t + b̃i s)mi =
r
Y
i=1
(s +
1 mi
t) .
b̃i
Qr
mi
Here we have used i=1 (b̃i ) = 1. From this we see that if M = M1 then
λ = 1, and γS (b, g) = γS (b̃, g̃) for some g whenever b−1
= b̃i for all i = 1, . . . , r.
i
Qr
˜
Similarly, we have f4 (M2 θ(s, t)) = i=1 (s + (ω3 b̃i )t)mi , so M = M2 implies
λ = 1, and γS (b, g) = γS (b̃,Q
g̃) for some g whenever bi = ω3 b̃i for all i = 1, . . . , r.
r
Finally, f˜4 (M3 θ(s, t)) = ω3 i=1 (s + b̃i t)mi , so M = M3 implies λ = ω32 , giving
2
A = ω3 M3 = I, the identity matrix, as expected. In summary, we see that
MS,b̃ = Mj for any b̃ ∈ BS .
To count the number of components in S we need to check five cases, namely
c = 2, 3, 4, 6, 12.
Qr
m /2
• If c = 2, then ϕS (BS ) has two components, defined by i=1 bi i = ±1.
1
1
Then the elements b̃ = (b̃1 , . . . , b̃r ), ( b̃ , . . . , b̃ ) and (ω3 b̃1 , . . . , ω3 b̃r ) all
1
r
belong to the same component, so S has two components.
Qr
m /3
• If c = 3, then ϕS (BS ) has three components, defined by i=1 bi i = ±ω3
Qr
mi /3
= 1. These will all be identified by ψ (where A = M2 ), so
and i=1 bi
S has one component.
82
CHAPTER 6. THE STRATA OF QUARTIC MONOIDS
Qr
m /4
• If c = 4, then ϕS (BS ) has four components, defined by i=1 bi i = ±1
Qr
Qr
mi /4
mi /4
and i=1 bi
= ±ω4 . The two components where i=1 bi
= ±ω4
will be identified by ψ (where A = M1 ), so S has three components.
Qr
m /6
• If c = 6, then ϕS (BS ) has six components, defined by i=1 bi i = ω6i
for i = 1, . . . , 6. Here ψ identifies the components where i = 1, 3, 5 and
the components where i = 2, 4, 6, so S has two components.
i
• If c = 12, then r = 1 and ϕS (BS ) has 12 components, defined by b1 = ω12
for i = 1, . . . , 12. Here ψ identifies the components where i = 2, 6, 10, the
components where i = 4, 8, 12 and the components where i is odd, so S
has three components.
To sum up we see that if c = 2e1 3e2 , then S has 1+e1 components. Furthermore,
the dimension of each component is the dimension of BS plus 12, or dim S =
12 + r − 1 = 11 + r.
Subcase 1b. Assume m ∈ {2, . . . , 12} and let m1 , . . . , mr be a partition of
12 − m. This defines a stratum S, and we define BS := Ar∆,0 .
For any point b ∈ BS there is a unique polynomial f4 such that
f4 (θ(s, t)) = stm−1
r
Y
(s + bi t)
i=1
and the coefficients of x1 x33 , x2 x33 and x43 in f4 are all zero. The morphism ϕS
is then defined by ϕS (b) = x0 f3 + f4 .
Qr
Observe that f4 (M1 θ(s, t)) = sm−1 t i=1 (bi s + t). When m > 2 the set
ϕS (BS ) does not contain monoids with all possible values of {f4 (θ(s, t))} (up to
multiplication with a nonzero constant) for the given stratum, but the projective
transformation corresponding to M1 ensures that the image γS (BS ) contains
monoids with all possible values of {f4 (θ(s, t))}. Then we also know that γS (BS )
is the whole stratum S.
We can say that the element M1 ∈ M1 correspond to the symmetry switching x2 with x3 and thereby switching the last to coordinates of θ(s, t). This
fulfills the promise that ϕS (BS ) would contain monoids with all possible values of {f4 (θ(s, t))} up to a multiplication with a nonzero constant and up to
symmetry.
Furthermore, we see that if m = 1, then MS,b̃ = M1 for any b̃ ∈ BS . If
m > 1 then MS,b̃ is the subgroup of M1 generated by M2 and M3 .
Since BS is open in Ar we know that BS , and thus S, is irreducible. Furthermore, equation (6.5) gives dim S = r + 12.
6.2. TYPES 1 TO 8
83
(2)
Case 2. The tangent cone is a cuspidal cubic, and we set f3 := f3 = x31 −x22 x3 .
The group M2 is isomorphic to the group k ∗ × Z3 and is generated by matrices
on the form




1 0
0
ω3 0
0
0  and M2 =  0 ω3 0 
M1 (a) = 0 a
0 0 1/a2
0
0 ω3
where a ∈ k ∗ . We see that M2 is of dimension 1 and has three components.
As in the first case, we have to analyze the cases m = 0 and m > 0 separately.
Recall that θ(s, t) = (s2 t, s3 , t3 ).
Subcase 2a. Assume m = 0 and let m1 , . . . , mr be a partition of 12. This
defines a stratum S. Let
BS := {b = (b1 , . . . , br ) ∈ Ar∆ |
r
X
mi bi = 0}.
i=1
For any b ∈ BS there is a unique polynomial Q
f4 whose coefficients of x41 ,
r
3
and x1 x3 in f4 are all zero and f4 (θ(s, t)) = i=1 (bi s + t)mi . Define ϕS
by writing ϕS (b) = x0 f3 + f4 . Now, by the analysis in Chapter 5, γS is a
morphism and its image is S. We see that the closure of BS as a subset of Ar
is isomorphic to Ar−1 , so S must have exactly one component. Similarly to
the previous cases, M ∈ MS,b̃ if and only if there exist a b ∈ BS and λ ∈ k ∗
such that f4 (θ(s, t)) = λf˜4 (M θ(s, t)). Now Q
M1 (a)θ(s, t) = θ(αs, t/α2 ) where
r
m1
3
4
˜
, so b = b̃/a
α = a. This gives λf4 (M1 (a)θ(s, t)) = λa
i=1 (s + (bi /a)t)
−4
and λ = a is an example showing that M1 (a) ∈ MS,b̃ for any b̃ ∈ BS . It
follows that MS,b̃ = M2 for any b̃ ∈ BS .
This proves that (6.5) is valid, and dim S = r + 10.
x31 x2
Subcase 2b. Assume m ∈ {2, 3} and let m1 , . . . , mr be a partition of 12 − m.
This defines a stratum S. Let
BS := {(b1 , . . . , br ) ∈ Ar | bi 6= bj for all i 6= j}.
For any b ∈ BS there is a unique polynomialQ
f4 whose coefficients of x41 , x31 x2 and
r
3
m
mi
x1 x3 in f4 are all zero and f4 (θ(s, t)) = s
. Define ϕS (b) = f4 .
i=1 (bi s + t)
r
Since BS is open in A we know that the stratum S has only one component.
Now M ∈ MS,b̃ if and only there exist a b ∈ BS and λ ∈ k ∗ such that
Qr
f4 (θ(s, t)) = λf˜4 (M θ(s, t)). We see that λf˜4 (M1 (a)θ(s, t)) = λa4 sm i=1 (s +
84
CHAPTER 6. THE STRATA OF QUARTIC MONOIDS
(bi /a)t)m1 . As in Subcase 2a, b̃ = b/a and λ = a−4 is and example showing
that M1 (a) ∈ MS,b̃ , and 6.5 gives its dimension, dim S = r + 11.
(3)
Case 3. The tangent cone is a conic and a line, and we set f3 := f3 =
x3 (x1 x2 + x23 ). The group M3 has six one dimensional components and is
generated by






a
0
0
0 1 0
ω3 0
0
M1 (a) = 0 1/a 0 , M2 = 1 0 0 and M3 =  0 ω3 0 
0
0
1
0 0 1
0
0 ω3
where a ∈ k ∗ .
In this case we have to study three subcases separately. Recall that θ1 (s, t) =
(s, t, 0) and θ2 (s, t) = (s2 , −t2 , st).
Subcase 3a. Assume j0 = j1 = k0 = k1 = 0, m1 , . . . , mr is a partition of 4 and
m0i , . . . , m0r0 is a partition of 8. This defines a stratum S. Define
0
0
0
mr
1
BS := {(b, b0 ) ∈ Ar∆,0 × Ar∆,0 | bm
= (b01 )m1 · · · (b0r0 )mr0 }.
1 · · · br
For each b := (b, b0 ) ∈ BS there is a unique polynomial Q
f4 whose coefficients
r
of x1 x33 , x2 x33 and x43 are all zero, such that f4 (θ1 (s, t)) = i=1 (s + bi t)mi and
Qr 0
0
f4 (θ2 (s, t)) = i=1 (s + b0i t)mi , and we define ϕS by writing ϕS (b) = x0 f3 + f4 .
Let u1 , . . . , ur be the coordinate functions on Ar and let u01 , . . . , u0r0 be co0
ordinate functions on Ar and let
c = gcd(m1 , . . . , mr , m01 , . . . , m0r0 ).
Then BS has c components, each defined by
m1 /c
u1
0
0
r /c
· · · um
= ωci (u01 )m1 /c · · · (u0r0 )mr0 /c
r
for i = 1, . . . , c. Since ui 6= 0 in BS we see that the components are disjoint.
Now M ∈ MS,b̃ if and only if there exist an λ ∈ k ∗ such that f4 (θ1 (s, t)) =
λf˜4 (M θ1 (s, t)) and f4 (θ2 (s, t)) = λf˜4 (M θ2 (s, t)). Since M1 (a)θ1 (s, t) = θ1 (as, t/a)
and M1 (a)θ2 (s, t) = θ2 (αs, t/α) where α2 = a we get
r
r
Y
Y
b̃i
b̃i
t
(as + t)mi = a4
(s + 2 t)mi
f˜4 (M1 (a)θ1 (s, t)) = f˜4 (θ1 (as, )) =
a
a
a
i=1
i=1
0
0
r
r
Y
Y
0
0
t
b̃0
b̃0
f˜4 (M1 (a)θ2 (s, t)) = f˜4 (θ2 (αs, )) =
(αs + i t)mi = a4
(s + i t)mi .
α
α
a
i=1
i=1
6.2. TYPES 1 TO 8
85
We see that M1 (a) ∈ MS,b for any b̃. This follows from b̃ = a2 b and
b̃ = ab0 with λ = a−4 , and we see that b = (b, b0 ) and b̃ = (b̃, b̃0 ) are in the
same component of BS .
Using M2 θ1 (s, t) = θ1 (t, s) and M2 θ2 (s, t) = −θ2 (t, −s) we get
! r
r
Y
Y
1
mi
˜
λf4 (M2 θ1 (s, t)) = λ
(b̃i )
(s + t)mi
b̃i
i=1
i=1
 0
 0
r
r
Y
Y
0
0
1
λf˜4 (M2 θ2 (s, t)) = λ  (b̃0i )mi  (s − 0 t)mi .
b̃
i
i=1
i=1
0
Qr
Qr 0
0
If M = M2 , then λ = i=1 (b̃i )−mi = i=1 (b̃0i )−mi , and setting bi = 1/b̃i and
b0i = −1/b̃0i shows that and M2 ∈ MS,b̃ for any b̃ ∈ BS . Furthermore, if (b, b0 )
m1 /c
is in the component of BS given by u1
mr /c
· · · ur
m /c
m0 /c
m0 /c
r0
1
· · · ur+r
= ωci ur+1
0 , then
m /c
m0 /c
m0 /c
r0
1
· · · ur+r
(b̃, b̃0 ) is in the “inverse” component given by u1 1 · · · ur r = ωc−i ur+1
0 .
We can safely ignore M3 θ1 (s, t): The identity is in MS,b̃ , so M3 must also be
for any b̃ ∈ BS . It follows that MS,b̃ = M3 for all b̃ ∈ BS , so (6.5) gives
dim S = r + r0 + 10.
We can now count the number of components of S. We see that γS only identify components when c = 4, and then exactly two components are identified.
Write c = 2e . Then S has 1 + e components.
Subcase 3b. Now assume that k0 = j0 = 0, k1 , j1 > 0, m1 , . . . , mr is a
partition of 4 − k1 and m1 , . . . , m0r0 is a partition of 8 − j1 , defining a stratum
S. (If we switch k0 with k1 and j0 with j1 we get the same stratum.) Define
0
BS = Ar∆,0 × Ar∆,0 and ϕS (b, b0 ) = x0 f3 + f4 where f4 is the unique polynomial
whose coefficients of x1 x33 , x2 x33 and x43 are all zero such that f4 (θ1 (s, t)) =
Qr
Qr 0
j1
0 m0i
sk1 i=1 (s + bi t)m
. Since BS is open in
i and f4 (θ2 (s, t)) = s
i=1 (s + bi t)
0
Ar+r , S must have one component. Q
r
Observe that f˜4 (M2 θ1 (s, t)) = tk1 i=1 (bi s + t)m
i and there is no b ∈ BS
such that f4 (θ(s, t)) = λf˜4 (M2 θ1 (s, t)) for any λ ∈ k ∗ . We can check that MS,b̃
is the subgroup generated by M1 and M3 for all b̃ ∈ BS , so dim S = r + r0 + 11.
The fact that M2 6∈ MS,b̃ is connected with the fact that ϕS (BS ) does not
contain monoids for all possible values of {f4 (θn (s, t))} for the given stratum.
Indeed, monoids that corresponds to switching j0 with j1 and k0 with k1 have
no monoid in ϕS (BS ) with the same values of {f4 (θn (s, t))}. However, these are
all projectively equivalent to monoids in ϕS (BS ) by a projective transformation
86
CHAPTER 6. THE STRATA OF QUARTIC MONOIDS
on the form (6.4) where A = M2 . This is because M2 corresponds to switching
x1 and x2 , which also corresponds to switching j0 with j1 and k0 with k1 .
Subcase 3c. Now assume that k0 , j0 , k1 , j1 > 0, m1 , . . . , mr is a partition of
4 − k0 − k1 and m1 , . . . , m0r0 is a partition of 8 − j0 − j1 , defining a stratum
0
S. Let BS := Ar∆,0 × Ar∆,0 × A10 and define ϕS by writing ϕ(b, b0 , b00 ) = x0 f3 +
f4 where f4 is the unique polynomial whoseQcoefficients of x1 x33 , x2 x33 and x43
r
are all zero such that f4 (θ1 (s, t)) = sk1 tk0 i=1 (s + bi t)m
i and f4 (θ2 (s, t)) =
0
Q
0
r
0 mi
00 j1 j0
. We see that, since BS has one component, S has one
b s t
i=1 (s + bi t)
Qr
˜
component. Since f˜4 (M2 θ1 (s, t)) = sk0 tk1 i=1 (sbi + t)m
i and f4 (M2 θ2 (s, t)) =
0
Q
0
r
b00 sj0 tj1 i=1 (b0i s + t)mi we see that M2 ∈ MS,f˜4 if and only if j0 = j1 and
k0 = k1 . Note that switching j0 with j1 and k0 with k1 will give the same
stratum, so we have two different definitions for BS when j0 6= j1 or k0 6= k1 .
However, these are symmetric, and the symmetry corresponds to switching x1
with x2 , which again corresponds to M2 . Thus, any of the two definitions of BS
will do for the construction of the stratum S. With this in mind we can check
that MS,f˜4 = M3 if j0 = j1 and k0 = k1 (when BS is well defined), and that
MS,f˜4 is generated by M1 and M3 for the rest of the strata (when we have two
symmetric definitions of BS ).
In every case MS,f˜4 is a subgroup of M3 of codimension zero, so (6.5) gives
dim(S) = r + r0 + 12.
(4)
Case 4. The tangent cone is a conic plus a tangent line, and we set f3 = f3
x3 (x1 x3 + x22 ). The group M4 is the set of matrices on the form


a2
a41 a2 − 4a24
1

a2 
M (a1 , a2 ) =  0 a1 − 2a
3
1
1
0 0
a2
=
1
∗
where a1 ∈ k and a2 ∈ k. We see that M4 has dimension 2. We have
to distinguish between three cases, (a) j0 = k0 = 0, (b) j0 = k0 = 1 and
(c) j0 , k0 ≥ 2. For each case we let m1 , . . . , mr be a partition of 4 − k0 and
m01 , . . . , m0r0 be a partition of 8 − j0 , defining a stratum S. For (a) we define BS
Pr
Pr0
0
as the set of b = (b, b0 ) ∈ Ar∆ × Ar∆ such that i=1 mi bi = i=1 m0i b0i . For (b)
0
BS is defined as the set of b = (b, b0 ) ∈ Ar∆ ×Ar∆ , and for (c) BS is defined as the
0
set of b = (b, b0 , b00 ) ∈ Ar∆ ×Ar∆ ×A10 . For all of the cases we define ϕS by writing
ϕS (b) = [x0 f3 +f4 ] where f4 is the unique polynomial whose coefficients
of x21 x23 ,
Qr
2
3
k0
mi
x1 x2 x3 and x1 x3 in f4 are all zero such that f4 (θ1 (s, t)) = t
i=1 (s + bi t)
0
Q
0
r
and, in subcases (a) and (b) f4 (θ2 (s, t)) = tj0 i=1 (s + b0i t)mi and, in subcase
6.2. TYPES 1 TO 8
87
Qr 0
0
(c) f4 (θ2 (s, t) = b00 tj0 i=1 (s + br+i t)mi . For all strata S the set BS will be
open in some affine space, so S will have one component.
Recall that θ1 (s, t) = (s, t, 0) and θ2 (s, t) = (s2 , st, −t2 ). Now M (a1 , a2 ) ∈
MS,b̃ if and only if f4 (θ1 (s, t)) = λf˜4 (M (a1 , a2 )θ1 (s, t)) and f4 (θ2 (s, t)) =
λf˜4 (M (a1 , a2 )θ1 (s, t)) for some λ ∈ k ∗ . We have
M (a1 , a2 )θ1 (s, t) = θ(a41 s + a2 t, a1 t),
M (a1 , a2 )θ2 (s, t) = θ2 (a21 s +
a2 t t
, ),
2a21 a1
giving
f˜4 (M (a1 , a2 )θ1 (s, t)) = (a1 t)k0
r
Y
(a41 s + a2 t + b̃i a1 t)mi
i=1
=
0
tk0 a16−3k
1
r
Y
a2
b̃i
+ 3
4
a1
a1
s+
i=1
f˜4 (M (a1 , a2 )θ2 (s, t)) = b̃
00
t
a1
j0 Y
r0
a21 s
i=1
! !mi
t
a2 t
b̃r+i t
+ 2+
2a1
a1
!m0i
0
= b̃00 tj0 a1 16−3j0
r
Y
s+
i=1
b̃r+i
a2
+ 3
4
2a1
a1
! !m0i
t
where we set b00 = 1 for (a) and (b).
Pr0
0 −16
In every case we seethat λ = a3k
. In case (a) wewe have i=1 m0i =
1
Pr0
Pr
Pr
br+i
a2
2 i=1 mi , so i=1 mi aa24 + abi3 = i=1 m0i 2a
. In case (c) we see
4 + a3
1
1
1
1
0 −3k0
that b00 = a3j
. To sum up, we see that M (a1 , a2 ) ∈ MS,b̃ for all a1 ∈ k ∗ ,
1
a2 ∈ k and b̃ ∈ BS . This gives dim S = r + r0 + 9 in subcase (a), dim S =
r + r0 + 10 in subcase (b) and dim S = r + r0 + 11 in subcase (c).
(5)
Case 5. The tangent cone is three lines, and we set f3 = f3
= x1 x2 x3 . The
88
CHAPTER 6. THE STRATA OF QUARTIC MONOIDS
group M5 consist of matrices on the form

a1
M1 (a1 , a2 ) =  0
0

a1
M3 (a1 , a2 ) =  0
0

0

a
M5 (a1 , a2 ) =
2
0

0
0 ,
0
a2
0
1
a1 a2
0
0
1
a1 a2
0
0
1
a1 a2

0
a2  ,
0

a1
0 ,
0

0 a1
0
0 
M2 (a1 , a2 ) = a2 0
0 0 a11a2


0
0 a1
a2 0 
M4 (a1 , a2 ) =  0
1
0 0
a a
 1 2

0
a1 0
0 a2 
M6 (a1 , a2 ) =  0
1
0
0
a1 a2

where a1 , a2 ∈ k, and is of dimension 2. We see that M5 has 6 disjoint components, each parameterized by Mi for i = 1, . . . , 6.
Recall that θ1 (s, t) = (0, s, t), θ2 (s, t) = (s, 0, t) and θ3 (s, t) = (s, t, 0). We
see that M1 (a1 , a2 )θ1 (s, t) = θ1 (a2 s, a1t,a2 ), M1 (a1 , a2 )θ2 (s, t) = θ2 (a1 s, a1t,a2 )
and M1 (a1 , a2 )θ3 (s, t) = θ3 (a2 s, a1 t), so M1 (a1 , a2 )θi (s, t) is a parameterization of Z(x1 ). However, M2 (s, t)θ1 (s, t) = (a1 s, 0, a1ta2 ) = θ2 (a1 s, a1ta2 ), so
M2 (s, t)θ1 (s, t) is a parameterization of Z(x2 ), not of Z(x1 ). This comes from
the fact that M2 (a1 , a2 ) sends Z(x1 ) to Z(x2 ) and vice versa. The six components of M corresponds to the permutation group of the 3 lines Z(x1 ), Z(x2 )
and Z(x3 ).
To avoid notation confusion, we rewrite the intersection numbers:
k1,2 := I(1,0,0) (f4 , x2 ),
k2,1 := I(0,1,0) (f4 , x1 ),
k3,1 := I(0,0,1) (f4 , x1 ),
k1,3 := I(1,0,0) (f4 , x3 ),
k2,3 := I(0,1,0) (f4 , x3 ),
k3,2 := I(0,0,1) (f4 , x2 ).
We have to consider 4 cases, depending on how many of the pairs (k1,2 , k1,3 ),
(k2,1 , k2,3 ) and (k3,1 , k3,2 ) are zero. In each of the subcases we will assume that
m1 , . . . , mr is a partition of 4 − k2,1 − k3,1 associated to the component x1 = 0,
m01 , . . . , m0r0 is a partition of 4 − k1,2 − k3,2 associated to the component x2 = 0
and m001 , . . . , m00r00 is a partition of 4 − k1,3 − k2,3 associated to the component
x3 = 0.
Subcase 5a. Assume k1,2 = k2,1 = k3,1 = k1,3 = k2,3 = k3,2 = 0. Define BS as
Qr
Qr00 00 m00
0
00
i
i ) =
the set of b = (b, b0 , b00 ) ∈ Ar∆,0 ×Ar∆,0 ×Ar∆,0 such that ( i=1 bm
i )( i=1 (bi )
Qr 0
0
0 mi
. Then, for each b ∈ BS there is a unique polynomial f4 whose coi=1 (bi )
6.2. TYPES 1 TO 8
89
efficients of x21 x2 x3 , x1 x22 x3 and x1 x2 x23 are all zero such that
f4 (θ1 (s, t)) =
r
Y
(bi s + t)mi ,
i=1
0
f4 (θ2 (s, t)) =
r
Y
0
(b0i s + t)mi and
i=1
r
Y
f4 (θ3 (s, t)) =
!
i
bm
i
00
r
Y
00
(b00i s + t)mi .
i=1
i=1
Define ϕS by writing ϕ(b) = [x0 f3 + f4 ]. Now we see that BS has
c := gcd(m1 , . . . , mr , m01 , . . . , m0r0 , m001 , . . . , m00r00 )
components, each given by
r
Y

!  r00
r0
Y
Y
00
mi /c 
m0i /c
m /c 
j
ur+r
=
ω
ur+i
ui i
0 +i
c
i=1
i=1
i=1
for j = 1, . . . , c.
We know that M ∈ MS,b̃ if and only if there exist an b ∈ BS and λ ∈ k ∗
such that f4 (θi (s, t)) = λf˜4 (M θi (s, t)) for i = 1, 2, 3. Looking at M1 (a1 , a2 ) we
see that
4 Y
r
1
t
(a1 a22 b̃i s + t)mi
=
f˜4 (M1 (a1 , a2 )θ1 (s, t)) = f˜4 θ1 a2 s,
a1 a2
a1 a2
i=1
f˜4 (M1 (a1 , a2 )θ2 (s, t)) = f˜4 θ2
t
a1 s,
a1 a2
f˜4 (M1 (a1 , a2 )θ3 (s, t)) = f˜4 (θ3 (a1 s, a2 t)) =
=
r
Y
i=1
1
a1 a2
!
i
b̃m
i
4 Y
r0
a42
0
(a21 a2 b̃0i s + t)mi
i=1
r 00
Y
i=1
(
00
a1 b̃00i
s + t)mi ,
a2
so b = (b, b0 , b00 ) = (a1 a22 b̃, a21 a2 b̃0 , aa12 b̃00 ) with λ = a41 a42 is an example showing
that M1 (a1 , a2 ) ∈ MS,b̃ for every b̃ ∈ BS . Furthermore, we see that b̃ and b is
in the same component of BS .
90
CHAPTER 6. THE STRATA OF QUARTIC MONOIDS
Considering M2 (a1 , a2 ) we see that
t
),
a1 a2
t
M2 (a1 , a2 )θ2 (s, t) = θ1 (a2 s,
),
a1 a2
M2 θ3 (s, t) = θ3 (a1 t, a2 s).
M2 (a1 , a2 )θ1 (s, t) = θ2 (a1 s,
This gives
˜
˜
f4 (M2 (a1 , a2 )θ1 (s, t)) = f4 θ2 a1 s,
f˜4 (M1 (a1 , a2 )θ2 (s, t)) = f˜4 θ1 a2 s,
f˜4 (M1 (a1 , a2 )θ3 (s, t)) =
!
r
Y
i
b̃m
i
i=1
t
a1 a2
t
a1 a2
r Y
i=1
=
b̃00i
=
mi
1
a1 a2
1
a1 a2
!
a41
4 Y
r0
0
(a21 a2 b̃0i s + t)mi
i=1
4 Y
r
r 00
Y
i=1
(a1 a22 b̃i s + t)mi
i=1
(
00
a2
s + t)mi .
00
a1 b̃i
Now M2 (a1 , a2 ) ∈ MS,b̃ only if there exist a b ∈ BS such that
r
Y
0
mi
(bi s + t)
=
i=1
r0
Y
i=1
0
(a21 a2 b̃0i s + t)mi ,
i=1
0
(b0i s + t)mi =
r
Y
(a1 a22 b̃i s + t)mi and
i=1
i=1
r 00
Y
r
Y
00
(b00i s
m00
i
+ t)
r
Y
00
a2
s + t)mi .
=
(
00
b̃
a
i=1 1 i
This can only happen if the partitions m1 , . . . , mr and m01 , . . . , m0r0 of 4 are equal.
This is in fact a sufficient condition: Assuming mi = m0i for i = 1, . . . , r = r0
we see that b = (b, b0 , b00 ) given by bi = a21 a2 b̃0i , b0i = a1 a22 b̃i and b00i = aab̃200 is
1
an element of BS , and f4 (θi (s, t)) = (a1 a2 )4 f˜4 (θi (s, t)) for i = 1, 2, 3. In other
words, each M2 (a1 , a2 ) ∈ MS,b̃ if and only if m1 , . . . , mr and m01 , . . . , m0r0 equal.
6.2. TYPES 1 TO 8
91
Furthermore, b and b̃ are in “inverse” components of S:
Q
Q 00
Q
Q 00 m00i r
r
r
r
mi
00 m00
2
0 mi
00
i
b
(b
)
(a
a
b̃
)
a
/(a
b̃
)
2
1 i
i=1 i
i=1 i
i=1 1 2 i
i=1
Q 0
Q 0
=
0
0
r
r
0 mi
2
mi
i=1 (bi )
i=1 (a1 a2 b̃i )
Q 0
r
0 m0i
i=1 (b̃i )
Q 00
= Q
r
r
mi
00 )m00
i
b̃
(
b̃
i=1 i
i=1 i
In a very similar way we can see that M3 (a1 , a2 ) ∈ MS,b̃ if and only if
m01 , . . . , m0r0 and m001 , . . . , m00r00 are equal partitions of 4, and M4 (a1 , a2 ) ∈ MS,b̃
if and only if m1 , . . . , mr and m001 , . . . , m00r00 are equal partitions of 4.
The elements of M5 on the forms M5 (a1 , a2 ) and M6 (a1 , a2 ) are members
of MS,b̃ if and only if all three partitions are equal, and we can easily check
that γS does not identify identify additional components of BS .
Similarly to the previous Subcases we get the same stratum if two of the
partitions of four are switched. Such a switch corresponds to the symmetry of
switching two of the variables x1 , x2 and x3 . The symmetry also corresponds
to one of the elements M2 (1, 1), M3 (1, 1) and M3 (1, 1) in M5 . The result is the
same as before: Any of the definitions of BS and ϕS will do for the construction
of the stratum S.
Note that if c = 4, then r = r0 = r00 = 1, m1 = m01 = m001 , and two
components are identified by γS . In all other cases c = 1 or 2 and no components
are identified. We se that as in previous cases, if c = 2e , then S has 1 + e
components. Furthermore, formula (6.5) gives dim S = r + r0 + r00 + 9.
Subcase 5bcd. In these subcases BS will be open in some affine space. In
Subcase 5b we assume k1,2 , k1,3 > 0 and k2,1 = k2,3 = k3,1 = k3,1 = 0, in
Subcase 5c we assume k1,2 , k1,3 , k2,1 , k2,3 > 0 and k3,1 = k3,1 = 0, and in
Subcase 5d we assume k1,2 , k1,3 , k2,1 , k2,3 , k3,1 , k3,1 > 0. For each stratum S we
assume m1 , . . . , mr is a partition of 4 − k2,1 − k3,1 , m01 , . . . , m0r0 is a partition of
4 − k1,2 − k3,2 and m001 , . . . , m00r00 is a partition of 4 − k1,3 − k2,3 . As in previous
cases, the following construction is not unique for some strata, but in each such
case one can select any definition of BS and ϕS .
0
00
For Subcase 5b we define BS = Ar∆,0 × Ar∆,0 × Ar∆,0 , for Subcase 3c we
0
00
define BS = Ar∆,0 × Ar∆,0 × Ar∆,0 × A10 and for Subcase 5d we define BS =
0
00
Ar∆,0 × Ar∆,0 × Ar∆,0 × A20 .
b =
We can define ϕS for Subcases 5b, 5b and 5c in one go by writing
Qr
mi
(b, b0 , b00 ) for 5b, b = (b, b0 , b00 , b000 ) for 5c and 5d, letting b000
in
1 =
i=1 bi
92
CHAPTER 6. THE STRATA OF QUARTIC MONOIDS
subcase 5b, and, for Subcase 5b and 5c, letting b000
2 = 1. Then we define ϕS by
writing ϕ(b) = x0 f3 + f4 where f4 is the unique polynomial whose coefficients
of x21 x2 x3 , x1 x22 x3 and x1 x2 x23 are all zero and such that
k3,1 k2,1
f4 (θ1 (s, t)) = s
t
r
Y
(bi s + t)mi ,
i=1
0
f4 (θ2 (s, t)) =
k3,2 k1,2
b000
t
2 s
r
Y
0
(b0i s + t)mi and
i=1
00
k2,3 k1,3
f4 (θ3 (s, t)) = b000
t
1 s
r
Y
00
(b00i s + t)mi .
i=1
In all subcases M1 (a1 , a2 ) ∈ MS,b̃ for all a1 , a2 ∈ k ∗ and b̃. Furthermore,
M2 ∈ MS,b̃ if and only if k1,2 = k2,1 , k1,3 = k2,3 , k3,1 = k3,2 and m1 , . . . , mr and
m01 , . . . , m0r0 are equivalents partitions. This comes from the fact that M2 (a1 , a2 )
sends Z(x1 ) to Z(x2 ) and vice versa. For the other components of M5 , similar
conditions apply: For each j = 2, . . . , 6, wether Mj (a1 , a2 ) is in MS,b̃ depends
only on the stratum S. We can easily check that (actually, it follows) that MS,b̃
is both a subgroup and a subvariety of M5 , and we can use (6.5) to calculate
the dimension of S. In subcase 5b dim S = r + r0 + r00 + 10, in subcase 5c
dim S = r + r0 + r00 + 11 and in subcase 5d dim S = r + r0 + r00 + 12.
Case 6. The tangent cone is three lines meeting in a point, and we set f3 =
(6)
f3 = x32 − x2 x23 . The group M6 consist, up to multiplication of third roots of
unity, of matrices on the form




a1 a2 a3
a1 a2 a3
0 ,
M1 (a1 , a2 , a3 ) =  0 1 0  , M2 (a1 , a2 , a3 ) =  0 1
0 0 1
0 0 −1




a1
a2
a3
a1
a2
a3
M3 (a1 , a2 , a3 ) =  0 −1/2 1/2 , M4 (a1 , a2 , a3 ) =  0 −1/2 1/2  ,
0
3/2 1/2
0 −3/2 −1/2




a1
a2
a3
a1
a2
a3
M5 (a1 , a2 , a3 ) =  0 −1/2 −1/2 , M6 (a1 , a2 , a3 ) =  0 −1/2 −1/2 ,
0 −3/2 1/2
0
3/2 −1/2
where a1 ∈ k ∗ and a2 , a3 ∈ k, so M6 has 18 disjoint components of dimension
3.
6.2. TYPES 1 TO 8
93
We have to distinguish between 3 subcases, (a) j1 = j2 = j3 = 0, (b)
j1 = j2 = j3 = 1 and (c) j1 > 1 and j2 = j3 = 1. Let m1 , . . . , mr be a partition
of 4 − j1 associated to the component Z(x2 ), let m01 , . . . , m0r0 be a partition of
4 − j2 associated to the component Z(x2 − x3 ) and let m001 , . . . , m00r00
0
00
For (a) we define BS as the set of b = (b, b0 , b00 ) ∈ Ar∆ × Ar∆ × Ar∆ such that
Pr00
Pr
Pr0
2 i=1 mi bi = ( i=1 m0i b0i )( i=1 m00i b00i ), and ϕ(b) = [x0 f3 + f4 ] where f4 is the
unique polynomial such that
f4 (θ1 (s, t)) =
r
Y
(s + bi t)mi ,
i=1
0
f4 (θ1 (s, t)) =
r
Y
0
(s + b0i t)mi and
i=1
00
f4 (θ1 (s, t)) =
r
Y
00
(s + b00i t)mi .
i=1
00
0
For (b) we define BS = Ar∆ × Ar∆ × Ar∆ , and ϕ(b) = ϕ(b, b0 , b00 ) = [x0 f3 + f4 ]
where f4 is the unique polynomial such that
f4 (θ1 (s, t)) = 2t
r
Y
(s + bi t)mi ,
i=1
0
f4 (θ1 (s, t)) = t
r
Y
0
(s + b0i t)mi and
i=1
00
f4 (θ1 (s, t)) = t
r
Y
00
(s + b00i t)mi .
i=1
0
00
For (c) we define BS = Ar∆ × Ar∆ × Ar∆ × k ∗ , and ϕ(b) = ϕ(b, b0 , b00 , b000 ) =
94
CHAPTER 6. THE STRATA OF QUARTIC MONOIDS
[x0 f3 + f4 ] where f4 is the unique polynomial such that
f4 (θ1 (s, t)) = b000 tj1
r
Y
(s + bi t)mi ,
i=1
f4 (θ1 (s, t)) = t
r0
Y
0
(s + b0i t)mi and
i=1
00
f4 (θ1 (s, t)) = −t
r
Y
00
(s + b00i t)mi .
i=1
For each stratum S we see that BS is open in some affine space, so each S
will be irreducible.
As in case 6, the families M1 , . . . , M6 represent permutations of the three
lines of Z(f3 ). Furthermore, wether Mi (a, b) ∈ MS,b̃ is only dependent on the
the stratum S: Mi (a, b) ∈ MS,b̃ if and only if the partitions are compatible
with the action of Mi on the lines. It follows that (6.5) applies and we see that
dim S = r + r0 + r00 + 8 in subcase (a), dim S = r + r0 + r00 + 9 in subcase (b)
and dim S = r + r0 + r00 + 10 in subcase (c).
(7)
Case 7. The tangent cone is a double line plus a line, and we set f3 = f3
x2 x23 . The group M7 is generated by matrices on the form


a1
a2
a3
 0 1/a24 0 
0
0
a4
=
where a1 , a4 ∈ k ∗ and a2 , a3 ∈ k, so M is of dimension 4.
We distinguish between two subcases, (a) where j0 = k0 = 0 and (b) where
j0 , k0 > 0. Let m1 , . . . , mr be a partition of 4 − j0 associated to the component
Z(x2 ) and m01 , . . . , m0r0 be a partition of 4 − k0 associated to the component
Z(x2 )
0
For Subcase (a) we define BS = Ar∆ × Ar∆ × A3 and for Subcase (b) we define
0
BS = Ar∆ × Ar∆ × A3 × k ∗ . We write b = (b, b0 , b00 , b000 ) for Subcase (b) and b =
(b, b0 , b00 ) with b000 = 1 for Subcase (a). We then define ϕ(b) = [x0 f3 + f4 ] where
f4 is the unique polynomial whose coefficients of x1 x2 x23 , x22 x33 and x2 x33 are all
zero, the coefficients of x21 x2 x3 , x1 x22 x3 and x32 x3 are b001 , b002 and b003 , respectively,
Qr
Qr 0
0
f4 (θ1 (s, t)) = tj0 i=0 (s + b01 t)mi and f4 (θ2 (s, t)) = b000 tk0 i=0 (s + b0i t)mi .
We easily check that γS (BS ) = S and that MS,b̃ = M7 for all S. Since
all BS are open in some affine space S we know that S has one component.
6.3. TYPE 9 - THE TANGENT CONE IS SMOOTH
95
Formula (6.5) gives the dimension of S, in case (a) dim S = r + r0 + 11 and in
case (b) dim S = r + r0 + 12.
(8)
Case 8. The tangent cone is a triple line, and we set f3 = f3
group M8 is the set of invertible matrices on the form


a1 a2 a3
a4 a5 a6 
0 0 ω3i
= x33 . The
where a1 , . . . , a6 ∈ k and i = 0, 1, 2.
In this case the tangent cone has no special points, and we choose a slightly
different approach to describing the strata. Let m1 , . . . , mr be a partition of 4.
This defines a stratum S. Let BS be the set of
b = (b1,1 , b1,2 , . . . , br,1 , br,2 , b01 , . . . , b07 ) ∈ (A2 \ (0, 0))r × A7
such that bi,1 bj,2 6= bi,2 bj,1 for each i 6= j. Define the map ϕS : BS → S as
follows:
ϕ(b) = [x0 x33 + (b1,1 x1 + b1,2 x2 )m1 · · · (br,1 x1 + br,2 x2 )mr + b01 x31 x3
+ b02 x21 x2 x3 + b03 x1 x22 x3 + b04 x32 x3 + b05 x21 x23 + b06 x1 x2 x23 + b07 x22 x23 ]
We see that the fibers of ϕS are of dimension r −1. As in cases 1 to 7, γS (b, g) =
γS (b̃, g̃) only if g −1 g 0 is on the form (6.4) where A = λM for some M ∈ M8
and some λ ∈ k ∗ . In fact, for any M ∈ M8 , λ ∈ k ∗ and b̃ ∈ BS there is exactly
one element h ∈ G on the form (6.4) such that ψ(ϕ(b̃), h) ∈ ϕ(BS ). Thus the
fibers of ψ|ϕS (BS )×G are of dimension 1 + dim M8 = 7, and the fibers of γS are
all of dimension (r − 1) + 7 = r + 6. From (6.2) we get
dim S = dim BS + 12 − dim γS−1 γS (b̃, g̃) = (2r + 7) + 12 − (r + 6) = r + 13.
6.3
Type 9 - the tangent cone is smooth
In this section we will use a lemma and linear algebra to calculate the dimension
of the strata of type 9.
Recall that each stratum of type 9 is characterized by a partition of 12: Two
monoids where the tangent cone is smooth (viewed as a curve in P2 ) are in the
same stratum if and only if their associated partitions of 12 are the same.
For any fixed f3 of type 9 and any point p ∈ Z(f3 ) we can find a local
power series parameterization ψ : (k, 0) → (A2 , p) of Z(f3 ), where A2 ⊂ P2
96
CHAPTER 6. THE STRATA OF QUARTIC MONOIDS
is a standard affine space containing p. The intersection multiplicity Ip (f3 , f4 )
is then the multiplicity of the factor t in the power series g := f4 (ψ(t)), and
the intersection multiplicity Iψ(t0 ) (f3 , f4 ) is the multiplicity of (t − t0 ) in g.
Furthermore, we see that Iψ(t0 ) (f3 , f4 ) ≤ m if and only if g (j) (t0 ) = 0 for
j = 0, . . . , m − 1, where g (j) denotes the jth derivative of g.
,x2 ,x3 ]4
Let e1 , . . . , e12 be a basis of the vector space k[x1(f
, where k[x1 , x2 , x3 ]4
3)
denotes the set of homogeneous polynomials in x1 , x2 , x3 over k of degree 4.
If the coefficient of x31 in f3 is different from zero we can, for example, choose
e1 , . . . , e12 as the set of monomials in x1 , x2 and x3 of degree four that are not
a multiple of x31 .
The following lemma will be helpful in determining the dimensions of the
strata of type 9.
Lemma 6.2. Let f3 ∈ k[x1 , x2 , x3 ] define a smooth cubic curve and let e1 , . . . , e12
be defined as above. For any local parameterization ψ of Z(f3 ), the power series
gi (t) := ei (ψ(t)) for i = 1, . . . , 12 are linearly independent and the Wronskian
determinant of g1 (t), . . . , g12 (t), equal to



det 

g1 (t)
g10 (t)
..
.
g2 (t)
g20 (t)
..
.
(11)
(11)
g1
(t) g2
···
···
..
.
(t) · · ·
g12 (t)
0
g12
(t)
..
.
(11)



,

g12 (t)
does not vanish identically.
Proof. The power series g1 (t), . . . , g12 (t) are linearly independent by construcP12
P12
,x2 ,x3 ]4
tion: If i=1 ci gi (t) = 0 where c1 , . . . , c12 ∈ k, then g̃ = i=1 ci ei ∈ k[x1(f
3)
is such that g̃(ψ(t)) ≡ 0. Since f3 is indecomposable, it follows that g̃ is a
,x2 ,x3 ]4
multiple of f3 and thus equal to 0 in k[x1(f
. Since e1 , . . . , e12 is a basis, it
3)
follows that c1 = · · · = c12 = 0.
The functions g1 , . . . , g12 are elements of a the differentiable field k((t)) with
constants k. Since g1 , . . . , g12 are linearly independent over k, Proposition 2.8
of [22] applies, so the Wronskian determinant of g1 , . . . , g12 is not identically
zero.
The lemma above follows from a specialization of [25, Theorem 2] (q = 4, p =
3 and F := f3 ) and its proof. Indeed, the proof presented above is practically
identical to the proof of [25, Theorem 2].
6.3. TYPE 9 - THE TANGENT CONE IS SMOOTH
97
Let X ∈ S be defined by F = x0 f3 + f4 , and S be characterized by the partition of 12 equal to m1 , . . . , mr . Furthermore, write {p1 , . . . , pr } = Z(f3 , f4 ) ⊂ P2
such that Ipj (f3 , f4 ) = mj for j = 1, . . . , r.
We will now show that in an open neighborhood of X the dimension of S is
equal to 12 + r. Write Sf3 = π −1 (f3 ) ∩ S, where π : S → P9 is the morphism
induced by the linear projection P24 99K P9 . Since type 9 corresponds to an
open subset of P9 it is sufficient to show that Sf3 is of dimension 3 + r.
We assume that pi 6∈ Z(x3 ) for i = 1, . . . , r and write x3 = 1 for the rest
of the section. Thus f3 , f4 define curves in A2 . Let ψi : (k, 0) → (A2 , pi ) for
i = 1, . . . , r be local parameterizations of Z(f3 ) and let ei for i = 1, . . . , 12 be
(j)
defined as above. Furthermore, let gi denote the row vector
!
∂j
∂j
e1 (ψi (ti )), . . . , j e12 (ψi (ti ))
∂tji
∂ti
of power series in ti .
(j)
Define M (t1 , . . . , tr ) to be the 12×12 matrix with rows gi where i = 1, . . . , r
and j = 0, . . . , mi − 1. We will see that this matrix completely describes Sf3 in
a neighborhood of X.
For any nonzero column vector c = (c1 , . . . , c12 )T ∈ k 12 let Sfc3 denote the
four dimensional set of monoids that can be written on the form Z(x0 f3 + f4 )
P12
where f4 is congruent to λ i=1 ci ei modulo f3 for some λ 6= 0. Note that Sfc3
is not necessarily a subset of Sf3 (contrary to what the notation may suggest).
If M (τ1 , . . . , τr )c = 0 for some τ1 , . . . , τr ∈ k and nonzero column vector c = (c1 , . . . , c12 ) ∈ k 12 , then it is clear that Sfc3 ⊂ Sf3 and that for
any monoid Z(x0 f3 + f4 ) ∈ Sfc3 we have Z(f3 , f4 ) = {ψ1 (τ1 ), . . . , ψr (τr )} and
Iψi (τi ) (f3 , f4 ) = mi for i = 1, . . . , r. In other words, the column vector c gives
us a four dimensional family of monoids in Sf3 that share intersection points
(with multiplicites) between f3 and f4 . These intersection points correspond to
τ1 , . . . , τr (by the local parameterizations ψ1 , . . . , ψr ).
We will now see that the family of monoids with these intersection points
and associated multiplicities is exactly the family Sfc3 . We need to show that
if f˜4 is such that Z(f3 , f˜4 ) = {ψ1 (τ1 ), . . . , ψr (τr )} and Iψi (Ti ) (f3 , f˜4 ) = mi for
i = 1, . . . , r, then Z(x0 f3 + f˜4 ) ∈ Sfc3 . We show this by a proof of contradiction:
Assume the opposite and fix any f40 such that Z(x0 f3 + f40 ) ∈ Sfc3 and any point
p ∈ A2 such that f˜4 (p), f40 (p) 6= 0. Consider the polynomial f400 = f˜4 (p)f40 −
f40 (p)f˜4 . Now we see that Iψi (τi ) (f3 , f400 ) ≥ mi for i = 1, . . . , r and Ip (f3 , f400 ) ≥ 1,
so, by Bézout’s theorem, f400 must be a multiple of f3 . However, this implies
98
CHAPTER 6. THE STRATA OF QUARTIC MONOIDS
Z(x0 f3 + f˜4 ) ∈ Sfc3 , a contradiction. It follows that for any (τ1 , . . . , τr ) such
that the equation M (τ1 , . . . , τr )c = 0 there exists exactly a four dimensional
family of monoids in Sf3 with intersection points ψi (τi ) and multiplicities mi .
Furthermore, we see that if M (τ1 , . . . , τr )c = 0 has no nonzero solution in c
there is no monoid in Sf3 with these intersection points and multiplicities.
Note that the equation M (τ1 , . . . , τr )c = 0 has a solution (in c) if and only
if det M (τ1 , . . . , τr ) = 0. By the remarks above, we see that the set defined by
det M (t1 , . . . , tr ) = 0
(6.6)
is of dimension four less than Sf3 . Thus, we need to show that the set defined
by equation (6.6) is of dimension r − 1. This is equivalent to showing that
det M (t1 , . . . , tr ) is not identically zero. The case r = 1, m1 = 12 is already
proven by Lemma 6.2, and this case is in fact a special case of the main result of
[25]. The rest of the cases follow from Lemma 6.2 and the following proposition.
Note that the proposition deals with column vectors to ease the notation, but
that we need the equivalent result for row vectors.
Proposition 6.3. Let n ∈ N and m1 , . . . , mr be a partition of n. If the column
vectors gi (ti ) ∈ k((ti ))n for i = 1, . . . , r are such that
(n−1)
det(gi (ti ), gi0 (ti ), . . . , gi
(ti )) 6= 0,
(j)
then the matrix with columns gi (ti ) where i = 1, . . . , r and j = 0, . . . , mi − 1
is also of full rank.
Proof. It is sufficient to show that if the proposition is true for the partition
m1 , . . . , mr , then it is also true for (a) the partition m1 , m2 , . . . , mr − 1, 1 and
(b) the partition m1 − 1, m2 , . . . , mr−1 , mr + 1. Let
(m1 −1)
e1 , . . . , en = g1 (t1 ), . . . , g1
(t1 ), g2 (t2 ), . . . , gr(mr −1) (tr ).
By assumption, this is a basis of k{t1 , . . . , tr }n and thus also of k{t1 , . . . , tr+1 }n .
For (a), let
Ψ := det (e1 , . . . , en−1 , gr+1 (tr+1 ))
and assume Ψ = 0 (with the intention of reaching a contradiction). It follows
that gr+1 (tr+1 ) is in the span of e1 , . . . , en−1 .
The power series Ψ is a determinant function. Recall that the derivative,
with respect to some variable, of a determinant function is the sum of determinants of the n matrices where one column is replaced with its derivative. This
follows from the product formula for derivation.
6.3. TYPE 9 - THE TANGENT CONE IS SMOOTH
99
The jth derivative of Ψ (with respect to tr+1 ) is equal to
∂j Ψ
∂tjr+1
(j)
= det e1 , . . . , en−1 , gr+1 (tr+1 ) = 0
(j)
for all j. Thus gr+1 (tr+1 ) is in the span of e1 , . . . , en−1 for j = 0, . . . , n − 1.
(n−1)
Since gr+1 (tr+1 ), . . . , gr+1 (tr+1 ) are linearly independent, this implies that a
space of dimension n is contained in a space of dimension n − 1, a contradiction.
Thus Ψ 6= 0, so the proposition is true for the partition m1 , m2 , . . . , mr − 1, 1.
For (b), let
(j)
Ψj := det e1 , . . . , ed
m1 , . . . , en , gr (tr )
(m −1)
where em1 = g1 1 (t1 ) is removed from the determinant. It is clear that
Ψj = 0 for j = 1, . . . , mr − 1 since these are determinants of matrices with
repeated columns. Assume Ψmr = 0 with the intention of getting a contradic(m )
tion. It follows that gr r (tr ) is in the span of e1 , . . . , ed
m1 , . . . , en . Furthermore,
(mr +1)
∂Ψmr
(tr ) is also in the span of e1 , . . . , ed
m1 , . . . , en . Con∂tr = Ψmr +1 , so gr
tinuing, we see that for j > mr we have
∂Ψj
(mr )
= Ψj+1 + det e1 , . . . , ed
(tr ), gr(j) (tr )
m1 , . . . , en−1 , gr
∂tr
(i)
If Ψi = 0 for i = 1, . . . , j > mr , then gr (tr ) is in the span of e1 , . . . , ed
m1 , . . . , en
for i = 1, . . . , j > mr , and thus all the columns of the determinant above are
contained in the span of e1 , . . . , ed
m1 , . . . , en . The determinant is therefore equal
to zero, giving Ψj+1 = Ψj = 0.
By induction Ψmi = 0 implies Ψi = 0 for all i, so the linearly independent
(j)
vectors gr (tr ) for j = 1, . . . , n are all in the span of e1 , . . . , ed
m1 , . . . , en . This
contradiction implies Ψmr 6= 0, so the proposition is true for the partition
m1 − 1, m2 , . . . , mr−1 , mr + 1.
With this we have proved that each determinant det M (t1 , . . . , tr ) does not
vanish identically, that dim Sf3 = r + 3 and that dim S = 12 + r. This concludes
the description of the strata of S in terms of dimension. However, the techniques
in this section are not sufficient to determine the number of components in any
of the strata of type 9. Some brute force calculations have been attempted, but
it seems that some additional insight is needed.
100
CHAPTER 6. THE STRATA OF QUARTIC MONOIDS
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