CMA seminar
University of Oslo
11.March 2010
An easy introduction to
Algebraic Geometry
and
Rational Cuspidal Plane Curves
Torgunn Karoline Moe
CMA/MATH
University of Oslo
Torgunn Karoline Moe
Cuspidal curves
Introduction
The most interesting objects in the world
Algebraic
geometry
Algebraic
curves
Singularity
theory
Cuspidal curves
References
• How many and what kind of cusps can a rational
cuspidal plane curve have?
Torgunn Karoline Moe
Cuspidal curves
Introduction
Algebraic
geometry
Algebraic
curves
Singularity
theory
Cuspidal curves
References
This happens today
• Basic algebraic geometry
• Plane algebraic curves
• Singularity theory
• Rational cuspidal plane curves
Torgunn Karoline Moe
Cuspidal curves
Introduction
Algebraic
geometry
Algebraic
curves
Singularity
theory
Cuspidal curves
References
Algebraic geometry in a nutshell
• The study of geometric objects using algebraic
methods.
• Can find new and surprising properties of both the
objects and the methods.
• Main tool: commutative algebra.
• Rings
• Ideals
• Main objects: varieties.
• Curves
• Surfaces
Torgunn Karoline Moe
Cuspidal curves
The worlds we work within
Introduction
• Algebraically closed fields – C.
Algebraic
geometry
• Affine spaces of dimension n – Cn ,
Algebraic
curves
• Projective spaces of dimension n – PnC ,
Singularity
theory
Cuspidal curves
References
(x1 , . . . , xn ).
(x0 : . . . : xn ).
•
Pn
can be constructed using Cn+1 , identifying points
in the affine space lying on the same line through the
origin.
Pn ∼
= (Cn+1 r {(0, . . . , 0)})/ ∼,
(a0 : . . . : an )∼(λa0 : . . . : λan ), ∀ λ ∈ C∗ .
Torgunn Karoline Moe
Cuspidal curves
Introduction
The objects we work with
• An algebraic set in Cn is the zero set V of a finite
set of polynomials in the ring C[x1 , . . . , xn ].
Algebraic
geometry
• An algebraic set in Pn is the zero set V of a finite set
Algebraic
curves
• In our worlds open sets are complements of algebraic
Singularity
theory
• An affine variety is a closed subset of Cn which can
Cuspidal curves
• A projective variety is an irreducible closed subset of
References
of homogeneous polynomials in C[x0 , . . . , xn ].
sets.
not be decomposed into smaller, closed subsets.
Pn .
• An algebraic set in a space of dimension n defined by
a single irreducible (homogeneous) polynomial is a
hypersurface – a variety of dimension n − 1.
Torgunn Karoline Moe
Cuspidal curves
Let’s get it all down to earth
• The projective plane P2 has coordinates (x : y : z).
• One irreducible homogeneous polynomial F (x, y , z)
Introduction
defines V(F ) – a curve in P2 .
Algebraic
geometry
• The degree of the curve is the degree of the
Algebraic
curves
• Letting z = 1, the polynomial f (x, y ) = F (x, y , 1)
Singularity
theory
Cuspidal curves
polynomial.
will define a curve V(f (x, y )) in a space isomorphic
to C2 .
• Letting y = 1, we get the curve V(f (x, z)) in
another affine plane.
• Letting x = 1, we get V(f (y , z)).
References
• These three affine curves constitute the projective
curve V(F ).
• Technically, we have covered P2 by three open affine
sets isomorphic to C2 ,
P2 r V(z),
P2 r V(y ),
P2 r V(x).
Torgunn Karoline Moe
Cuspidal curves
A typical conic I - V(x 2 + y 2 − z 2 )
Introduction
Algebraic
geometry
Algebraic
curves
Singularity
theory
z =1
y =1
x =1
Cuspidal curves
References
• Remember that this is just the real picture - the
complex world hides its secrets.
Torgunn Karoline Moe
Cuspidal curves
Introduction
A typical conic II - V(x 2 + y 2 +(z − 1)2 )
Algebraic
geometry
Algebraic
curves
Singularity
theory
Cuspidal curves
References
• All conic curves in P2 (circles, ellipses, hyperbolas
and parabolas) are equivalent when we work over C.
Torgunn Karoline Moe
Cuspidal curves
Introduction
The nodal cubic - V(zx 2 − zy 2 − x 3 )
Algebraic
geometry
Algebraic
curves
Singularity
theory
Cuspidal curves
• This curve has one obviously interesting point in
References
(0 : 0 : 1).
Torgunn Karoline Moe
Cuspidal curves
The cuspidal cubic - V(zy 2 − x 3 )
Introduction
Algebraic
geometry
Algebraic
curves
Singularity
theory
Cuspidal curves
References
• This curve also has an interesting point in (0 : 0 : 1),
but it is different from the point in the previous
example.
Torgunn Karoline Moe
Cuspidal curves
Let’s compare
Introduction
Algebraic
geometry
Algebraic
curves
Singularity
theory
Cuspidal curves
The nodal cubic
The cuspidal cubic
References
• How and why are these interesting and different -
even over C?
Torgunn Karoline Moe
Cuspidal curves
Introduction
Algebraic
geometry
Some more theory I
• A point a = (a0 : a1 : a2 ) on a curve V(F ) is called
Algebraic
curves
singular if it is in the zero set of all the partial
derivatives of F ,
V(Fx (a), Fy (a), Fz (a)).
Singularity
theory
• A curve can only have a finite number of singular
Cuspidal curves
• The other points on the curve are called smooth.
References
points.
Torgunn Karoline Moe
Cuspidal curves
Some more theory II
Introduction
Algebraic
geometry
Algebraic
curves
• Every smooth point a on a curve has a uniqe tangent
line given by
V(Fx (a)x + Fy (a)y + Fz (a)z).
• Every singular point p has one or more tangent
line(s).
• For p = (0 : 0 : 1) singular,
Singularity
theory
Cuspidal curves
F (x, y , 1) = fm (x, y ) + fm+1 (x, y ) + . . . + fd (x, y ).
• The tangent line(s) of C at p is given by the zero
set(s) of each reduced linear factor of fm (x, y ).
References
• A tangent line is special because it touches the curve
at the given point a bit more than other lines.
Torgunn Karoline Moe
Cuspidal curves
More about the interesting points
Introduction
Algebraic
geometry
Algebraic
curves
Singularity
theory
Cuspidal curves
References
• Unbelieveably many different kinds of singularities.
• Can classify singularities using invariants:
• Branches – counting the number of times the curve
passes through the point.
• A singularity with more than one branch is called a
multiple point.
• A singularity with only one branch is called a cusp.
• Multiplicity – the amount of intersection between a
general line and the curve at the point.
• Is equal to the m in fm (x, y ) for p = (0 : 0 : 1).
• Tangent intersection – the intersection multiplicity
of the tangent line and the curve at the point.
Torgunn Karoline Moe
Cuspidal curves
Introduction
Algebraic
geometry
Algebraic
curves
Singularity
theory
Cuspidal curves
References
Multiplicity 2
Torgunn Karoline Moe
Cuspidal curves
Introduction
Algebraic
geometry
Detonating the algebraic bomb
• Can investigate the inside of a singularity by blowing
it up.
• Replace the singularitiy with a projective line.
• In an affine neighbourhood of the singularity, look at
Algebraic
curves
all the lines through the point.
• Lift each line to a height corresponding to the slope
Singularity
theory
of the line.
• Observe that the curve is practically unchanged
Cuspidal curves
References
outside the singularity.
• Close the curve and get a new curve.
• Look at the point(s) of the new curve corresponding
to the blown up singularity.
Torgunn Karoline Moe
Cuspidal curves
Introduction
Algebraic
geometry
Algebraic
curves
Singularity
theory
Cuspidal curves
References
Blowing up the cuspidal cubic
Torgunn Karoline Moe
Cuspidal curves
Introduction
Algebraic
geometry
Algebraic
curves
Singularity
theory
Cuspidal curves
References
•
Yes, the blown up space is strange and funny. And it doesn’t really
look like that. But don’t worry.
Torgunn Karoline Moe
Cuspidal curves
Introduction
Useful properties of a cusp
• When a cusp is blown up, we have only one point
Algebraic
geometry
corresponding to the singularitiy.
• This point might still be singular.
Algebraic
curves
Singularity
theory
Cuspidal curves
References
• Then we blow up again.
• Let mi denote the multiplicity of the remaining
singularity after i blowing-ups.
• For a cusp we define the multiplicity sequence m
• m = (m, m1 , . . . , ms ).
• Have m ≥ m1 ≥ . . . ≥ ms .
• There are more restrictions here.
Torgunn Karoline Moe
Cuspidal curves
Let’s go back to start
• How many and what kind of cusps can a rational
cuspidal curve have?
Introduction
Algebraic
geometry
Algebraic
curves
• A curve is called cuspidal if all its singular points are
cusps.
• A curve of degree d is rational ⇐⇒
(d − 1)(d − 2)
=
2
X
singular points
X mi (mi − 1)
(
).
2
i
Singularity
theory
• A rational curve can be given by a parametrization.
Cuspidal curves
• By the formula, the cuspidal cubic is the only
References
rational cuspidal curve of degree 3.
• A rational cuspidal plane curve of degree d must also
satisfy
• Bézout: mp + mq ≤ d.
• Matsuoka–Sakai: d < 3 · m,
where m is the highest multiplicity of the cusps.
Torgunn Karoline Moe
Rational cuspidal curves of degree 4
Cuspidal curves
Introduction
Algebraic
geometry
Algebraic
curves
(2), (2), (2)
(22 ), (2)
Singularity
theory
Cuspidal curves
References
(3)
(23 )
Torgunn Karoline Moe
Cuspidal curves
Introduction
Algebraic
geometry
Rational cuspidal curves of degree 5
# Cusps
1
Algebraic
curves
2
Singularity
theory
3
Cuspidal curves
References
4
Curve
C1
C2
C3
C4
C5
C6
C7
C8
Cuspidal configuration
(4)
(26 )
(3, 2), (22 )
(3), (23 )
(24 ), (22 )
(3), (22 ), (2)
(22 ), (22 ), (22 )
(23 ), (2), (2), (2)
# Curves
3 – ABC
1
2 – AB
1
1
1
1
1
Torgunn Karoline Moe
Cuspidal curves
Conjecture [Piontkowski (2007)]
Introduction
• There is only one rational cuspidal plane curve with
Algebraic
geometry
more than three cusps – the curve of degree 5 with
cuspidal configuration [(23 ), (2), (2), (2)].
• The only tricuspidal curves are
Algebraic
curves
• [Fenske, Flenner & Zaidenberg (1996-1999)]
Singularity
theory
Cuspidal curves
References
Series
I
II
III
d
d
2a + 3
3a + 4
m̄p
(d − 2)
(d − 3, 2a )
(d − 4, 3a )
m̄q
(2a )
(3a )
(4a , 22 )
m̄r
(2d−2−a )
(2)
(2)
For d
d ≥4
d ≥5
d ≥7
• The curve of degree 5 with cuspidal configuration
[(22 ), (22 ), (22 )].
Result [Tono (2005)]
• A rational cuspidal curve has ≤ 8 cusps.
Torgunn Karoline Moe
Cuspidal curves
Introduction
Algebraic
geometry
Algebraic
curves
Singularity
theory
Cuspidal curves
References
A world of opportunities for me
• Use Cremona transformations to give new
restrictions.
• Construct cuspidal curves by projecting a rational
smooth curve in Pn .
• There is a connection between the number of cusps
and the centre of projection that is used.
• This is linked to the tangents of the smooth curve in
Pn .
• Try to interpret the problem in other worlds; i.e.
toric geometry or tropical geometry.
Torgunn Karoline Moe
Cuspidal curves
Introduction
Algebraic
geometry
Algebraic
curves
Singularity
theory
Cuspidal curves
References
Useful literature
T. Fenske
Rational cuspidal plane curves of type (d, d − 4) with
χ(ΘV hDi) ≤ 0.
H. Flenner, M. Zaidenberg
On a class of rational cuspidal plane curves.
H. Flenner, M. Zaidenberg
Rational cuspidal plane curves of type (d, d − 3).
R. Hartshorne
Algebraic Geometry.
M. Namba.
Geometry of projective algebraic curves.
J. Piontkowski.
On the Number of the Cusps of Rational Cuspidal
Plane Curves.
I hope there’s more cake!