Geometry of Families of Curves
Fall 2012
Course taught by Joe Harris
Notes by Atanas Atanasov
One Oxford Street, Cambridge, MA 02138
E-mail address: nasko@math.harvard.edu
Contents
Lecture
1.1.
1.2.
1.3.
1. September 5, 2012
Parameter spaces
The Hilbert scheme
Twisted cubics
6
6
6
7
Lecture 2. September 7, 2012
2.1. Key definition and theorem
2.2. Constructing the Hilbert scheme
9
9
10
Lecture 3. September 10, 2012
3.1. The geometry of Hilbert schemes
3.2. More on twisted cubics
11
11
12
Lecture 4. September 12, 2012
4.1. Generalizations of the Hilbert scheme
4.2. Hilbert schemes of curves
14
14
14
Lecture 5. September 14, 2012
5.1. The restricted Hilbert scheme
5.2. Linkage
16
16
17
Lecture 6. September 17, 2012
6.1. Examples
6.2. Picard groups of surfaces and monodromy
19
19
20
Lecture 7. September 19, 2012
7.1. The theorem on cohomology and base-change
7.2. Relations with the Hilbert scheme
22
22
23
Lecture 8. September 21, 2012
8.1. Tangent spaces to Hilbert schemes
8.2. An application
25
25
26
Lecture 9. September 24, 2012
9.1. Mumford’s example
28
28
Lecture 10. September 26, 2012
10.1. Estimating the dimension of the restricted Hilbert scheme
10.2. A parametric approach to studying H in genus g ≥ 2
31
31
31
Lecture 11. September 28, 2012
11.1. Liaison
11.2. The Hartshorne-Rao module
33
33
34
Lecture 12.
36
October 1, 2012
3
4
CONTENTS
12.1. Deformation theory
12.2. Deformations and obstructions
36
37
Lecture 13. October 3, 2012
13.1. Deformation theory
13.2. Hurwitz space
39
39
40
Lecture 14. October 5, 2012
14.1. Hurwitz space
14.2. Compactifying the small Hurwitz space
42
42
43
Lecture 15. October 10, 2012
15.1. Severi varieties
15.2. Smoothness and dimension of Severi varieties
45
45
45
Lecture 16. October 12, 2012
16.1. Wrapping up parameter spaces
16.2. Moduli spaces
48
48
48
Lecture 17. October 15, 2012
17.1. Moduli of elliptic curves
17.2. Deligne-Mumford moduli spaces
51
51
51
Lecture 18. October 17, 2012
18.1. The moduli space of genus g curves
18.2. Compactifying the moduli space of genus g curves
54
54
55
Lecture 19. October 19, 2012
19.1. Compactifying Mg
19.2. Examples of stable reduction
56
56
56
Lecture 20. October 22, 2012
20.1. Base change and normalization
20.2. The cuspidal example
59
59
60
Lecture 21. October 24, 2012
21.1. More examples of stable reduction
61
61
Lecture 22. October 26, 2012
22.1. Remarks on stable reduction
22.2. Geometry of singular curves
64
64
65
Lecture 23. October 31, 2012
23.1. Stratifying Mg by topological type
23.2. Deformation of planar curve singularities
67
67
68
Lecture 24. November 2, 2012
24.1. Examples of (mini)versal deformation spaces
24.2. Dualizing sheaves
70
70
71
Lecture 25. November 5, 2012
25.1. A proof of Riemann-Roch
25.2. Dualizing sheaves
73
73
74
Lecture 26.
76
November 7, 2012
CONTENTS
26.1.
26.2.
Planar singularities and dualizing sheaves
Kontsevich space
5
76
77
Lecture 27. November 9, 2012
27.1. The Kontsevich space and the Hilbert scheme
27.2. More examples of Kontsevich spaces
79
79
79
Lecture 28. November 12, 2012
28.1. Complete subvarieties of Mg
28.2. Diaz’s Theorem
82
82
82
Lecture 29. November 14, 2012
29.1. Arbarello’s proof
29.2. Diaz’s proof
84
84
85
Lecture 30. November 16, 2012
30.1. The birational geometry of Mg
30.2. Unirationality in low genera
87
87
87
Lecture 31. November 19, 2012
31.1. Unirationality and Mg
31.2. Divisors on Mg
89
89
89
Lecture 32. November 26, 2012
32.1. Divisor class theory for the moduli space of curves
32.2. A pencil of plane quartics
91
91
92
Lecture 33. November 28x, 2012
33.1. Divisor class theory for Mg
33.2. A pencil of hyperplane sections on a quartic surface
94
94
94
Lecture 34. November 30, 2012
34.1. Basic facts about Pic(Mg )Q
34.2. Grothendieck-Riemann-Roch and its applications
97
97
97
Lecture 35. December 3, 2012
35.1. More about divisors on Mg
35.2. A pencil on a K3 surface
100
100
101
LECTURE 1
September 5, 2012
1.1. Parameter spaces
What is a parameter space?
A priori, a parameter space is a set of objects in algebraic geometry. In other subjects such as topology
and differential geometry, there are too many degrees of freedom and there is no way to endow such space
with the structure of a finite dimensional space. Since varieties are somewhat rigid, one of the distinguishing
features of algebraic geometry such finite dimensional spaces do exist. Our aim is to develop tools suitable
for studying varieties in families.
For example, consider
{ subvarieties/subschemes in Pn with a given Hilbert polynomial p } .
Similar construction are
{ sheaves/vector bundles on a given X }
and
{ subschemes with given numerical invariants } /isomorphism.
Our goal is to identify these with the points of a variety or scheme in a natural way. Giving the right meaning
to “natural” gives rise to several difficulties.
Example 1.1.
{hypersurfaces of degree d in Pn } = PN = P H0 (OPn (d)).
1.2. The Hilbert scheme
Naturality, in the world of varieties, means the following. Given a family of objects
X ⊂ B × Pn
B
with fibers subschemes Xb ⊂ Pn of specified Hilbert polynomial p, we get a map
ϕX : B −→ Hp = {subschemes of Pn with Hilbert polynomial p}.
We want to give Hp the structure of a variety such that ϕX is regular.
For various reasons, we need to do this in the category of schemes. This introduces one main complication:
a map of varieties is determined by its values on points (i.e., on the underlying sets), but a map of schemes
is not. We fix this by restricting our attention to flat families.
Remark 1.2. Flatness always implies constant Hilbert polynomial of the fibers. The converse is true
for a reduced base though not in general.
6
1.3. TWISTED CUBICS
7
In the world of schemes, we want to specify for all flat families of schemes
X ⊂ B × Pn
π
B
with Hilbert polynomial p, a map ϕπ : B → Hp and these commute with base change. The last statement
means that if we have a fiber square
X 0 ⊂ B 0 × Pn
π0
B0
X ⊂ B × Pn
u
π
B
then ϕπ0 = ϕπ ◦ u.
Remark 1.3. If we take B = Spec K we get a bijection of the points of Hp and the subschemes of Pn
with Hilbert polynomial p.
For any family π : X 0 → B as above, by a base change with B 0 = pt we deduce that ππ (b) = [Xb ] for all
b ∈ B. In other words, we have a functor
Sch −→ Set,
B 7−→ {X ⊂ B × Pn flat over B with Hilbert polynomial p}.
We say H is a fine parameter space if we have an isomorphism of functors with Hom(−, H).
Remark 1.4. In particular, if H is a finite parameter space we can take B = H and idH : H → H to get
a family X ⊂ H × Pn . This family is universal in the sense that any other family is a base change of this
one.
Theorem 1.5. There exists a fine parameter space Hp for subschemes in Pn with Hilbert polynomial p.
1.3. Twisted cubics
Let us consider the space of twisted cubics, that is, rational degree 3 curves in P3 :
P1 −→ C ,→ P3 .
A standard way such curve is given by t 7→ [1, t, t2 , t3 ]. More generally, any basis f0 , . . . , f3 of H0 (OP1 (3))
determine a twisted cubic given parametrically by [f0 , . . . , f3 ].
A better way to describe C is as the vanishing locus of a set of polynomials. In the case of twisted
cubics, three quadrics suffice, that is, C = V (Q1 , Q2 , Q3 ). For example, in the case of the standard curve,
the quadrics
XZ − W 2 ,
XW − Y Z,
Y W − Z2
are sufficient. It is important to note the Qi are not unique, but their span in H0 (OP3 (3)) is. In other words,
to a twisted cubic C we associate a 3-dimensional vector space of quadrics.
Digression on notation. Consider the graded ring S = K[X0 , . . . , Xn ] where deg Xi = 1 for all i.
The d-th homogeneous part Sd consists of all degree d homogeneous polynomials in the n + 1 variables
X0 , . . . , Xn . To a projective variety X ⊂ Pn , we can associate its ideal I(X) ⊂ S, and its d-th graded piece
I(X)d = I(X) ∩ Sd . Geometrically Sd corresponds to H0 (OPn (d)) and I(X)d to H0 (IX,Pn (d)).
8
1. SEPTEMBER 5, 2012
Back to twisted cubics, H0 (OP3 (2)) has dimension 10. Starting with a twisted cubic C, we produce a
point of the Grassmannian G(3, 10), that is, we get a map
{twisted cubics} −→ G(3, 10).
Note that the latter space has dimension 21 and the image is of relatively small dimension (actually, of
dimension 12). One issues is that the image is locally closed but not closed. This means that a twisted cubic
may degenerate to something else.
Example 1.6. Start with a twisted cubic C ⊂ P3 . Define linear transformations of P3 given by


1
 1

,
At = 

1 
t
and Ct = At (C). The curves Ct form a family of twisted cubics (over the punctured line) with special fiber
C0 a planar nodal cubic.
One of our goals next time will be to remedy this problem.
LECTURE 2
September 7, 2012
2.1. Key definition and theorem
We will start by reviewing the ideas behind the Hilbert scheme expanding on previous discussion.
Definition 2.1. Given a set Σ ⊂ {subschemes X ⊂ Pn } we say a scheme H is a fine parameter space
for Σ if we have an isomorphism of functors from
Sch −→ Set
B 7−→ {flat families X → B, X ⊂ B × Pn with fibers in Σ}
to HomSch (−, H).
There are several immediate consequences worth mentioning.
(i) If H is a fine parameter space, we have a bijection Σ → {points of H} (to see this take B = Spec C).
(ii) Consider a family
X ⊂ B × Pn
π
B
and let the induced map ϕπ : B → H. Then for all points bnB, we have an equality ϕπ (b) = [Xb ].
(iii) There exists a universal family
C ⊂ H × Pn
H
and all other families are realizable as pullbacks from C → H in a unique way.
Theorem 2.2. Let p = p(m) be a polynomial and set
Σ = {subschemes X ⊂ Pn with Hilbert polynomial p}.
Then there exists a fine parameter space Hp,n for Σ.
Our next aim is to describe some of the key ideas in the proof of this theorem. This discussion is
somewhat independent of what will follow. Our long term goal is to understand the geometry of these spaces
and use this knowledge to answer questions about the schemes in question. Thinking about our example of
twisted cubics, here are two such problems.
Question 2.3. Given 12 lines L1 , . . . , L12 ⊂ P3 , how many twisted cubics meet all 12?
Question 2.4. Do there exist non-constant families of twisted cubics over a complete base?
9
10
2. SEPTEMBER 7, 2012
2.2. Constructing the Hilbert scheme
The main idea behind the construction is given X ⊂ Pn (of class Σ), to consider its ideal I(X). Then we
can view the m-th graded piece of this ideal I(X)m ⊂ Sm as a point in a Grassmannian of linear subspaces
of Sm .
Example 2.5. For Σ = {twisted cubics}, define
C 7−→ (I(C)2 ⊂ S2 ) ∈ G(3, 10).
There are several problems. A degenerate twisted cubic may not be cut out by quadrics, or, the space of
quadrics passing through such a curve may not have dimension 3.
These issues are resolved if we pass to I(C)3 ⊂ S3 . Then, however, there are subschemes of P3 with
Hilbert polynomial 3n + 1 (that of a twisted cubic) which are not cut out by cubics. Such an example is the
union of a planar cubic and a single point elsewhere in P3 .
We claim there is a high enough degree which makes this work.
Theorem 2.6. Given a polynomial p, there exists an integer m0 such that for all m ≥ m0 and all
X ⊂ Pn with Hilbert polynomial p, we have:
(a) dim I(X)m = m+n
− p(m),
m
(b) I(X)m generates I(X) up to saturation.
Suppose we have chosen a value of m as in the previous result. We only get a map of sets
m+n
m+n
Σ −→ G
− p(m),
,
m
m
X 7−→ (I(X)m ⊂ Sm ).
Theorem 2.7. The image of this map is closed and admits a scheme structure H satisfying the fine
parameter space property.
The difficulty lies in writing equations which cut out the image. To do so, we consider the multiplication
morphisms
µk : Λ ⊗ Sk −→ Sm+k
for Λ a subspace of Sm . If Λ generates an ideal as desired, then
m+k+n
rank µk ≤
− p(m + k).
n
These conditions define a determinental subvariety of the Grassmannian. To carry this out globally, let
U denote the tautological bundle over the Grassmannian, and Sk the trivial bundle with fiber Sk . The
multiplication maps we considered glue to give a morphism of vector bundles
µk : U ⊗ Sk −→ Sm+k .
We can then impose the necessary rank conditions and intersect these loci over all k. In fact, one can further
show that finitely many k suffice.
LECTURE 3
September 10, 2012
3.1. The geometry of Hilbert schemes
Recall we already constructed the Hilbert scheme
Hp,n = {subschemes X ⊂ Pn with Hilbert polynomial p}.
Equality holds in the stronger sense of being a fine parameter space. The only remaining thing to show is
given a family
X ⊂ B × Pn
π
B
with Hilbert polynomial p, we get a map ϕπ : B → Hp,n . On the level of sets, it is clear how such a map is
defined. The question is to do this even when B is non-reduced. Doing so involves using the cohomology and
base-change theorem in a crucial way. We will abstain from saying more about this and put this question
aside, at least for now.
As an instructional example, let us discuss punctual Hilbert schemes, that is, Hilbert schemes of 0dimensional subschemes. Such subschemes have constant Hilbert polynomial. In other words, we are interested in
Hd,n = {subschemes of dimension 0 and degree d in Pn }
which contains
H0 = {reduces subschemes} == (Pn )d \ ∆ /Sd .
We may think that H = Hd,n is a compactification of H0 , but H may be reducible. The subscheme H0 is
dense in one of the components of H.
Observation. If n ≥ 3 and d 0, then H has other (larger) components.
To see this start by noting that dim H0 = dn, so it suffices to exhibit a family with strictly greater
dimension. For the sake of concreteness, take n = 3, and look at Γ ⊂ P3 supported at a point p such that
mk+1 ⊂ IΓ ⊂ mk . Here m = mp is the maximal ideal corresponding to p ∈ Pn . Of course, Γ determines a
subspace V = IΓ /mk+1 ⊂ mk /mk+1 . The point is any such subspace determines an ideal. Without loss of
generality, we may look at a chart around p isomorphic to A3 . Then
k+2
k+3
k
k+1
dim m =
,
dim m
=
.
3
3
Choose I = (mk+1 , V ) for a subspace V ⊂ mk /mk+1 . Then
k+2
d = deg V (I) =
+ codim V.
3
If
1 k+2
k2
codim V ∼
∼
,
2
2
4
11
12
3. SEPTEMBER 10, 2012
then the dimension of the family of such subschemes is the dimension of the Grassmannian, which is asymptotically k 4 /16. But d ∼ k 3 /6, so for k 0, the Hilbert scheme H has another component. We can then
write
0
H = H ∪ other components.
Things are much better behaved for n = 1, 2.
0
(i) When n = 1, the Hilbert scheme is H = H = Pd .
0
(ii) When n = 2, the Hilbert scheme H = H is smooth, so it provides a canonical desingularization of
Symd P2 .
3.2. More on twisted cubics
Let us look back on Hilbert schemes of curves. We already discussed
H = H3n+1,3 ⊃ {twisted cubics} = Ht .
We will see Ht is irreducible of dimension 12. Let H0 = Ht .
Proposition 3.1. We have a decomposition H = H0 ∪ H1 where
H1 = {C = C0 ∪ {p} | C0 is a plane cubic, p ∈ P3 \ C0 is a point}.
Furthermore, the open subset inside H1 is irreducible of dimension 15.
To see the last point, note that the open in question is dense in the product of P3 and a projective
bundle over (P3 )∨ (the fibers of the bundle are cubic curves lying in a given plane).
Lemma 3.2. Let C ⊂ P3 be a subscheme of dimension 1 and degree 3. Then pa (C) ≤ 1 with equality if
and only if C is a plane cubic.
Proof of Lemma. Consider a hyperplane section Γ = C ∩ H. The short exact sequence
0
/ OC (m − 1)
/ OC (m)
/ OΓ (m)
/0
gives an inequality
hC (m) ≥ hC (m − 1) + hΓ (m).
A general hyperplane section Γ consists of three points. They can either be collinear, in which case
hΓ (m) = 1, 2, 3, 3, . . . ,
or non-collinear, in which case
hΓ (m) = 1, 3, 3, 3, . . . .
It follows that
hC (m) ≥ 1, 3, 6, 9, 12, . . . ,
which implies pa (C) ≤ 1. We have equality if and only if a general hyperplane section is collinear; this
happens only is the curve is planar.
This result allows us to analyze both components as stated in the Proposition above.
We would like to understand the intersection H0 ∩ H1 . For example, given a subscheme with the right
Hilbert polynomial, how can we check if it is the limit of twisted cubics. A specific such curve is the union
of a planar conic and a line (not contained in the plane) but intersecting the conic.
3.2. MORE ON TWISTED CUBICS
13
It can be shown such curves are limits of smooth twisted cubics. To see this note that the curve in question
is a (2, 1) curve on a smooth quadric surface.
A more challenging problem is to ask whether a triple line and a planar embedded point
C0 = V (z, y 4 , y 3 x, y 3 w)
is in H0 .
LECTURE 4
September 12, 2012
4.1. Generalizations of the Hilbert scheme
We will start by mentioning some generalizations of the Hilbert scheme. Our main goal is to record
the objects which can be parametrized by a fine moduli space, so we will not attempt to go through the
constructions (most of them are modifications of the original Hilbert scheme construction).
(i) If Z ⊂ Pn is a closed subscheme, we can construct the Hilbert scheme
Hp,Z = {subschemes of Z with Hilbert polynomial p}.
To carry this out, we need to make sure all ideals we consider contain I(Z).
(ii) Given X and Y projective, we can parametrize morphisms f : X → Y . To do this, we can choose and
embedding X × Y ,→ Pn , and then look at graphs of morphisms {Γ ⊂ X × Y ⊂ Pn }. (It is possible to
relax the projectivity hypothesis slightly.)
(iii) It is possible to construct an incidence correspondence sitting in a product of two Hilbert schemes
which parametrizes pairs X ⊂ Y ⊂ Pn with specified Hilbert polynomials for X and Y .
(iv) Combining the previous two observations, we can parametrize triples (X, Y, f : X → Y ) with suitable
fixed data.
(v) Let X and Y be projective, and we have a morphism π : X → Y . There is a relative Hilbert scheme
Hp,X/Y = {subschemes of X contained in a fiber of π}.
The following is an open question related to the generalizations we presented.
Question 4.1. Given smooth curves C, D of genera g, h ≥ 2, how many non-constant maps f : C → D
can there be?
The reason to expect finiteness is the graph of such a morphism Γ ⊂ C × D has negative self-intersection.
We don’t know the answer in general, but there are various bounds in terms of g and h.
4.2. Hilbert schemes of curves
We have already discussed H = H3n+1,3 = H0 ∪ H1 , where
H0 = {twisted cubics},
H1 = {C ∪ {p} | C a plane cubic, p a point in P3 \ C}.
It is a nice exercise to verify that
H0 ∩ H1 = {C where C is a nodal plane cubic with a spatial embedded point at the node}.
In general, H = Hdn−g+1,n will have many extraneous components.
Example 4.2. There are components whosegeneral member is C ∪Γ, where C is a planar curve of degree
d, and Γ is a 0-dimensional scheme of degree d−1
2 −g. Since we do not understand well 0-dimensional schemes
in isolation, it is hard to study these components.
14
4.2. HILBERT SCHEMES OF CURVES
15
Example 4.3. Ribbons often form extraneous components. These are curves which are double along
their reduced locus. For example, consider the line L = V (X, Y ) in P3 . A ribbon X will be a scheme of
degree 2 supported on L. We assume X is contained in the scheme cut out by the square of the ideal of L,
that is,
I(X) = (X 2 , XY, Y 2 , F (Z, W )X + G(Z, W )Y )
for two homogeneous polynomials F, G of degree m ≥ 0 The tangent space of X winds around L m times.
(i) If m = 0, we plane a planar double line of genus g = 0.
(ii) If m = 1, we get a double line on a quadric of genus g = −1.
In general, the genus of X is −m. It is also worth noting that for m ≥ 2, the general ribbon is not smoothable.
To sum up the situation, the Hilbert scheme is a mess and we do not want to be dealing with it in full
generality.
Definition 4.4. The restricted Hilbert scheme H0 is the union of the components of H whose general
point corresponds to a reduced, irreducible, non-degenerate curve.
We asked a question about the number of twisted cubics which meet each of 12 lines. This was solved
by Schubert a long time ago by doing intersection theory on H0 , the relevant component of the Hilbert
scheme H3n+1,3 . To do so, we need to know H0 is smooth. We can calculate the tangent space at a given
[C] ∈ H0 \ H1 and it has dimension 12, so H0 is smooth at [C]. It is much harder to verify smoothness of H0
along H0 ∩ H1 . The tangent space there has dimension 16. This verification was completed more recently
by Piene and Schlessinger.
Our next goal is to study the restricted Hilbert scheme H0 ⊂ Hdn−g+1,3 for small values of d and g. For
example, we will look at
d = 3, g = 0,
d = 4, g = 0,
d = 4, g = 1,
and others.
LECTURE 5
September 14, 2012
5.1. The restricted Hilbert scheme
Our aim today will be to describe (as explicitly as possible), the restricted Hilbert schemes in low degree
and genus. We will denote
H = Hdn−g+1,n = {the Hilbert scheme of curves of degree d and genus g in Pn },
H0 = {the restricted Hilbert scheme}
[
=
components of H whose general point corresponds to reduced, irreducible curves in Pn .
Question 5.1. For given (d, g, n) find the irreducible components of H0 and their dimensions.
We will focus on n = 3, that is, on curves embedded in P3 . We will discuss three methods of answering
the question above, and apply them to the case of twisted cubics d = 3, g = 0.
Parametric method. Consider
Φ = {(F0 , F1 , F2 , F3 ) | Fi is a basis for H0 (OP1 (3))}/scalars
which is an open subset of P15 . We have a map Φ → H whose image consists of all smooth twisted cubics,
hence is dense in H0 . All fibers are isomorphic to PGL(2). It follows that H0 is irreducible of dimension 12.
Cartesian method I. Let us think about the surfaces which contain a twisted cubic. In this particular
case, it suffices to look at quadrics. We construct the following incidence correspondence.
{(Q, C) | C ⊂ Q, Q a smooth quadric} ⊂ U × Ht ⊂ P9 × H0
*
P9 ⊃ U t
Ht ⊂ H 0
Here, U ⊂ P H0 (OP3 (2)) ∼
= P9 denotes the locus of smooth quadric surfaces, and Ht denotes the locus of
twisted cubics in the restricted Hilbert scheme H0 .
Pick a smooth quadric Q ∼
= P1 × P1 and let us try to understand the fiber over it on the left. This
amounts to asking for the space of twisted cubics which lie on Q, and these can be shown to be (1, 2) or
(2, 1) curves. It follows that all fibers are isomorphic to
P H0 (OQ (1, 2)) t P H0 (OQ (2, 1)) ∼
= P5 t P5 ,
so the incidence correspondence has dimension 14.
Next, let us think about the fibers on the right. Given a twisted cubic C, we are asking for the space of
quadrics which contain it. It is easy to see all such quadrics form a copy of P2 , so
dim H0 = dim Ht = 12.
The following is an interesting issue related to understanding the fibers on the right.
Question 5.2. Given C, what do the singular quadrics look like in the P2 of all quadrics containing C?
Our discussion has a defect, namely, we did not recover the fact H0 is irreducible.
16
5.2. LINKAGE
17
5.2. Linkage
The first Cartesian method we presented above used intricately our knowledge of the geometry of quadric
surfaces. We also understand relatively well cubic surfaces, but in degrees 4 and higher the analysis becomes
considerably harder. Unfortunately, curves of high degree and genus will not, in general, lie on low degree
surfaces, so this is a significant shortcoming. We present an alternative approach to studying curves better
suited for higher values of d and g. The main ingredient is a technique known as linkage or liaison.
Start with a curve C ⊂ P3 and consider two surface S, T which contain it. Ideally, C = S ∩ T though
this is generally not the case. Let us assume S ∩ T = C ∪ D where D is a curve sharing no components with
C. We call such pairs of curves C, D linked.
In many cases of interest, it is possible to find suitable surfaces for which D is simpler than D. We can
then transfer information about the Hilbert scheme of curves D and deduce statements about the Hilbert
scheme of curves C.
We would like to explore how the geometric properties of C and D are related. Assuming S ∩ T = C ∪ D,
let
s = deg(S),
t = deg(T ),
c = deg(C),
d = deg(D),
g = genus(C),
h = genus(D).
Bezout’s theorem immediately implies
c + d = st.
To simplify our discussion, assume S is smooth, and let H denote the class of a hyperplane section on S. By
adjunction, we know that KS = (s − 4)H. Then
2g − 2 = C · C + KS · C = C 2 + (s − 4)C,
so
C 2 = 2g − 2 − (s − 4)c.
A similar computation reveals that
C · D = C(tH − C) = tc + (s − 4)c − (2g − 2),
in other words, the number of intersection points of C and D is determined by the numerical invariants of
the bond. In a similar spirit,
D2 = D(tH − C) = td − tc − (s − 4)c + 2g − 2.
Applying adjunction to D, we get
2h − 2 = D2 + Ks · D = D2 + (s − 4)d = (d − c)(s + t − 4) + 2g − 2.
It is common to rewrite this expression as
s+t−4
.
2
We are now ready to apply these observations to our study of twisted cubics.
h − g = (d − c) ·
Cartesian method II. Going through the linkage computations for two quadric surfaces, we see that
twisted cubics are linked to lines. We are lead to consider the following incidence correspondence.
Q, Q0 are smooth quadrics, C a twisted
0
(Q, Q , C, L) cubic, L a line, Q ∩ Q0 = C ∪ L
G(1, 3) = Hm+1,3
s
+H
t
We know G(1, 3) is irreducible of dimension 4. Computing the fiber over a line [L] ∈ G(1, 3) amounts to a
choice of two distinct quadrics containing it. The space of such pairs is an open in
{quadrics containing L}2 = (P6 )2 ,
18
5. SEPTEMBER 14, 2012
so the incidence correspondence is irreducible of dimension 16. The fibers on the right amount to a choice
of two distinct quadrics containing a given twisted cubic. These form an open in (P2 )2 , so Gt is irreducible
of dimension 12.
It is possible to use a parametric approach in positive genera g ≥ 1, but this requires more machinery.
We will cover this later, but, for now, we are restricted to using the cartesian approach when dealing with
positive genus curves.
Example 5.3. Consider the case d = 6 and g = 3. We start by asking what surfaces contain such a
curve C ⊂ P3 . For quadrics, we are interested in computing the kernel of the map
H0 (OP3 (2))
/ H0 (OC (2)).
The domain has dimension 10. By Riemann-Roch, the codomain has dimension 12 − 3 + 1 = 10. A priori,
this map need not have a non-trivial kernel. Let us look at degree 3 surface:
H0 (OP3 (3))
/ H0 (OC (3)).
The domain has dimension 20, and the codomain 18 − 3 + 1 = 16. This guarantees, there is at least a three
dimensional family of cubic surfaces containing C. Applying the linkage formulas, we see that C is linked to
a twisted cubic D via two cubic surfaces. This leads us to consider the following incidence correspondence.
S, S 0 smooth cubics, C of degree 6 and
0
(S, S , C, D) genus 3, D a twisted cubic, S ∩S 0 = C ∪D
πD
H3m+1,3
s
We will discuss this example later on.
πC
+H
LECTURE 6
September 17, 2012
6.1. Examples
We would like to apply the techniques we already discussed in several examples.
It is worth going through the implications of the parametric approach for genus g = 0 in greater
generality. We will look at the restricted Hilbert scheme H0 ⊂ Hmd+1,n . Consider the space
F ∈ H0 (O 1 (d)), Fi linearly independent, with no comP
.
Ψ = (F0 , . . . , Fn ) i
mon zeros, and defining a very ample linear series
Note that Φ is a dense open in P(H 0 (OP1 (d))⊕(n+1) ) = P(d+1)(n+1)−1 . Let us briefly discuss the open
conditions we impose on the (n + 1)-tuple (F0 , . . . , Fn ). Firstly, being linearly independent ensures the
rational map P1 99K Pn they define is non-degenerate (i.e., the image does not lie in a proper linear subspace
of Pn ). Asking for no common zeros is equivalent to saying the aforementioned map is well-defined, that
is, it is an honest regular map. Finally, defining a very ample linear series amounts to saying the map in
question is an embedding. Remembering only the image of a map P1 → P bbn defined a dominant morphism
Ψ → H0 all of whose fibers are 3-dimensional and isomorphic to Aut(P1 ) = PGL(2). It follows that H0 is
irreducible of dimension (d + 1)(n + 1) − 4. For example, taking n = 3, we obtain the standard formula
dim H0 = 4d.
Example 6.1 (Curves on quadrics). Consider a smooth quadric Q ⊂ P3 , Q ∼
= P1 × P1 . Let L and M
denote the two rulings.
A curve C ⊂ Q has type (a, b) if [C] = aL + bM , or, equivalently, if it is the vanishing locus of a polynomial
of bidegree (a, b). It follows that deg(C) = a + b. Note that KQ = −2L − 2M , so
2ab − 2a − 2b
C · C + KQ · C
+1=
+ 1 = (a − 1)(b − 1).
2
2
Suppose we fix a pair of integers a, b and we want to compute the dimension of the space H of space curves
C which lie on a smooth quadric as curves of type (a, b). Consider the following incidence correspondence.
Q smooth quadric, C of
Φ = (Q, C) type (a, b) on Q
genus(C) =
P H0 (OP3 (2)) = P9 ⊃ U
)
u
H
To compute the dimension of the fibers on the right, we start by fixing a quadric. We need to compute
the dimension of the space of bihomogeneous polynomials of degree (a, b). Note that if a = b this space is
19
20
6. SEPTEMBER 17, 2012
irreducible; otherwise, it consists of two components (each corresponding to a choice of rulings). It is easy to
compute h0 (OQ (a, b)) = (a + 1)(b + 1), so the fibers on the left have dimension (a + 1)(b + 1) − 1. It follows
that Φ has dimension (a + 1)(b + 1) + 8. Furthermore, since the monodromy acts transitively on the rulings,
we also deduce Φ is irreducible.
6.2. Picard groups of surfaces and monodromy
An integral part of our last argument required understanding the monodromy of the rulings. Note that
Pic(Q) = ZhL, M i, and the monodromy of rulings lifts to a monodromy action on the Picard group. This
point of view is more useful for surfaces of higher degree, or, higher dimensional varieties.
Let us consider the degree 3 case. A smooth cubic surface S is isomorphic to the blowup of the plane
P2 at 6 points p1 , . . . , p6 , no three collinear, and not lying on a conic.
If l denotes the class of a line on P2 pulled back to S, and ei denote the classes of the exceptional fibers,
then
Pic(S) = Zhl, e1 , . . . , e6 i.
The embedding S ,→ P3 is given by the linear system 3l − e1 − · · · − e6 . Note that KS = OS (−1) =
−3l + e1 + · · · + e6 .
We also want to find an analogue of the monodromy statement. That is, given such a surface S, it is not
uniquely expressible as a blowup S → P2 . In fact, there are 72 such morphisms corresponding to 6-tuples
of disjoint lines on S. Monodromy acts on Pic(S) as the fill symmetry group of the lattice preserving KS .
The order of this group is 72 · 6! = 51840, 72 coming from line labellings, and 6! from S6 . This allows us to
produce irreducibility arguments similar to the case of quadrics we just discussed.
Unfortunately, this is where the story stops. A very general surface S ⊂ P3 of degree m ≥ 4 has
Pic(S) ∼
= Z. Let P H0 (OP3 (m)) denote the set of all surface of degree m in P3 , and U the open set of smooth
ones. The surfaces with rank(Pic(S)) ≥ 2 form a countable union of subvarieties in U .
Example 6.2. Let us return to an example we already discussed briefly: d = 6, g = 3. Such a curve
C ⊂ P3 falls in one of two cases:
(a) C lies on a (smooth) quadric and has type (2, 4), or
(b) C doesn’t lie on a quadric but lies on a cubic.
Case (a) gives rise to a 23 dimensional family of curves (there are 9 dimensions of quadrics, and 14 dimensions
of (2, 4) curves on any given quadric). To analyze (b), start with two general cubics S and S 0 containing C.
Then S ∩ S 0 = C ∪ T where T is a twisted cubic satisfying
T 2 = 1,
T · KS = −3,
T · C = 8.
As before, we consider an incidence correspondence.
{(S, S 0 , C, T )}
{T }
x
&
{C} = H
6.2. PICARD GROUPS OF SURFACES AND MONODROMY
21
The space of twisted cubics on the left has dimension 12. Given a twisted cubic T , there is a pullback map
H0 (OP3 (3))
/ H0 (OT (3)) ∼
= H0 (OP1 (9))
whose codomain has dimension 10. The fiber on the left has dimension 9 + 9 = 18, and we can deduce the
incidence correspondence is irreducible of dimension 30. On the right, the fibers have dimension 3 + 3 = 6,
so H is irreducible of dimension 24. Details of this argument will appear in the homework.
Remark 6.3.
(i) Any curve on a quadric cone is a limit of curves lying on smooth quadrics.
(ii) Recall our discussion of curves of type (a, b) on quadrics. If a, b ≥ 3, then the curves in question form
a components of the Hilbert scheme. If a = 1, 2 and b ≥ 4 this is not the case.
LECTURE 7
September 19, 2012
7.1. The theorem on cohomology and base-change
There result in question deals with a family of sheaves parametrized by a scheme B, that is, a family
of schemes π : X → B and a sheaf F on X flat over B. For points b ∈ B, we denote Xb = π −1 (b) and
Fb = F|Xb .
Question 7.1. Do the vector spaces H0 (Fb ) fit together to form the fibers of a vector bundle (or coherent
sheaf) on B?
The answer is not exactly, but there is a “best possible approximation”. This approximation, denoted
π∗ F, is defined as (π∗ F)(U ) = H0 (Fπ−1 (U ) ). Notice that we have maps
/ H0 (Fb )
ϕb : (π∗ F)b
but these need not be isomorphisms. Similarly for Hi (Fb ), we have higher pushforwards Ri π∗ F defined by
(Ri π∗ F)(U ) = Hi (F|π−1 (U ) ). It is useful to investigate the question we just outlines by referring to several
examples.
Example 7.2. Consider an elliptic curve E and let p ∈ E be a point. Line bundles on E are parametrized
by E, namely, they are all of the form OE (q−p) for some q ∈ E. We can fit these in a family. Take X = E ×E
with projection π = π1 : X → E. Consider divisors ∆ ⊂ E × E the diagonal, and Γ = E × {p}. The sheaf of
interest is F = OE×E (∆ − Γ).
Example 7.3. We would like to consider the possible ideals of three points in P3 . Consider the points
p = [−1, 0, 1],
q = [1, 0, 1],
rt = [0, t, 1]
for t ∈ A1t . Set
X = A1t × P2 ,
Γ = A1 × {p, q} ∪ {(t, rt )},
To summarize, the points p, q, rt are collinear only for t = 0.
22
F = IΓ (1).
7.2. RELATIONS WITH THE HILBERT SCHEME
23
There are several basic facts we can state about the general setup we described..
(1) h0 (Fb ) is upper semi-continuous. The same is true for all hi (Fb ).
(2) In the open set U ⊂ B where h0 (Fb ) is constant, π∗ F is locally-free and ϕb is an isomorphism.
(3) Where hi (Fb ) jump, they jump in adjacent pairs.
(4) Where hi (Fb ) and hi+1 (Fb ) both jump, this behavior is reflected in Ri+1 π∗ F, but not Ri π∗ F.
To illustrate the last point, note that in the first example above the jump can be seen in (R1 π∗ F)p , i.e.,
1
R π∗ F is a skyscraper sheaf supported over p with rank 1 there.
The idea behind the proof, and a mnemonic to remember these results, is to note there exists a complex
of locally-free sheaves on B
/ K1
/ K2
/ ···
/ K0
0
i
whose cohomology are exactly R π∗ F. A good way to illustrate the behavior in (4) is by looking at B = A1t
and the map of sheaves
·t /
OA1
OA1 .
t
t
Its kernel, as a sheaf, is trivial, but its cokernel is a skyscraper sheaf supported at t = 0.
Question 7.4. Why is this called the theorem on cohomology and base-change?
The general setup involves the following cartesian diagram.
F0
F
X ×B B 0
/X
B
/ B0
The maps ϕb arise when we take B 0 = {b} ,→ B.
7.2. Relations with the Hilbert scheme
We have already noted the Hilbert scheme H = Hp,n is a fine parameter space for subschemes of Pn
with Hilbert polynomial p. This means that for each scheme B there is a natural bijection between suitable
families over B and morphisms B → H. Recall that the construction of H involved Grassmannians. We
never verified the universal property after the construction.
Remark 7.5. A morphism B → G(k, V ) is the same as an inclusion of vector bundles ϕ : E → V ⊗ OB
over B. We ask for |Ec to be locally free of rank k and ϕb be injections for all b ∈ B. The last condition
is stronger than saying E → V ⊗ OB is an injective morphism of sheaves. (The last issue becomes a little
easier if we use quotients.)
24
7. SEPTEMBER 19, 2012
Given a family π : X → B, X ⊂ B × Pn , we get an inclusion of sheaves
IX (m)
/ OB×Pn (m).
Taking direct images, we obtain a morphism
π∗ IX (m)
/ Sm ⊗ OB
as necessary to define a map into the appropriate Grassmannian.
LECTURE 8
September 21, 2012
8.1. Tangent spaces to Hilbert schemes
Recall the universal property of the Hilbert scheme H = Hp,n : for each scheme B there exists a natural
bijection
/ HomSch (B, H).
{flat families X ⊂ B × Pn over B with Hilbert polynomial p}
We would like to investigate this correspondence for
B = I = Spec C[ε]/(ε2 ).
Digression. For any scheme Z and a point p ∈ Z, there is an identification
Tp Z = {f : IZ | f (Spec C) = p}.
The tangent space Tp Z is naturally a vector space, but on the right we just have a set. It is a nice exercise
to interpret the vector space structure on the right geometrically.
In the case of H, we can say that for every X ⊂ Pn we have
T[X] H = {X ⊂ I × Pn flat over I such that X ∩ (Spec C × I) = X}.
That is, we identified the tangent space at [X] ∈ H with flat families over I with central fiber X. Such
objects are called first order deformations of X.
We will start our discussion locally by fixing a chart An ⊂ Pn . Fix a first order deformation X ⊂ I × An
e where Ie is an ideal in the ring C[x1 , . . . , xn , ε]/(ε2 ).
with central fiber X ∩ Spec C × An = X. Let X = V (I)
Assume
Ie = ({fα + εgα })
for
fα , gα ∈ C[x1 , . . . , xn ].
n
The condition X ∩ Spec C × A = X means
{f ∈ C[x1 , . . . , xn ] | f + εg ∈ Ie for some g} = I(X).
Lastly we need to interpret flatness.
Digression. Suppose we have a module M flat over C[ε]/(ε2 ). Flatness means that tensoring with a
short exact sequence preserves exactness. Fortunately, there are very few interesting sequences. The one we
are interested in is the following.
0
/ (ε)
∼
=
C
/ C[ε]/(ε2 )
:
·ε
25
/C
/0
26
8. SEPTEMBER 21, 2012
e
Interpreting this for Ie as a C[x1 , . . . , xn , ε]/(ε2 ) module, flatness is equivalent to saying that if εg ∈ I,
then g ∈ I(X). Reinterpreting again, flatness means that for all f ∈ I(X), there exists g ∈ C[x1 , . . . , xn ]
such that f + εg ∈ Ie and g is unique modulo I(X). In other words, a first order deformation X flat over I
determines a homomorphism
/ C[x1 , . . . , xn ]/I(X)
I(X)
e Working globally with sheaves of ideals, a
which sends a polynomial f ∈ I(X) to g such that f + εg ∈ I.
n
first order deformation of X ⊂ P is equivalent to a morphism of sheaves
/ OPn /IX = OX ,
IX
that is, a global section of Hom(IX , OX ) = NX/Pn , the normal bundle of X in Pn . Finally, a first order
deformation corresponds to an element of H0 (NX/Pn ). We conclude that
T[X] H = H0 (NX/Pn ).
Remark 8.1. The normal bundle NX/Pn = Hom(IX , OX ) bears its name since for smooth points p ∈ X,
we have
(NX/Pn )p = TPn ,p /TX,p .
Remark 8.2. If X ⊂ Z ⊂ Pn are smooth subschemes, we have a short exact sequence
0
/ NX/Z
/ NX/Pn
/ NZ/Pn |X
/ 0.
8.2. An application
We had previously asserted that if a, b ≥ 3, the locus of smooth curves of type (a, b) lying on a smooth
quadric is open in the appropriate Hilbert scheme. Our goal will be do demonstrate this claim using the
tools we just developed.
Consider a smooth curve C of type (a, b) lying on a smooth quadric Q ⊂ P3 . Following the previous
remark, there is a short exact sequence
0
/ NC/Q
/ NC/P3
/ NQ/P3 |C
/ 0.
The first and third terms are line bundles on C we can identify:
NC/Q = OC (a, b),
NQ/P3 |C = OC (2, 2).
Our aim is to show
h0 (NC/P3 ) = dim[C] H = dim{curves of type (a, b) on a smooth quadric} = (a + 1)(b + 1) + 8,
where the last equality has already been established through other means. To do so, it suffices to prove the
following three statements.
(i) The sequence above is exact on global sections, that is, h1 (NC/Q ) = h1 (OC (a, b)) = 0.
(ii) h0 (NC/Q ) = (a + 1)(b + 1) − 1
(iii) h0 (OC (2, 2)) = 9
For (i), since C has class (a, b) there is a short exact sequence
0
/ OQ
/ OQ (a, b)
/ OC (a, b)
/ 0.
Remark 8.3. Since Q ∼
= P1 × P1 , we can apply the Künnth formula to deduce that h0 (OQ (a, b)) = 0
1
if a, b ≥ −1. Similarly, h (OQ (a, b)) = 0 if a, b ≤ −1. Serre duality and Riemann-Roch also imply these
equalities.
8.2. AN APPLICATION
27
We also know that h2 (OQ ) = 0, so h1 (OQ (a, b)) = 0. This also takes care of (ii) implying that
h0 (NC/Q ) = h0 (OC (a, b)) = (a + 1)(b + 1) − 1.
Finally, we focus on (iii). Consider the short exact sequence
0
/ OQ (2 − a, 2 − b)
/ OQ (2, 2)
/ OC (2, 2)
/ 0.
We know that a, b ≥ 3, so 2 − a, 2 − b ≤ −1 and the remark about h1 above applies. We deduce that
h0 (OC (2, 2)) = h0 (OQ (2, 2)) = 9.
LECTURE 9
September 24, 2012
9.1. Mumford’s example
Our goal will be to describe Mumford’s example, a component of the restricted Hilbert scheme H0 that
0
is everywhere non-reduced. That happens for d = 14 and g = 24. Set H = H14m−23,3
and assume C ⊂ P3 is
a smooth curve corresponding to a point of H. For every m, we have a restriction map
/ H0 (OC (m)).
H0 (OP3 (m))
If C was a curve of type (a, b) lying on a smooth quadric, then a + b = 14, (a − 1)(b − 1) = 24. These
Diophantine equations admit no solutions. Similarly, it is possible to verify C does not lie on a singular
quadric.
Remark 9.1. A smooth curve of even degree lying on q quadric cone is a complete intersection. This
allows us to compute its genus by adjunction.
m
h0 (OP3 (m)) h0 (OC (m))
3
4
5
6
20
25
56
84
19 or 20
33
47
61
For m ≥ 4, the line bundles OC (m) are non-special so we can compute h0 (OC (m)) by Riemann-Roch
directly. For m = 3, there is some ambiguity; it is this feature that leads to the example we are interested
in. We will consider two cases:
(a) C lies on a cubic,
(b) C does not lie on a cubic.
Let us first consider (b). We know that C lies on two independent quadric surfaces S and S 0 , so we can
write S ∩ S 0 = C ∪ Q. By Bezout’s theorem,
deg(Q) = 16 − 14 = 2.
By the linkage formulas
4+4−4
(2 − 14) = 0,
2
so Q is a plane conic. Consider the typical liaison incidence correspondence.
genus(Q) = 24 +
Φ = {(C, Q, S, S 0 )}
{Q}
w
'
{C}
Given a planar conic Q, there is a 25-dimensional space of quartics containing it. It follows that the fibers
on the left are irreducible of dimension 25 + 25 = 50, so Φ is irreducible of dimension 58. Finally, the locus
of curves C not lying on a cubic surface is irreducible of dimension 56.
28
9.1. MUMFORD’S EXAMPLE
29
Remark 9.2. We are using the fact a curve C as above lies only on a pencil of quartics. To see this we
have to show
/ H0 (OC (4))
H0 (OP3 (4))
is surjective, or h1 (IC (4)) = 0.
Next, consider case (a), that is C lies on a cubic surface S. Assume S is smooth. (This is true for
a general curve C but requires verification.) Bezout’s theorem shows that C does not lie on quartics not
containing S (otherwise deg C ≤ 3 · 4 = 12). Similarly, C does not lie on a quintic not containing S. In
degree 6, the curve C may lie on a “new” sextic surface not containing S. Let us pick such a sextic T ⊂ P3 .
Say S ∩ T = C ∪ C 0 , and we can compute
3+6−4
(4 − 14) = −1.
deg(C 0 ) = 4,
genus(C 0 ) = 24 +
2
The curve C 0 can either the union of two disjoint conics or a line and a twisted cubic intersecting in a single
point. The conclusion we are leading to is that H has three components:
(a) C does not lie on a cubic, H0 ,
(b) C lies on a cubic and is residual to two disjoint conics, H1 ,
(c) C lies on a cubic and is residual to a line and a twisted cubic, H2 .
The second of these possibilities H1 will give us Mumford’s example, so we will focus on it. For contrast, it
is interesting to see what goes differently with H2 .
A conic Q on a cubic surface is residual to a line L, so Q ∼ H − L where H is the hyperplane class. If
we have two disjoint conics Q and Q0 then Q, Q0 ∼ H − L for the same line L. In other words, we have a
pencil of planes with base L and we are picking two of the residual conics Q and Q0 for two members of this
pencil. It follows that
C ∼ 6H − Q − Q0 ∼ 4H + 2L,
so
KC = (C + KS )|L = OC (3H + 2L).
By Riemann-Roch
h0 (OC (3)) = 19 + h1 (OC (3)) = 19 + h0 (KC (−3)) = 19 + h0 (OC (2L)) = 20.
It looks like C does not lie on a cubic but it still does! (In the case of H2 , we get h0 (OC (3)) = 19.)
Let us compute the dimension of H1 . As usual, we look at the liaison incidence correspondence.
{(C, C 0 , S, T )}
v
{C 0 = Q ∪ Q0 }
'
{C} = H1
The space of cubics containing C 0 has dimension 19 − 7 − 7 = 5. The space of sextics containing C 0
has dimension 83 − 13 − 13 = 57. In conclusion, the incidence correspondence is irreducible of dimension
8 + 8 + 5 + 57 = 78. We said there is a 22-dimensional family of sextics containing a given C, so H1 is
irreducible of dimension 78 − 22 = 56.
30
9. SEPTEMBER 24, 2012
Finally, we need to compute the tangent space to the Hilbert scheme H1 , that is, H0 (NC/P3 ). We know
C is smooth and C ∼ 4H + 3L in S ⊂ P3 . We have a short exact sequence as follows.
0
/ NC/S
/ NC/P3
OC (4H + 2L)
/ NS/P3 |C
/0
OC (3)
Note that KS = OS (−1) and KC = (C + KS )|C , so deg(NC/S ) = (2 · 24 − 2) + 14 = 60. Then
h0 (NC/S ) = 60 − 24 + 1 = 37,
h1 (NC/S = 0.
Finally,
dim T[C] H1 = h0 (NC/P3 ) = h0 (NC/S ) + h0 (NS/P3 |C ) = 37 + 20 = 57,
hence H1 is everywhere non-reduced. It happens that non-integrable first order deformations of C lose the
cubic, i.e., they do not lie on first order deformations of a cubic. It is also possible to show that H1 has
multiplicity 2.
LECTURE 10
September 26, 2012
10.1. Estimating the dimension of the restricted Hilbert scheme
The dimension of the tangent space dim T[C] H = h0 (NC/Pr ) serves as a good first approximation to
the actual dimension of the restricted Hilbert scheme H at C. Next, we can approximate h0 (NC/Pr ) by
χ(NC/Pr ). Note that the inequalities go in opposite directions, namely,
dim[C] H ≤ dim T[C] H = h0 (NC/Pr ) ≥ χ(NC/Pr ).
We will address this discrepancy later. For now, note there is a short exact sequence
0
/ TC
/ TPr |C
/ NC/Pr
/ 0.
The degrees of the first two vector bundles are 2 − 2g and (r + 1)d respectively, so
c1 (NC/Pr ) = (r + 1)d + 2g − 2,
and
χ(NC/Pr ) = (r + 1)d + 2g − 2 − (r − 1)(g − 1) = (r + 1)d − (r − 3)(g − 1) = h(d, g, r).
We will call the number h(d, g, r) the expected dimension of H. In fact, deformation theory can show this is
a lower bound for the dimension of any component of the restricted Hilbert scheme H.
Remark 10.1. Castelnuovo’s genus bound is
g ≤ π(d, r) ∼
d2
2(r − 1)
and this is sharp.
(i) When r = 3, the second term in h(d, g, r) vanishes, so we get h(d, g, r) = 4d.
(ii) When r ≥ 4, the second term can become very negative, hence h(d, g, r) < 0 eventually.
Definition 10.2. A curve C ⊂ Pr is called rigid if its only deformations are obtained by automorphisms
P . In other words, PGL(r + 1) acts on H with a dense orbit.
r
Question 10.3. Are there rigid curves?
There are some obvious examples – rational normal curves in any linear subspace (assuming we are
interested in degenerate curves). It is not known whether other examples exist.
10.2. A parametric approach to studying H in genus g ≥ 2
For g ≥ 2, we will assert there exists a variety Mg parametrizing abstract smooth curves of genus g up
to isomorphism. The space Mg is irreducible of dimension 3g − 3. Since we are interested in embedding
curves in projective spaces, we are naturally lead to study line bundles on curves. We consider the following
31
32
10. SEPTEMBER 26, 2012
tower of spaces.
H
Gdr = {(C, L, V ) | V r+1 ⊂ H0 (L)}
Pd,g = {(C, L) | L ∈ Picd (C)}
Mg = {C}
The space Pd,g parametrizes pairs of a genus g curve and a line bundle L on it of degree d. The fiber of the
morphism Pd,g → Mg over a point [C] ∈ Mg is isomorphic to Picd (C), an abelian variety of dimension g. It
follows that Pd,g is irreducible of dimension 4g − 3. The next morphism in the tower Gdr → Pd,g is far more
mysterious. Depending on the values of g, r and d its behavior can vary from being a birational embedding
to dominant with positive dimensional fiber. It is also possible that Gdr may be reducible. One of the basic
results of Brill-Noether theory says that
dim Gdr ≥ dim Pd,g − (r + 1)(g − d + r)
Finally, the fibers of H → Gdr are PGL(r + 1), so their dimension is r2 + 2r. It follows that
dim H ≥ 4g − 3 − (r + 1)(g − d + r) + r2 + 2r = h(d, g, r).
This is where things get interesting. We have two estimates of dim H which agree. In fact, if d is relatively
large to g, the estimate is precise.
Question 10.4. When is the estimate dim H ≥ h(d, g, r) an equality? When is is a strict inequality?
Interestingly enough, for many curves we can write down, such as complete intersections and curves on
quadrics, there is a strict inequality. Brill-Noether theory said that any components of H that dominates
Mg has the expected dimension. We can then ask how special do the curves of a component have to be to
before the estimate is not exact.
Conjecture 10.5. The previous statement is true for any component of H whose image in Mg had
codimension less than g − 4.
The codimension bound we suggested comes from studying examples. One interesting case is curves of
type (4, 4) on a quadric.
LECTURE 11
September 28, 2012
11.1. Liaison
We have already encountered the topic of liaison which was used in a central way to define certain
incidence correspondences. It is possible to go one step further and introduce an equivalence relation.
We will say two curves C and D in P3 are linked if C ∪D is a complete intersection. This statement is not
completely rigorous and less general than we would like. It is possible that a component of the intersection of
two surfaces S and T has high multiplicity. We would like to allow ourselves to distribute such multiplicities
over the curves C and D.
Let C ⊂ P3 be a locally Cohen-Macaulay curve, S and T surfaces with no common component. If C lies
in S ∩ T we define the residual curve D as
IC
D = V Ann
.
IS∩T
This formulation allows us to extend linkage in the situation we just described. We are interested in the
equivalence relation generated by C ∼ D for C and D linked. Linkage allows us to “go down” and reduce
the geometric complexity of the curves we are studying, hence can be a valuable tool in analyzing Hilbert
schemes.
Theorem 11.1 (Gaeta). For a locally Cohen-Macaulay curve C, the following conditions are equivalent:
(i)
(ii)
(iii)
(iv)
C is linked to a complete intersection,
C is arithmetically Cohen-Macaulay (i.e., the affine cone over C is Cohen-Macaulay at the vertex),
the restriction maps H0 (OP3 (m)) → H0 (OC (m)) are surjections for all m,
H1 (IC (m)) = 0 for all m.
Remark 11.2. Projective normality is equivalent to being smooth and arithmetically Cohen-Macaulay.
Proof. The equivalence of (ii)-(iv) is fairly standard. We will demonstrate the relation between (i) and
(ii). Let the degrees of S and T be s and t respectively. For the sake of convenience, assume C ⊂ S is a
Cartier divisor (certainly true if S is smooth). We will use the following short exact sequence.
/ IS/P3
0
/ IC/P3
/ IC/S
/0
OP3 (−s)
Twisting by m, we get the following.
0
/ IS/P3 (m)
/ IC/P3 (m)
OP3 (m − s)
33
/ IC/S (m)
/0
34
11. SEPTEMBER 28, 2012
The first of these shaves has no cohomology in degrees 1 and 2. Note that KS = OS (s − 4), so
H1 (IC/P3 (m)) = H1 (IC/S (m))
= H1 (OS (m)(−C))
= H1 (KS (m − s + 4)(−C))
= H1 (OS (s + t − 4 − m)(−D))∨
= H1 (ID/P3 (s + t − 4 − m))∨ .
11.2. The Hartshorne-Rao module
Hartshorne has the idea to collect the information of all H1 (IC/P3 (m)) for varying m in one object.
Definition 11.3. The Hartshorne-Rao module associated to C is
M
M
H0 (OC (m))/{restrictions of polynomials of degree m on P3 }.
MC =
H1 (IC/P3 (m)) =
m
m
Note that MC is a finite module over S = C[x, y, z, w], the homogeneous coordinate ring of P3 . The
main observation is up to twisting and dualizing MC is liaison invariant.
Example 11.4. Let C be the union of two disjoint lines L and L00 in P3 .
We can compute that
(
C if m = 0,
H (IC/P3 (m)) =
0 otherwise,
1
so MC = C0 . The question is whether we can link C to anything else. If we use two quadrics, a brief
argument shows the residual curve is another pair of disjoint lines. Using a quadric and a cubic the curve C
is linked to D, a (3, 1) curve on the quadric. Alternatively, we may describe D as a rational quadric curve.
(
C if m = 1,
1
H (ID/P3 (m)) =
0 otherwise.
Theorem 11.5 (Hartshorne-Rao). The Hartshorne-Rao module construction defines a bijection
n
o
finite graded modules
twisting,
/ liaison classes of .
M− :
over S = C[x, y, z, w]
dualizing
curves in P3
Example 11.6. It is worth noting that liaison classes also occur in continuous families. Consider a curve
C given as the union of three skew lines L, L0 and L00 .
11.2. THE HARTSHORNE-RAO MODULE
35
The Hartshorne-Rao module is given by
(MC )0 = C2 ,
(MC )1 = C2 ,
(MC )2 = 0,
and all higher (MC )i = 0. The first equality follows from the fact C has three connected component. To see
the second note that there are 2 + 2 + 2 = 6 dimensions of linear forms on the lines and only 4 on P3 . For
the third one, we use the fact any three skew likes lie on a unique smooth quadric Q.
We would like to know when two triples of skew lines are linked. We can determine this by fleshing
out the multiplication map of the module. One piece of data contained in the multiplication structure is a
morphism
/ Hom((MC )0 , (MC )1 ) = Hom(C2 , C2 ).
H0 (OP3 (1))
The locus in H0 (OP3 (1)) where the image has determinant zero cuts out a quadric dual to Q ⊃ L ∪ L0 ∪ L00 .
The conclusion is two triples of skew lines are linked if and only if they lie on the same smooth quadric.
Recall sextic curves of genus 3 are linked to twisted cubics in a complete intersection of cubic surfaces.
0
We obtained an incidence correspondence. Using the unirationality of H3m+1,3
, it is possible to use that
0
and show H6m−2,3 is also unirational. Severi had the idea one can use this construction to show Mg is
unirational for all g. Juan Migliore showed that a general genus g curve embedded in P3 is minimal in its
liaison class, hence we cannot use linkage inductively as Severi had suggested.
LECTURE 12
October 1, 2012
12.1. Deformation theory
Deformation theory is applicable in many contexts. In order to be concrete, we will focus on deforming
subschemes of a given scheme. We will also introduce several other useful examples.
Definition 12.1. Let X ⊂ Pn be any scheme. A deformation of X is an étale germ of a pointed scheme
(B, b) and a subscheme X ⊂ B × Pn , flat over B, and satisfying X ∩ {b} × Pn = X.
In other words, two deformations X ⊂ B × Pn and X 0 ⊂ B 0 × Pn are equivalent if there are étale
neighborhoods over which the two families are isomorphic. If thinking in complex analytic terms, we can
imagine étale neighborhoods as opens in the complex analytic topology.
As we said earlier, there are several other settings for deformation theory worth mentioning.
(i) Let f : X → Pn be a morphism with X fixed. Deformations of f are morphisms fe satisfying a commutative triangle
/ B × Pn
fe
B×X
#
B
{
such that fe|{b}×X = f .
(ii) Let E be a vector bundle over X. A deformation of E is a vector bundle E over B × X such that
E|{b}×X ∼
= E.
(iii) A deformation of an abstract scheme X is a family
X
⊃
Xb
B
3
b
together with an isomorphism ϕb : Xb → X.
Returning to our original example, our goal is to describe the space of deformations of X ⊂ Pn . Before
getting any further, we should formulate precisely what we mean by the “space of deformations”. In many
practical situations, it is hard to come by a space satisfying universality in a strong sense, but one can get
away with weaker hypotheses.
A versal deformation is one
X ⊃ X
∆ 3 0
such that any other deformation of X can be presented as the pullback of this one. Without asking for
uniqueness, this is not a well-defined space, e.g., we can always take the product with a space ∆0 and obtain
a new versal deformation X × ∆0 → ∆ × ∆0 . If we have uniqueness of first order deformations, that is,
T0 ∆ = {first order deformations},
36
12.2. DEFORMATIONS AND OBSTRUCTIONS
37
we say the deformation X → ∆ is miniversal. As an étale germ, a miniversal deformation is unique.
12.2. Deformations and obstructions
Given a deformation over A1 = Spec C[ε], for any n ≥ 2, we can restrict it to a deformation over
In = Spec C[ε]/(εn ). Due to the sequence of inclusions
I1
/ I2
/ I3
/ ···
/ In
/ In+1
/ ···
/ A1 ,
it follows that given a deformation over any In , we can restrict it to a deformation over Im for any m < n.
The practice of deformation theory often follows a pattern which reverses this process. We start with
a deformation over I2 —a first order deformation—and ask if we can extend it to one over I3 . If that goes
through, we try to extend it to a deformation over I4 and so on. If this process succeeds for arbitrary n, we
can invoke a theorem of Artin producing an honest deformation over A1 . Let us go through this program a
little more carefully.
(i) We start by describing the space of first order deformations. When working with a subscheme X ⊂ Pn
this is H0 (NX/Pn ). In the other three settings we mentioned the spaces of first order deformations
respectively are
(a) H0 (f ∗ TPn ),
(b) H1 (Hom(E, E)) (if E is a line bundle this is H1 (OX )),
(c) H1 (TX ) (assuming X is smooth).
(ii) Given a first order deformation
X
⊂
X
I1
⊂
I2
⊂
I2 × Pn
we ask if we can extend it to a second order deformation, that is, a deformation over I3 . We can then
repeat this process one step at a time.
For convenience, set the deformation space to be
Def(X) = {first order deformations of X}.
To carry out our program, we introduce a second space Obs(X) called the obstruction space. It turns out
there is a map
/ Sym2 (Def(X)∨ ).
ϕ2 : Obs(X)
Given a first order deformation η ∈ Def(X), it extends to second order if and only if ϕ2 (η)(τ ) = 0 for all
τ ∈ Obs(X).
Note that in all examples of interest we have been able to identify the deformation space with a cohomology group. In each of these cases, the obstruction space is the next cohomology group, respectively
H1 (NX/Pn ),
H1 (f ∗ TPn ),
H2 (Hom(E, E)),
H2 (TX ).
In the next, step we get a map
ϕ3 : Ker ϕ2
/ Sym3 (Def(X)∨ )/(Im ϕ2 )3 .
Extending the process further and further, we obtain a sequence of maps
ϕk : Ker ϕk
/ Symk (Def(X)∨ )/(Im ϕ2 , . . . , Im ϕk−1 )k .
for all k ≥ 2. In general, the images generate an ideal I in the graded ring Sym• (Def(X)∨ ) and we are
taking its respective graded pieces. After identifying Def(X) with Spec Sym• (Def(X)∨ ), we are saying a
given η ∈ Def(X) can be extended to all orders if and only if η ∈ V (I).
38
12. OCTOBER 1, 2012
There is one question remaining: assuming we can extend a given deformation to Xk → Ik all orders k,
we can then find an actual deformation (i.e., over a smooth curve) that agrees with all Xk . In particular, if
X → ∆ is miniversal, then the tangent cone to ∆ at 0 is exactly V (I). From this we can conclude that
dim ∆ ≥ dim Def(X) − dim Obs(X).
In particular, if X ⊂ Pn is a smooth curve, then
dim[X] H ≥ h0 (NX/Pn ) − h1 (NX/Pn ) = χ(NX/Pn ).
This justified a heuristic approximation we had made previously.
LECTURE 13
October 3, 2012
13.1. Deformation theory
Given a deformation problem (e.g., deforming a subscheme X ⊂ Pn ) out goal is to describe the versal
deformation space ∆.
X ⊂ X
0
∈
∆
In the process, we introduced two vector spaces
Def(X) = {first order deformations}
(in our case H0 (NX/Pn )),
Obs(X) = {obstructions}
(in our case H1 (NX/Pn )).
There is also a sequence of obstruction maps
ϕ2 : Obs(X) −→ Sym2 (Def(X)∨ ),
..
.
ϕk : Ker(ϕk−1 ) −→ Symk (Def(X)v ee)/(Im ϕ2 , . . . , Im ϕk−1 ).
If we let I be the ideal in Sym• (Def(X)∨ ) generated by all images Im ϕk for k ≥ 2, then V (I) is the tangent
cone to ∆ at 0. In particular, if Obs(X) = 0, then ∆ is smooth of dimension dim Def(X). One of the main
problems is the maps ϕk can be very difficult to compute.
Example 13.1. Recall the Hilbert scheme H = H3m+1,3 parametrizing twisted cubics. We saw that
H = H0 ∪ H1 where H0 is the closure of the locus of twisted cubics, and H1 is the closure of the locus of the
union of a plane cubic and a point. Their dimensions are 12 and 15 respectively. It is possible to see that
H0 is smooth at any point in H0 \ H1 . In fact, we can compute the tangent spaces for [C] ∈ H0 ∩ H1 but
h0 (NC/P3 ) = 16,
h1 (NC/P3 ) = 4.
It remains to see that H0 is smooth there. Piene and Schlessinger computed ϕ2 for such a curve C. In terms
of coordinates x1 , . . . , x16 for Def(C), they found that
Im ϕ2 = hx1 x2 , . . . , x1 x5 i
and ϕ2 is injective. This means there are no more obstruction maps and the tangent cone to H is
V (x1 ) ∪ V (x2 , . . . , x5 ).
This verifies the smoothness of both components at a point of their intersection.
Remark 13.2. Even if Obs(X) is not zero, it is possible that all obstruction maps ϕi are zero.
Example 13.3. Let C = S ∩ T ⊂ P3 be the complete intersection of two surfaces S and T , respectively
of degrees s and t. Note that
KC = OC (s + t − 4),
NC/P3 = NC/S ⊕ NC/T = OC (t) ⊕ OC (s).
39
40
13. OCTOBER 3, 2012
If either s ≥ 4 or t ≥ 4, then h1 (NC/P3 ) > 0. For simplicity, assume s = t ≥ 4. Then the Hilbert scheme H0
parametrizing such smooth curves is an open in G(2, H0 (OP3 (s))) and
s+3
0
0
dim H = dim G(2, H (OP3 (s))) = 2
− 4.
3
We also have
s+3
h (NC/P3 ) = 2h (OC (s)) = 2
−2 ,
3
so the two expressions are equal. This means that despite the presence of potential obstructions, there are
no obstructions imposing any conditions.
0
0
13.2. Hurwitz space
Fix integers d, g and consider the space
simply branched covers f : C → P1 of degree d,
0
Hd,g
=
.
C a connected curve of genus g
Simply branched means the branch locus is reduced of degree b = 2d + 2g − 2. Pictorially, the preimage of
a small neighborhood in P1 looks like one of the following two diagrams.
(0)
Remark 13.4. Despite the deceiving notation Hd,g is different from the Hilbert scheme.
Note that we have a map
0
π : Hd,g
/ Symb P1 \ ∆ = Pb \ ∆.
0
We claim π is finite (meaning all fibers are finite sets), and we can give Hd,g
the structure of a variety so π
is étale.
Let us start by fixing b distinct points p1 , . . . , pb in P1 . We would like to investigate what additional
date we need to provide in order to specify a branched cover of P1 simply branched over the pi . Let us also
fix a basepoint p0 and loops γi around each pi .
If we start with a cover as desired, we obtain permutations σi ∈ Sd corresponding to the monodromy of the
cover around γi . Furthermore, the elements σi satisfy three simple conditions.
(i) All ramification points are simple, so σi are transpositions.
13.2. HURWITZ SPACE
41
(ii) The curve C is connected, hence σi generate a transitive subgroup of Sd . In our case, this is equivalent
to saying
Q the σi generate all of S
Qd .
(iii) Since i γi = 1, it follows that i σi = 1 in Sd .
Finally, the choice of basepoint p0 introduces ambiguity in the σi up to simultaneous conjugation. We
conclude the fiber of π over {p1 , . . . , pb } is
σi ∈ Sd transpotisions,
simultaneous
Q
(σ1 , . . . , σb ) .
hσi i = Sd , i σi = 1
conjugation
If we vary the branch points a little, we can also deform the branched cover along. It follows that Hd,g has
the structure of a variety (complex manifold) of dimension 2d − 2g − 2. Recall that
h(d, g, r) = (r + 1)d − (r − 3)(g − 1).
Setting r = 1, we get h(d, g, 1) = 2d − 2g − 2, so we can think of Hurwitz space as “curves in P1 ” (interpreted
as branched covers).
Given a branched cover f : C → P1 , we can remember either the abstract curve C or the collection of
branch points. This observation gives rise to the following incidence correspondence.
Hd,g
ϕ
Mg
|
π
#
Pb \ ∆
A basic observation states that when d g (actually when d ≥ 2g + 1), the morphism ϕ is surjective, that
is, any curve of genus g can be realized as a d-sheeted branched cover.
Question 13.5. Is the moduli space of genus g curves Mg irreducible?
We propose to answer this question positively by proving the irreducibility of Hd,g .
0
Theorem 13.6 (Clebsch). The space Hd,g
is connected (hence irreducible). In particular, the space Mg
is also irreducible.
We can always match branch points, but then we need to prove one can go among the points in a given
fiber of π. We can analyze by letting the branch points vary.
If we keep all but two branch points fixed and interchange the remaining pair, the monodromy data
(σ1 , . . . , σb ) is sent to
−1
(σ1 , . . . , σi−1 , σi+1 , σi+1
σi σi+1 , σi+2 , . . . , σb ).
It is then possible to show repeated applications of this operation can always reduce the monodromy data
to a single normal form, hence showing one can go between any two sheets of the finite map π.
LECTURE 14
October 5, 2012
14.1. Hurwitz space
The main object we are interested in is
0
Hd,g
= f : C → P1 C smooth curve of genus g, f simply branched of degree d .
This space is evidently non-compact. For example, even while keeping C smooth, we may move the branch
0
points and make them collide. Because of this restricted nature of Hd,g
, it is often called the small Hurwitz
space.
Remark 14.1. It is noteworthy that we are not dividing by automorphisms of P1 . In fact, there are
several flavors of Hurwitz space useful in different contexts. It is easy to explain at least for different ones,
each depending on two options – dividing by automorphisms of P1 and marking the branch points.
We have the following incidence correspondence.
0
Hd,g
ϕ
Mg
π
"
Pb \ ∆ = Symb P1 \ ∆
}
The morphism π is an unramified cover. Let us fix a set of b points B = {p1 , . . . , pb } ∈ Pb \ ∆. To construct
a degree d cover of P1 simply branched exactly above B, it suffices to specify some combinatorial data, that
is, the monodromy of the d sheets around each of the branch points. Then
σ are transpositions,
simultaneous
.
π −1 (B) = (σ1 , . . . , σb ) ∈ (Sd )b Qi
σi = 1, hσi i = Sd
conjugation
The degree of π is called the Hurwitz number. We already mentioned the following result.
0
Theorem 14.2 (Clebsch). The small Hurwitz space Hd,g
is connected, hence irreducible. In particular,
Mg is also irreducible.
One of the advantages of Hurwitz space is it allows is to vary (deform) abstract curves. It is possible
to do so analytically, but the arcs obtained are highly transcendental. Via Hurwitz space, we can vary the
branch points (each in a little disc, all disconnected), and that would induce a deformation of the abstract
curve as a d-sheeted cover. Via a sequence of such small motions, it is also possible to make two branch
points exchange places. If pi and pi+1 switch places while all others remain fixed, we get a loop in Pb \ ∆.
Interpreting this change in terms of monodromy, we get arrive at the transformation
/ (σ1 , . . . , σi−1 , σi+1 , σ −1 σi σi+1 , σi+2 , . . . , σb ).
(σ1 , . . . , σb )
i+1
The main point is that given a solution to the combinatorial problem (σ1 , . . . , σb ), we can always apply a
sequence of the aforementioned moves to arrive at the following normal form:
( (1 2), . . . , (1 2), (2 3), (2 3), . . . , (d − 1 d), (d − 1 d) ).
|
{z
} | {z }
|
{z
}
2g+2
We conclude
0
Hd,g
2
2
is connected which proves the desired result.
42
14.2. COMPACTIFYING THE SMALL HURWITZ SPACE
43
14.2. Compactifying the small Hurwitz space
As we previously mentioned, it is quite simple to produce families of d-sheeted covers of P1 which are
not complete. We would like to briefly discuss some approaches to compactifying the small Hurwitz space.
Let us imagine we have a cover f : C → P1 which has two branch points p and q in P1 . We would like to
investigate what happens when we move p and q so they coincide. If the ramification above p and q involves
disjoint sheets, we have no problem of arriving at a cover with two points of ramification over p = q. The
source curve remains smooth.
The next case is when the ramification above p and q shares a single sheet in common. Again, it is
possible to fill in the family with a smooth curve but a ramification of higher order in the central member.
Finally, if the sheets involves in the ramification above p and q coincide, we can use a nodal curve for
the central member of the family.
Of course, dealing with nodal curves is not very difficult so we can make this allowance. The problem is,
we can further move two or more nodes together and obtain very singular central members. In other words,
if we allow points to coincide and we allow for one simple singularity, it automatically brings more and more
degenerate singularities along. It is not clear if there is a natural place to stop, and, in general, this is a very
hard problem.
There is an alternative solution which does not allow any branch points to coincide. The resulting space
Hd,g is called the space of admissible covers. To motivate this approach, let us forget about the curve C and
focus on the base P1 and the branch points p1 , . . . , pb . Since P1 is rigid, a family of the above objects over
a base ∆ amounts to b sections ∆ → P1 × ∆ of the projection on the second factor. Let us pick a family in
which two of the sections cross transversely.
44
14. OCTOBER 5, 2012
One possible solution is to blow up the point of intersection. The proper transforms of the two incident
sections are disjoint but the fiber containing the exceptional divisor is now a nodal curve. This simple
example suggests we can use M0,b to parametrize the possible bases and branch points. We will expand
more on this idea later on.
LECTURE 15
October 10, 2012
15.1. Severi varieties
Suppose we would like to parametrize maps f : C → P2 from a smooth genus g curve C such that
the image of f has degree d. Looking at the Hilbert scheme, at least directly, does not help since it only
remembers the image There are constructions such as the Kontsevich space which carry out this procedure
fairly well, but they come at a significant price in the complexity of the object we are dealing with. Note
that, if we additionally require the image to be a nodal curve in P2 , then we can recover C by normalizing
the image. In other words, as long as we impose some conditions on the image, disposing of the morphism
f is not a huge waste. This idea is carried out via what we call the Severi variety.
Let us fix d ≥ 1 and g ≥ 0, and consider the projective space of all degree d plane curves
PN = P H0 (OP2 (d)) = {plane curves of degree d}
where N =
d+2
2
− 1. We define a locally closed subset
Vd,g = {reduced, irreducible curves C ⊂ P2 of degree d and geometric genus g}
We can impose a further condition, hence defining an even smaller set of curves
d−1
d,δ
2
Ud,g = U = {C ⊂ P | C reduced and irreducible degree d curve with δ =
− g nodes} ⊂ Vd,g
2
called the Severi variety. Note that we have a sequence of inclusions
Ud,g ⊂ Vd,g ⊂ V d,g ⊂ PN
so we have an embedding of Ud,g in a projective space PN for relatively small N . (The case of comparison
here is the Hilbert scheme for which the dimension of the ambient space is much larger.) There are several
basic facts about Severi varieties worth mentioning. We will present three in increasing level difficulty:
(i) Ud,g is smooth of dimension N − δ = 3d + g − 1,
(ii) Ud,g is dense in Vd,g ,
(iii) Ud,g is irreducible.
15.2. Smoothness and dimension of Severi varieties
Our next goal will be to present a self-contained proof of fact (i) above. In the beginning, let us focus
on the case δ = 1. Start by introducing the following incidence correspondence.
Φ = {(C, p) | C singular at p}
PN ⊃ V
d,1
=V
(
v
45
P2
46
15. OCTOBER 10, 2012
We would like to write things down in coordinates. Let us choose
for PN where i, j ≥ 0 and i + j ≤ d. Then

F = P aij xi y j ,

P
Φ = (aij , x, y) G = P iaij xi−1 y j ,

H=
jaij xi y j−1
affine coordinates x, y for P2 and aij


.

In other words, a point (aij , x, y) lies in Φ if the polynomial defined by the aij and its partial derivatives with
respect to both variables all vanish at the point (x, y). Let C denote the curve defined by the polynomial
with coefficients aij . We claim that if C has a node at p = (x, y), then Φ is smooth at (C, p). In fact, we
will see that Φ is always smooth but, the condition of having a node is necessary to ensure its image V is
smooth at C. Without loss of generality, we may assume x = y = 0. (In terms of the coefficients, this means
a00 = a10 = a01 = 0.) Next, we record the derivatives if F , G, H with respect to x, y, a00 , a10 , a01 at (C, p).
The partial derivatives with respect to all other aij vanish.
∂/∂x
∂/∂y
∂/∂a00
∂/∂a10
∂/∂a01
F
G
H
0
0
1
0
0
a20
a11
0
1
0
a11
a02
0
0
1
It is immediate that Φ is always non-singular. Asking for p ∈ C to be a node is equivalent to the non-vanishing
of the minor
a20 a11
a11 a02
In turn this is equivalent to the morphism π : Φ → V being an immersion, so we deduce V is smooth at C.
Remark 15.1. Note that Φ is a PN −2 -bundle over P2 which implies smoothness. This approach, however,
does not allow us to analyze the morphism Φ → V .
Furthermore, the tangent space to a point C ∈ V can be identified with the hyperplane a00 = 0. In
conclusion, if C is a plane curve with a node at p and no other singularities, then V is smooth at C with
tangent hyperplane {B ∈ PN | p ∈ B} consisting of all other curves which contain the point p.
Our next goal is to relax the assumption
δ = 1. 0Consider a curve C with nodes at p1 , . . . , pδ and no other
singularities. Note that genus(C) = d−1
− δ = h (KC ). We can build a similar incidence correspondence
2
including a morphism π : Φ → PN with image V . According to our setup, the preimage π −1 (C) contains
e → C = V (f ) to write
precisely δ points (C, p1 ), . . . , (C, pδ ). We can use the resolution of singularities ν : C
e
down canonical differentials for C. The differential
ω = g(x, y)
dx
∂f /∂y
e if and only if g has degree d − 3 or less and g(pα ) = 0 for 1 ≤ α ≤ δ. Then
is regular on C
d−1
e = h0 (K e ) = dim polynomials of degree ≤ d − 3 ≥ d − 1 − δ,
− δ = genus(C) = genus(C)
C
vanishing at p1 , . . . , pδ
2
2
and we have equalities throughout. It follows that p1 , . . . , pδ impose independent conditions on polynomials
of degree d − 3, hence also on polynomials of degree d. (This requires a small argument.) As a result V ⊂ PN
around C looks as follows.
15.2. SMOOTHNESS AND DIMENSION OF SEVERI VARIETIES
d,δ
47
There are δ smooth hyperplanes crossing normally at C and V
is the intersection of these sheets, hence
is smooth of the correct dimension. In conclusion, in a neighborhood of C the variety U d,δ is smooth of
codimension δ in PN and has tangent space {B ∈ PN | p1 , . . . , pδ ∈ B} at C.
LECTURE 16
October 12, 2012
16.1. Wrapping up parameter spaces
We have recently been focusing on the following three parameter spaces:

2

if r = 1,
simply branched covers of P
e
Hd,g,r = Severi variety of nodal curves
if r = 2,


Hilbert scheme of smooth curves if r ≥ 3.
In all three cases the expected dimension is
h(d, g, r) = (r + 1)d − (r − 3)(g − 1).
In fact, for r = 1, 2 the space is always
(i) of the expected dimension,
(ii) smooth, and
(iii) irreducible.
All three properties fail for r ≥ 3. We have also been interested in finding compactifications which are
parameter spaces. For example, the (restricted) Hilbert scheme is a projective variety. On the other hand,
the varieties it parametrizes may include arbitrarily bad singularities and degenerations. The situation is
quite similar for r = 2. For example, given a (non-reduced) plane curve we do not have a good method of
deciding whether it lies in the closure of the small Severi variety. For r = 1, there is a nice compactification
via admissible covers.
Problem 1 Find a good compactification of the spaces in question.
Remark 16.1. Kontsevich space is somewhat better than the Hilbert scheme, but it is still very badly
behaved.
Even for r = 1, 2, there are many subtle open problems.
e d,g,r denotes the open parameter space as given above, then Pic(H
e d,g,r ) ⊗ Q = 0.
Conjecture 16.2. If H
This can be shown to be false for r = 1, 2. For r ≥ 3, it is also false in general, but may be true for d
e d,g,r that dominate Mg .
large and for components of H
16.2. Moduli spaces
We would like to find a fine moduli space Mg for smooth complete curves of genus g, that is, we would
like to find a scheme Mg and an isomorphism of functors
HomSch (−, Mg ) ∼
= {families of smooth curves of genus g}.
Unfortunately, it is not possible to achieve this goal. Our plan will be to go through a potential construction
in the genus 1 case and see how the requirement above fails. Fortunately, there is a way to encode and
contain this failure via the notion of a coarse moduli space.
Let us fix g = 1. Instead of working with abstract genus 1 curves, we will assume each is endowed with
a point, what is called an elliptic curve. Since the automorphism group of a genus 1 curve acts transitively
on its points, this is not an issue.
48
16.2. MODULI SPACES
49
Given a point p on a smooth genus 1 curve C, we can look at the linear series |2p| and construct a
morphism
/ P1
|2p| : C
which is a double cover ramified at 4 points. It follows that we can give C as the vanishing locus of
y 2 = (x − a)(x − b)(x − c)(x − d)
where a, b, c, d ∈ P1 are the ramification points of |2p|. Then, we can send three of these points to 0, 1, ∞,
hence normalizing the equation of the elliptic curve to
C = Cλ : y 2 = x(x − 1)(x − λ).
Note that λ depends on a choice of the three points to send to 0, 1, ∞. For example, if we switch the
points corresponding to 0 and 1, we are sending λ to 1/λ. Analyzing all combinations carefully, we conclude
Cλ ∼
= Cλ0 if and only if
1
λ−1
λ
1
,
,
.
λ0 ∈ λ, , 1 − λ,
λ
1−λ
λ
λ−1
In other words, given an abstract branched cover, all possible choices of picking which points to send to
0, 1, ∞ produce only 6 values of λ. It looks like the symmetric group S3 is acting on U = P1 \ {0, 1, ∞} with
orbits as given above. We would like to say the moduli space of genus 1 curves is the quotient U/S3 .
Remark 16.3. The choice we are making amounts to ordering 4 points, hence we expect to see an
S4 -action. This is the case but the Klein 4-group acts trivially, so the action factors through the quotient
S3 .
Define the j-function as
(λ2 − λ + 1)3
.
λ2 (λ − 1)2
Note that the fibers of j are precisely the orbits of the S3 -action we mentioned earlier. The following diagram
summarizes this fact.
j
/ A1
U
= j
j(λ) = 256
!
U/S3
∼
=
We are lead to take M1 = A1j but it turns out this is not a fine moduli space. Our goal would be to
investigate why this is the case. Certainly, the points of M1 are in bijection with the isomorphism classes:
M1 (C) = {isomorphism classes of smooth genus 1 curves}.
Given a family C → B of smooth genus 1 curves, we get a map B → M1 . The idea is to pick sections
locally and use these to produce the map in question.
50
16. OCTOBER 12, 2012
Suppose we have a section a ∈ OB (∆) for some open ∆ ⊂ B. This allows us to present all curves over ∆ and
2-sheeted ramified covers. One of the points of ramification is given by the section a while the remaining
three yield sections b, c, d ∈ OB (∆). The family over ∆ can then be written as
y 2 = (x − a)(x − b)(x − c)(x − d).
By normalizing, we can get it in the form
y 2 = x(x − 1)(x − λ)
for λ ∈ OB (∆). Compose the following maps:
∆
λ
/ P1 \ {0, 1, ∞}
/ A1 = M1 .
j
It remains to show these compositions glue well as we vary the open ∆ and we obtain the desired morphism
B → M1 . If F denotes the functor parametrizing smooth genus 1 curves, we have shown there is a natural
transformation
/ HomSch (−, M1 ).
F
This is a bijection when evaluated on Spec C but not an isomorphism of functors.
Recall the definition of the j-function
(λ2 − λ + 1)3
.
λ2 (λ − 1)2
Recall that given a family C → B as in the previous paragraph, we constructed four sections a, b, c, d.
j(λ) = 256
But the map j : P1 \ {0, 1, ∞} → A1j has triple ramification over the roots of λ2 − λ + 1. In particular, when
evaluated on a family of curves the j-function must always have triple zeros over these roots. This shows
that F → HomSch (−, M1 ) is not in general surjective.
We can also show the natural transformation is not injective in general either. Rick an unramified double
cover B 0 → B and choose a genus 1 curve E. Let τ : B 0 → B 0 denote the automorphism exchanging the two
sheets of the cover B 0 → B. Let ι : E → E be the automorphism defined by negation in some group law of
E. We can then construct a family of genus 1 curves over B given as
B 0 × E/(τ, ι)
/ B.
All fibers are isomorphic to E, so the associated map to A1j is constant. On the other hand, it is possible to
show this is not the trivial family B × E.
LECTURE 17
October 15, 2012
17.1. Moduli of elliptic curves
Recall we attempted to construct the moduli space of elliptic curves. Any such curve can be represented
as
Cλ = {y 2 = x(x − 1)(x − λ)}.
Furthermore Cλ ∼
= Cλ0 if and only if
λ0 ∈
1
λ−1
λ
1
,
,
λ, 1 − λ, ,
λ 1−λ
λ
λ−1
.
The sets in question can be realized as the orbits of an S3 -action on U = P1 \ {0, 1, ∞}. Finally, we can take
M1 = U/S3 = A1j where
(λ2 − λ + 1)2
.
j(λ) =
λ2 (λ − 1)2
We also showed this is not a fine moduli space but a lot of the necessary properties are there.
(i) There is a natural transformation of functors
/ HomSch (−, M1 ).
ϕ : F = {elliptic curves}
(ii) There is an induced bijection
ϕ(Spec C) : F (Spec C)
∼
=
/ M1 (Spec C).
Unfortunately, in general ϕ is neither surjective nor surjective.
(i) Surjectivity: A map B → M1 is the j-function associated to a family C → B only is all zeros are triple.
In particular, there does not exist a universal family over M1 .
(ii) Injectivity: There exist non-trivial families with constant fiber (so called isotrivial families).
17.2. Deligne-Mumford moduli spaces
Our next goal is to formalize the properties of M1 we observed, and define a suitable class of moduli
spaces.
Definition 17.1. Let F : Schop → Set be a moduli functor (a functor which encodes data about families
of some objects), and M be a scheme. We say M is a Deligne-Mumford moduli space for F (abbreviated a
DM space) if
(i) there exists a natural transformation ϕ : F → HomSch (−, M),
(ii) ϕ(Spec C) is a bijection,
(iii) ϕ is an isomorphism up to finite (potentially ramified) covers. The last statement means the following.
(a) Given a map f : B → M there exists a finite cover i : B 0 → B such that f ◦ i is in the image of
ϕ(B 0 ).
(b) Given two families C → B and D → B such that
ϕ(C) = ϕ(D) : B
51
/ M,
52
17. OCTOBER 15, 2012
then there exists a finite cover B 0 → B which induces the following isomorphism of families over
B0.
∼
=
/ D ×B B 0
C ×B B 0
$
B0
z
Remark 17.2. Being a DM moduli space is stronger being a coarse moduli space.
Theorem 17.3 (Deligne-Mumford ’69, Part I). There exists a DM moduli space Mg for smooth curves
of genus g curves.
There are several avenues we should investigate further:
(i) generalizing to a DM moduli space of n-pointed curves Mg,n ,
(ii) deducing basic properties of Mg (dimension, irreducibility),
(iii) how to handle the failure of Mg to be a fine moduli space,
(iv) how do we construct Mg , i.e., how is the theorem proved.
We can define the generalization in question as
C smooth genus g curve,
Mg = (C, p1 , . . . , pn ) .
p1 , . . . , pn ∈ C distinct point
Formally, a family over B would be
CV V
σ1 ,...,σn
B
a family of curves C → B as before and n disjoint sections σi : B → C.
Remark 17.4. Distinct points are necessary to arrive at a nice compactification. Otherwise, it is an
irrelevant condition.
To establish the basic properties of Mg we will look at the incidence correspondence
0
Hd,g
π
{
Pb \ ∆
!
Mg
where b = 2g − 2 + d and
0
Hd,g
=
simply branched covers f : C → P1 of degree d,
.
C a smooth curve of genus g
For d ≥ 2g + 1 the right map is surjective. Let us compute the dimension of the fibers of π. Given a smooth
curve C of genus g, then we have an inclusion of an open set
π −1 (C) ⊂ {rational functions f on C of degree d},
that is, for d g a general rational function yields a simply branched map to P1 . To choose such a rational
function f we start by choosing a polar divisor D = (f )∞ which is a d-dimensional family of choices. By
Riemann-Roch, choosing an actual function f ∈ H0 (OC (D)) adds dimension d + g − 1. The fibers have
dimension 2d − g + 1, so
dim Mg = dim H0d,g −(2d − g + 1) = (2g − 2 + 2d) − (2d − g + 1) = 3g − 3.
It is possible to carry over an analogous computation with the Severi variety instead of Hurwitz space. The
basic point is to study Mg we need to do so in the context of a second space which he understand better.
17.2. DELIGNE-MUMFORD MODULI SPACES
By a similar token
dim Mg,n = 3g − 3 + n.
This holds whenever the automorphism group of (C, p1 , . . . , pn ) is finite. This excludes the cases
M0,0 , M0,1 , M0,2 , M1,0 .
53
LECTURE 18
October 17, 2012
18.1. The moduli space of genus g curves
We will start by recalling a concept introduced last time.
Definition 18.1. Given a moduli functor F : Schop → Set, we say a scheme M is a Deligne-Mumford
moduli space for F if the following conditions hold.
(i) There exists a natural transformation ϕ : F → HomSch (−, M),
(ii) ϕ(Spec C) is bijective,
(iii) Given a morphism f : B → M, there exists a finite (potentially ramified) map π : B 0 → B such that
f ◦ π lies in the image of ϕ(B 0 ).
(iv) Given two families C → B and D → B such that ϕ(B)(C) = ϕ(B)(D), then there exist a finite morphism
B 0 → B for which the pullbacks are isomorphic.
∼
=
C ×B C 0
$
B0
/ D ×B B 0
z
Theorem 18.2 (Deligne-Mumford). There exists a DM-moduli space Mg for smooth curves of genus g.
There are several questions worth listing.
(i) What do we do about the fact Mg is not a fine moduli space? For example, there is no universal family.
(ii) How do we construct Mg ?
(iii) Can we compactify Mg ?
There are three approaches to answering the first of these questions.
(a) The crux lies in identifying the reason representability fails, namely, that there are curves with automorphisms. We can restrict to the space of automorphism-free curves M0g . This is an opens subset
of Mg whose compliment has codimension g − 2. It is not hard to see M0g is a fine moduli space for
automorphism free-curves.
(b) Alternatively, we can rigidify the moduli space. This amounts to looking at curves with some finite
additional structure which prevents automorphisms.
One such approach is the study of level structure. The space of genus g curves with level m ≥ 3
structure is
{(C, σ1 , . . . , σ2g ) | σ1 , . . . , σ2g is a symplectic basis for H1 (C, Z/m)}.
[m]
This kills all automorphisms and we arrive at a fine moduli space Mg of such objects (C, σi ). It is a
finite (Galois) cover of Mg whose ramification locus lies over the locus of curves with automorphisms.
Remark 18.3. Note that the definition of level structure makes sense for m = 2, but the hyperelliptic
curves retain their involution even after adding the level 2 structure. We assume m ≥ 3 in order to make
sure all objects are automorphism free.
While level 2 structure has its shortcomings, we have already seen it appear in a previous discussion.
54
18.2. COMPACTIFYING THE MODULI SPACE OF GENUS G CURVES
55
Example 18.4. Recall the space U = P1λ \ {0, 1, ∞} of elliptic curves with a specified ordering of the
branch points. Note that the branch points consist of the order 2 torsion points which in turn correspond
[2]
to elements of H1 (C, Z/2). In other words U = M1 . Note that while U is not a fine moduli space due
to the existence of automorphisms, there exists a universal family given by {y 2 = x(x − 1)(x − λ)}.
(c) Finally, we can resort to the language of stacks to remedy the failure of representability. The philosophy
is to put all blame on the category of schemes, namely, it is not large enough to contain a suitable fine
moduli space. We are naturally lead to look for a newer notion of space which encompasses all schemes
but is strictly more inclusive than it. We will not attempt to carry further this thread of through.
18.2. Compactifying the moduli space of genus g curves
We would like to find a projective variety Mg which includes Mg as an open subset. Furthermore, we
would like produce a modular compactification, i.e., the points of Mg \ Mg should correspond to curves. In
other words, we would like Mg to be a moduli space for a larger class of curves.
Recall the valuative criterion for properness. Let ∆× ⊂ ∆ be the inclusion of the punctured disc in the
disc. In an algebraic setting, ∆ would be the spectrum of a DVR and ∆× the compliment of its maximal
ideal. A variety X is proper if and only if all morphisms ∆× → X extend uniquely to a morphism ∆ → X.
The key is in interpreting this in the context of Mg . Given a family C × → ∆× of smooth curves, after
a finite base change ∆× → ∆× , we should be able to extend it to a family C → ∆ in the broader class.
Allowing a finite base change is necessary since Mg is not a fine moduli space.
The existence and uniqueness in the statement above are competing notions. Of course, we can allow
all singular curves, but then uniqueness does not hold, hence the moduli space in question is not separated.
That is, we need to allow some singular curves in a limited fashion. Deligne and Mumford found a solution
of this problem. They called a connected curve C stable if
(i) C has only nodes as singularities,
(ii) # Aut(C) < ∞.
Theorem 18.5 (Deligne-Mumford). There exists a proper (projective) DM moduli space Mg for stable
curves of genus g.
LECTURE 19
October 19, 2012
19.1. Compactifying Mg
Our aim is to compactify the moduli space of genus g curves Mg . We would like to find a larger class Σg
of curves for which there exists a DM moduli space Mg for curves of this class. Furthermore, we require that
Mg is proper. By the valuative criterion for properness, we can translate the last condition. Given a family
C × → ∆× of smooth curves over ∆× , after a finite base change ∆× → ∆× , there should exist a unique way
to fill to a family over ∆. The central fiber C0 should be in Σg and unique under the given conditions.
Remark 19.1. Initially, it might seem we need to fill any family in Σg but it suffices to do so for smooth
curves only (assuming Mg ⊂ Mg is going to be a dense open).
Deligne and Mumford proposed a solution to this problem:
C connected, of arithmetic genus g,
Σg = {stable curves} =
.
nodal, and # Aut(C) < ∞
Finiteness of the automorphism group is equivalent to two conditions:
(i) the normalization of every rational component should have at least three points which are preimages
of nodes in C,
(ii) the normalization of every genus 1 component should have at least one point which is the preimage of
a node in C.
We can introduce markings to arrive at a class Σg,n of stable pointed curves:


C connected, of arithmetic genus g,

Σg,n = {stable pointed curves} = (C, p1 , . . . , pn ) p1 , . . . , pn ∈ Csm distinct points, .


# Aut(C, p1 , . . . , pn ) < ∞
The two conditions we listed above translate similarly to the pointed setting.
19.2. Examples of stable reduction
Example 19.2. Suppose we have a smooth curve C and two points p and q approach each other. For
simplicity, assume p is fixed and q is varying.
The underlying curve of the family will stay constant, so only the sections p and q matter. To keep the
sections p and q disjoint, we can blow their intersection. The exceptional divisor is a rational component
sprouted off the point at which p and q collide. The proper transforms of the sections are then two distinct
points in the exceptional fiber. The following diagrams illustrate the blow-up process and the resulting
family.
56
19.2. EXAMPLES OF STABLE REDUCTION
57
Remark 19.3. If p and q did not intersect transversely, their proper transforms will still intersect a
blow-up. We repeat the procedure until the proper transforms are disjoint. All exceptional fibers until the
last one can be contracted, leading to a result analogous to the simple transverse intersection requiring a
single blow-up.
Remark 19.4. The picture can get more interesting if more than two points approach each other. For
example, if three points p, q and r approach each other at “constant speeds”, blowup up once suffices to
arrive at a stable central fiber. The rational exceptional curve intersects the original curve in one point, and
contains the proper transforms of p, q and r. Four points on P1 have moduli, and the particular configuration
we arrive at depends on the ratio of the speeds of approach.
It is possible to have even richer geometry in the central fiber if more marked points approach each
other.
Example 19.5. A similar scenario unveils if a marked point approaches a node. For simplicity, assume
we start with a curve C with a unique node p0 as singularity. A point p is moving towards p and we are
interested in the stable limit of this family. The normalization C ν of C has two points q and r lying over
p0 . We can normalize all curves in the family at the same time to arrive at a family with constant fiber
C ν and three sections, two constant ones q and r, and one moving p. Depending on the branch p is using
to approach p0 , its preimage in the normalization may be approaching either of q or r. Without loss of
generality, assume p is intersecting with q.
58
19. OCTOBER 19, 2012
Blowing up the point of intersection sprouts off a rational component E ∼
= P1 in the central fiber. Under
mild hypotheses, the proper intersections of p and q will not intersect in E. To get back to the original
family of nodal curves, we can identify the sections q and r. The central fiber is a copy of C ν with a rational
component attached at q and r which contains the marking p as the third special point on it. Note that
three points on P1 do not vary in families, so the description we provided is unambiguous.
Example 19.6. Suppose we have a family C → ∆ with smooth fibers for t 6= 0 and C0 has a single cusp.
For simplicity, let us assume the total space C is smooth. Let C0ν denote the normalization of the central
fiber C0 .
Blowing up the cusp results in a central fiber consisting of C0ν meeting to order 2 the exceptional divisor E1
(this singularity is called a tacnode). Blowing up the common point forces C0ν and E1 to meet transversely.
The new exceptional divisor E2 also passes through the same point. Finally, blowing up the common point
makes C0ν , E1 and E2 disjoint, and they all cross the newest exceptional divisor E3 . We might be tempted to
think this completes the process but all exceptional curves we produced have higher multiplicities (indicated
in circles next to the curves in the diagram above). To get rid of the multiplicities we have to make base
changes and normalize. We will finish this procedure next time.
LECTURE 20
October 22, 2012
20.1. Base change and normalization
Reinterpreting the statement of stable reduction through the valuative criterion of properness, we conclude that the Deligne-Mumford moduli space of stable curves is proper. Last time we started talking about
applying stable reduction to a family with a cuspidal fiber. By performing three blowups. we were lead to
the following family whose central fiber consists of four components.
At this point, the central fiber is set-theoretically nodal. Three of the components have higher multiplicities
2, 3 and 6. To bring these down and arrive at a family with a reduced central fiber, we perform a sequence
of base changes and normalization. Instead of focusing on the details in the case of the cuspidal family, we
will discuss the procedure in slightly greater generality.
Suppose we have a family of curves with central fiber C0 . Furthermore, let us assume the total space C
of this family is smooth. Consider a component D of C0 and a point p on it. Let t denote the coordinate on
the base of this family ∆t .
If D has multiplicity 1 in C0 , then we can find coordinates (x, y, z) around p such that D = V (y) and
y = t. Performing a base change of order 2 amounts to introducing a variable s satisfying s2 = t. In the
new set of coordinates (x, y, s), we have s2 = t = y. This is a smooth 2-sheeted cover of C branched along
D. In other words, D remains unchanged after the base change. The space we arrive at is smooth around
p, so normalization does not change anything.
Next, suppose D has multiplicity 2 in C0 . This means we have local coordinates (x, y, t) satisfying
y = y 2 . Applying the order 2 base change as above, we arrive at s2 = y 2 . Drawing a diagram around p,
we see there are two sheets crossing transversely. These are separated after normalization. It follows that
locally around p, we see two copies of D after the base change and normalization.
59
60
20. OCTOBER 22, 2012
Finally, let us see what happens if D has multiplicity 3 in C0 . Originally, we have t = y 3 , after the
base change, we arrive at s2 = y 3 . The resulting surface looks like a cuspidal curve crossed with the affine
line. Normalizing produces a smooth surface, hence D appears once, just as in the multiplicity one case. In
general, the behavior of the curve D under an order 2 base change depends on the parity of the multiplicity
of D in C0 .
The analysis above carries through if we consider a base change of prime order p.
(i) If the multiplicity of D is not divisible by p, then performing the base change leads to a cover ramified
over D. This means that D remains unchanged, and so does its multiplicity.
(ii) If the multiplicity of D is divisible by p, then performing the base change leads to a cover which is p
to 1 onto D. In other words, we replace D by a degree p cover and divide its multiplicity by p.
20.2. The cuspidal example
We are now ready to complete the stable reduction we already started.
(1) The first step is to perform an order 2 base change and normalization. The curve E3 is replaced by
a double cover E30 branched at 2 points, hence it is a smooth rational curve. The preimage of E1
consists of two disjoint copies E10 and E100 , both smooth and rational. Finally, E2 is left untouched.
(2) The next step is to perform an order 3 base change and normalization. The curve E2 is replaced
by an unbranched triple cover, hence three disjoint copies E20 , E200 , E2000 . The curves E10 and E100
stay put. Finally, E30 is replaced by a triple cover branched at 3 points, hence is a smooth genus 1
curve E = E300 . Furthermore, it can be shown that the smoothness of the total space of the family
implies j(E) = 0.
(3) At this point we have arrived at a family with reduced nodal fiber. The only issue is the central
fiber is not stable. To do that, we blow down the offending components E10 , E100 , E20 , E200 and E2000 .
To summarize, we performed stable reduction to a family with a cuspidal central fiber C0 and smooth
total space. The resulting family has central fiber C0ν and a genus 1 curve E with j(E) = 0 attached at the
cusp point.
LECTURE 21
October 24, 2012
21.1. More examples of stable reduction
Last time we applied stable reduction to a family of curves C → ∆ with central fiber a cuspidal curve.
This required several birational transformations and a degree 6 base change. The central fiber of the resulting
family was identified to be the union of C0ν , the normalization of the central fiber beforehand, with an elliptic
curve E attached at the preimage of the cusp. If C is smooth, then j(E) = 0. Otherwise, the value j(E)
may be arbitrary.
Remark 21.1. The stable limit, i.e., the central fiber in the stable family, does not only depend on the
central fiber before but on an infinitesimal neighborhood.
We would like to discuss several other families of interest.
Example 21.2 (Tacnode). Consider a family of curves C → ∆ with smooth total space whose general
fiber is smooth, and the central fiber contains a tacnode. We start by blowing up the tacnode once to arrive
an exceptional divisor E1 of multiplicity 2. The resulting point of intersection has multiplicity 4, so blowing
it up produced a multiplicity 4 exceptional divisor E2 . We can now make a base change of order 2 and
normalize. The last operation leaves the curve C0ν untouched. The preimage of E2 , denoted E20 , is a branch
cover of E2 at 2 points, so E20 is still rational. The preimage of E1 is the disjoint union of two copies of it
E10 ∪ E100 . The only higher multiplicity in the resulting central fiber is 2 occurring along E20 . We are lead
to perform another order 2 base change. It can be shown the preimage of E20 , denoted by E, is an elliptic
curve. The curves E10 and E200 remain the same, so we can blow them down. The resulting central fiber is the
normalization of C0ν with a genus 1 curve attached at the two preimages of the tacnode in the normalization.
Such a curve is called an elliptic bridge.
If the total space C we started with is smooth, one can show that j(E) = 1728, otherwise, it may be
arbitrary.
61
62
21. OCTOBER 24, 2012
In general, an elliptic bridge is a genus 1 components attached to the remainder of the curve in two
points.
Remark 21.3. It can be shown the miniversal deformation space of a cusp is two dimensional and can
be realized (locally) as the family
y 2 = x3 + ax + b
over the 2-disc ∆a,b . We get a map
∆× = ∆a,b \ {(0, 0)}
/ Mg
which we can treat as a rational map
∆a,b
/ Mg .
Resolving this rational map amounts to understanding the dependence of the stable limit on the family. (It
takes three blowups to resolve this rational morphism.)
Example 21.4 (Triple point). Applying stable reduction to a family with central fiber C0 having a
(planar) triple point p produces a family with central fiber consisting of C0ν and a genus 1 curve E attached
at the three preimages of p. As before, j(E) = 0 if the total space is smooth, and may be arbitrary otherwise.
Remark 21.5. Suppose that in the process of stable reduction we encounter a rational curve with 2
special points. Blowing it down leads to a nodal central fiber. Locally around the node the total space of
21.1. MORE EXAMPLES OF STABLE REDUCTION
63
the family has the form xy − t = 0, so it is smooth. This is may not be the case however. More generally,
we can encounter a chain of k rational curves as shown below (k ≥ 2). Blowing them down, we get a
Ak−1 -singularity in the total space of the family.
Finally, it is interesting to work through an example whose central fiber is non-reduced.
Example 21.6. Consider a family of plane quartics specializing to a double conic. In other words, we
pick a quadric Q(x, y, z) and a quartic F (x, y, z) then consider the family
{Q2 + tF } ⊂ ∆t × P2 .
Note that the total space of the family is not smooth. Precisely, it is singular at the 8 base points of the
pencil over the central fiber.
Blowing up these 8 points produces 8 multiplicity 1 exceptional curves E1 , . . . , E8 . The total space is smooth,
so we can perform an order 2 base change and normalize. The resulting central fiber consists of a genus 3
curve C (the double cover of the conic {Q = 0} ramified at the 8 base points) and 8 exceptional rational
curves E1 , . . . , E8 attached at the points of ramification. We can blow down the exceptional divisors to
arrive at a family with smooth central fiber. The only difference is the corresponding central fiber is a genus
3 hyperelliptic curve.
Remark 21.7. The process of replacing a ribbon by a hyperelliptic curve may be reversed by looking
at the canonical model of the hyperelliptic curve.
LECTURE 22
October 26, 2012
22.1. Remarks on stable reduction
We worked through several examples of stable reduction so far. To summarize, and to indicate how the
general result is proved, we outline the procedure.
(1)
(2)
(3)
(4)
Start with a family whose general fiber is smooth and so is the total space.
Perform blow ups until the central fiber is (set-theoretically) nodal.
Apply base change and normalize until all multiplicities in the central fiber are 1.
Blow down any resulting unstable curves in the central fiber.
Remark 22.1. As presented above, there is one detail missing in step (3). It is possible that a normal
singularity of the form z p = xl y k arises and we have to handle this separately. We will not discuss this in
more detail.
Remark 22.2. It is possible to adapt the procedure above to a setting where the the general fiber is
nodal and not necessarily smooth. To do so, we can first “undo” the nodes while remembering the locus of
points lying over them, perform stable reduction, and then identify the node points back.
Suppose C → ∆ is a 1-parameter of curves whose general fiber is smooth and the central fiber C0 is
nodal at p ∈ C0 . Then there exists a local coordinate in C near p such that
C = V (xy − tl )
for some k ≥ 1. If k = 1 the total space is smooth. Otherwise, C has an Ak−1 -singularity at p. When k = 2,
C looks like a cone over a conic locally around p. Blowing up p we obtain a (−2)-exceptional curve E.
When k = 3, blowing up p leads to a chain of two rational curves and the resulting surface is smooth.
For k > 3, blowing up also introduces a chain of two rational curves, but the resulting surface is not smooth.
Actually, around the common point of these two rational curves, the new surface has the form V (xy − tk−2 ),
that is, there is a Ak−3 -singularity. This is strictly simpler and we can repeat the process until we arrive at
a smooth surface. The result is a chain of k − 1 exceptional curves.
64
22.2. GEOMETRY OF SINGULAR CURVES
65
Note that while the total space is smooth, the central fiber is not stable. In general there is a trade-off
between these two competing notions. Our next goal will be to describe a variant of stable reduction which
always produces smooth total spaces.
Definition 22.3. We say a curve C is semistable if C is nodal and every smooth rational component
of C meets the rest of C in ≥ 2 points.
Note that the example above shows the moduli space of semistable curves is not separated. If however,
we are interested in a smooth total space (e.g., for the purposes of intersection theory), then it could be
useful to work with semistable curves. A result analogous to stable reduction holds in this setting.
Theorem 22.4 (Semistable reduction). Given any family of curves C → ∆ whose general fiber is smooth,
then after a sequence of birational transformations and base changes, we can arrive at a family C 0 → ∆ whose
fibers are semistable and the total space C 0 is smooth.
Finally, there is one more flavor of stable reduction worth mentioning.
Theorem 22.5 (Nodal reduction). Given a family of curves
C ⊂ ∆ × Pn
∆
whose general fiber is smooth, after a sequence of birational modifications and base changes we can arrive at
a family
/ C ⊂ ∆ × Pn
C0
∆
/∆
whose fibers are nodal.
22.2. Geometry of singular curves
Let C be a reduced curve singular at p ∈ C, and suppose C ν denotes the normalization of C at p (if p is
the unique singularity of C, then C ν is the normalization of C). There is a natural morphism π : C ν → C.
We would like to compare the arithmetic genera pa (C ν ) and pa (C). To do so, consider the short exact
sequence
/ OC
/ π∗ OC ν
/ Fp
/ 0.
0
We know that Fp is a skyscraper sheaf supported over p. We define the δ-invariant of the singularity p as
δ = rank Fp = h0 (Fp ).
This is easy to compute in small cases.
(i) If p is a node, then δ = 1.
(ii) If p is a cusp, then δ = 1.
66
22. OCTOBER 26, 2012
(iii) If p is a tacnode, then δ = 2.
(iv) If p is a planar triple point, then δ = 3.
(v) If p is a spatial triple point, then δ = 2.
Since π is finite, it preserves cohomology, hence χ(OC ν ) = χ(π∗ OC ν ). Combining this observation with the
definition of the arithmetic genus pa (C) = 1 − χ(OC ), we obtain
χ(OC ν ) = χ(OC ) + δ(p),
hence
pa (C) = pa (C ν ) + δ(p).
S
Let
us assume C is nodal with δ nodes, and write C = Ci , a union of k irreducible components. Then
F
C ν = Ciν , and
X
pa (C) = pa (C ν ) + δ =
g(Ci ) − k + 1 + δ,
where g(Ci ) = pa (Ciν ). Put differently
X
g(Ci ) = pa (C) + k − 1 − δ ≤ pa (C),
the inequality depending on connectedness of C.
A more sensitive tool for investigating the topology of a nodal curve is its dual graph. Given a curve C,
the vertices of this graph Γ correspond to irreducible components of C. For each node of C, there is an edge
of Γ joining the vertices corresponding to the components of the two branches of the node. For example,
here is the diagram of a nodal curve and its dual graph.
Furthermore, we can enhance the information carried by the graph by weighing each vertex by the geometric
genus of the associated component.
LECTURE 23
October 31, 2012
23.1. Stratifying Mg by topological type
S
Let C beFa nodal curve with δ nodes, and C = Ci be its irreducible decomposition into k components.
Then C ν = Ciν and
X
g(C) =
g(Ciν ) − k + 1 + δ.
P
It is worth noting the first three terms on the right
g(Ciν ) − k + 1 provide the genus of C ν , the disjoint
ν
union of the normalizations Ci .
The dual graph of a nodal curve C is a weighted graph ΓC such that:
(i) for every irreducible component Ci of C, we have a vertex with weight gi = g(Ciν ),
(ii) for every nodal of C we draw an edge between the vertices corresponding to the two components the
node touches.
We can stratify Mg by the combinatorial type of the dual graph, or, put equivalently, by topological type.
We set
MΓ = {C ∈ Mg | ΓC = Γ} ⊂ Mg
which are locally closed subsets of Mg . The closure of one MΓ is the union of other such which precisely is
the meaning of being a stratification.
Example 23.1. We list all possible strata of M2 by drawing a real-analytic picture, a complex-analytic
doodle, and the associated dual graph for each one.
The stratification looks as follows.
67
68
23. OCTOBER 31, 2012
Suppose we would like to compute the dimension of a stratum of type Γ = ΓC . Let ni denote the number
of points of Ciν lying over nodes. It is not hard to see the desired dimension is
X
X
dim MΓ =
(3gi − 3 + ni ) = 3
gi − 3k + 2δ = 3g − 3 − δ.
In other words, the dimension of a stratum classifying curves with δ nodes is precisely δ. In particular,
codimension 1 strata correspond to curves with 1 node. These divisors will play a crucial role when we study
the geometry of Mg more closely.
23.2. Deformation of planar curve singularities
Without the planar hypothesis, arbitrary singularities can be very hard to deal with. If, however, we
assume a singular point p on a curve C is planar, that is, dim Tp C = 2, there is a lot more we can say about
its deformations. The hypothesis dim Tp C = 2 means that C can be locally embedded in a surface, hence
the name planarity.
As before, a deformation of a singular curve (C, p) is a family
C0
0
⊂
C∼
=C
∈
∆
modulo isomorphisms over étale neighborhoods of the base and isomorphisms of étale neighborhoods of
p∈C∼
= C0 .
23.2. DEFORMATION OF PLANAR CURVE SINGULARITIES
69
There is a key result which describes the miniversal deformation space of a planar singularity. Since the
issue is local, we consider a polynomial f ∈ k[x, y] such that C = V (f ) ⊂ A2 is a reduced curve with isolated
singularity at p = (0, 0) ∈ C. We introduce the Jacobian ideal
∂f ∂f
J=
,
⊂ C[x, y](x,y) = OA2 ,p .
∂x ∂y
The fact p ∈ C implies that J has a finite index in OA2 ,p . We are now ready to state the main result.
Theorem 23.2. Using the notation as above, the miniversal deformation space of the pair (C, p) is
/ ∆t1 ,...,t
k
V (f + t1 f1 + · · · tk fk )
where f1 , . . . , fk ∈ C[x, y] form a basis of OA2 ,p /J.
Example 23.3. The local equation of a node is f (x, y) = xy, so the Jacobian ideal is J = (x, y).
Applying the result above, the miniversal deformation space is
V (xy − t)
/ ∆t
In particular, any family of curves C → B with C0 nodal at p is locally a pullback of the one above. More
specifically, around p the variety C is locally given by an equation xy − α where α is a regular function on
B. From here, we can deduce that in any family of nodal curves the locus of nodal curves has codimension
≤ 1 in B. Furthermore, if the codimension is 1 and B is reduced, then the locus is a Cartier divisor in B.
LECTURE 24
November 2, 2012
24.1. Examples of (mini)versal deformation spaces
We have already seen the miniversal deformation space of a node V (xy) is given as follows.
C = V (xy − t)
∆t
Our next example of interest is a cuspidal singularity V (y 2 − x3 ). The partial derivatives are 2y and 3x2 ,
so the quotient of the Jacobian ideal is generated by 1 and x as a vector space. We deduce the miniversal
deformation space is 2-dimensional.
C = V (y 2 − x3 − ax − b) ⊂ ∆a,b × A2x,y
∆a,b
The general fiber is smooth, the fiber over (0, 0) is cuspidal, and nodes occur exactly over the discriminant
locus 4a3 + 27b2 = 0. The following diagram describes the situation pictorially.
Suppose we restrict this family to a line through (0, 0) and apply stable reduction. If the line has nonzero slope, then the total space of the family is smooth and stable reduction replaces the central fiber by
its normalization and an elliptic tail with j-invariant 0 attached at the cusp point. A line of slope 0 results
in a stable central fiber which attaches a rational nodal curve to the cusp point. This procedure can be
interpreted as a blowup of ∆a,b at (0, 0) and is the first step in resolving the rational map
/ Mg .
∆a,b
70
24.2. DUALIZING SHEAVES
71
On the other hand, it is we still do not have an honest regular map. In fact, it takes two more blowups to
reach the final result.
As our next example, let us tackle a tacnode V (y 2 − x4 ). The miniversal deformation space is the
following.
C = V (y 2 − x4 − ax2 − bx − c)
∆a,b,c
We can ask the same set of questions.
Question 24.1. What fibers appear in he family? Describe the locus where each fiber appears.
As before, singularities in the fibers are governed by the roots of x4 + ax2 + bx + c. We summarize the
possible root structures, the fibers they correspond to, and the dimensions of the corresponding loci in the
following table.
Root structure
Fiber
Dimension in ∆a,b,c
since quadruple root
tacnodal curve
0
triple root
cuspidal curve
1
two double roots
curve with two nodes 1
one double root, two single ones curve with one node 2
distinct roots
smooth curve
3
The really difficult questions remains resolving the rational map ∆a,b,c 99K Mg .
Note that in the previous examples the polynomial defining the singularity contained only y 2 so the versal
deformation space and the corresponding degenerations are governed by a polynomial in a single variable x.
This is not the case in general. We will consider the smallest example which breaks away from the patters,
that is, a planar triple point V (y 3 − x3 ). A brief computation shows the versal deformation space is the
following.
C = V (y 3 − x3 − axy − bx − cy − d)
∆a,b,c,d
The analysis in this case is considerable more involved. A good question to start with is understanding what
singular fibers occur. Here is a list containing several of the possibilities:
(1) three nodes,
(2) two nodes,
(3) one node,
(4) a cusp,
(5) a cusp and a node,
(6) two cusps, etc.
24.2. Dualizing sheaves
We have already seen that in order to work with a compact moduli space of curves Mg , we have to allow
some singular curves. We have a large menagerie of tools to approach smooth curves, but it is not clear
which of these port over to a singular case, even if we are only interested in nodal curves. One of the most
important tools we would like to employ is the canonical bundle. For a large family of singular curves, there
exists an analogue called the dualizing sheaf. We present a list of several of the most important properties
of the canonical bundle which carry over to the dualizing sheaf:
(i) uniqueness,
72
24. NOVEMBER 2, 2012
(ii) the Riemann-Roch Theorem,
(iii) good behavior in families.
Definition 24.2. Let C be a reduced curve and π : C ν → C the associated normalization morphism.
Consider an isolated singular point p ∈ C and U a neighborhood of it. The sections of the dualizing sheaf
ωC of C on U are defined to be
ω is a meromorphic differential on π −1 (U ) such
P
.
ωC (U ) = ω that q∈π−1 (p) Resq (f ω) = 0 for all f ∈ OC (U )
LECTURE 25
November 5, 2012
25.1. A proof of Riemann-Roch
One of the key properties the dualizing sheaf possesses is it allows a version of Riemann-Roch to hold
for singular curves. In order to appreciate the definition of the dualizing sheaf and see how it fits in such a
result, we will go through a bogus proof of Riemann-Roch.
P
pi of degree d. For simplicity, we assume each point pi appears with
Proof. Consider a divisor D =
multiplicity 1. We study the vector space of meromorphic functions with allow at worst simple poles over
the pi .
L(D) = H0 (OC (D)) = {f ∈ M(C) | (f ) + D ≥ 0}
Above M(C) denotes the vector space of meromorphic functions on the curve C. Given f ∈ L(D) and a
coordinate zi around pi , we can express f as
a−1
f (zi ) =
+ a0 + a1 zi + · · · .
zi
The polar part of f at pi refers to a−1 /zi . Suppose we fix coordinates zi around each pi . Given a function
f ∈ L(D), we have a collection of d polar parts a−1 /zi . Reversing the question, we can ask for which
collections of polar parts there exists a function f . In this way we obtain an inequality
dim L(D) ≤ 1 + d − (g − dim L(K − D)).
Before going any further, let us explain where each term comes from. Note that adding a scalar does not
change the polar parts of a function, hence the first term 1. Next, specifying a polar part amounts to a
producing a single coefficient, so specifying d polar parts gives us the second term d. To understand the last
term, we recall that any rational differential ϕ on C satisfies
X
Resp ϕ = 0.
p∈C
0
In particular, given f ∈ L(D) and ω ∈ H (K), we get an equality
X
Resp (f ω) = 0
p∈C
which is a linear relation on the polar parts. Since dim H0 (K) = g, we have g such relation, though some
may be redundant. For example, if we start with a differential ω vanishing at all pi the resulting relation
is trivial. The dimension of such differentials is dim H0 (K − D) = dim L(K − D), hence the inequality we
wrote above. Using the same logic to the residual divisor K − D we arrive at a second inequality
dim L(K − D) ≤ 1 + (2g − 2 − d) − (g − dim L(D)).
Adding the two inequalities, we obtain
dim L(D) + dim L(K − D) ≤ dim L(D) + dim L(K − D)
which is an equality, hence both inequalities we used are also forces to be equalities.
73
74
25. NOVEMBER 5, 2012
Remark 25.1. The only issue with the proof above is we assumes that both D and K − D are effective.
Besides that, with a little more work one can get rid of the hypothesis all points appearing in D have
multiplicity 1.
25.2. Dualizing sheaves
We are now ready to define the dualizing sheaf on a (possibly) singular reduced curve C.
Definition 25.2. Let C be a reduced curve and π : C ν → C the normalization morphism. The dualizing
sheaf ωC of C has sections given by
rational differentials ϕ on π −1 (U ) such
P that for all regular functions .
ωC (U ) =
f ∈ OC (U ) and all p ∈ U , we have q∈π−1 (p) Resq (f ϕ) = 0
Example 25.3. Let p ∈ C be a node. From the definition of ωC it may seem as if we are allowing
rational differentials of arbitrary poles, but that is not the case. On a neighborhood of the node, we can
have a function vanishing simply on one branch and at arbitrary order on the other. Using such a function
in the definition, it follows that a section of ωC can only have simple poles.
If we consider the node at (0, 0) of V (y 2 − x2 ), then dx/y is an example of a suitable differential.
Example 25.4. Next, we would like to look at a cusp. Suppose the cusp is locally of the form V (y 2 − x3 )
and its normalization is t 7→ (t2 , t3 ). The functions upstairs are power series in t, while those downstairs are
those without a t term. The definition prevents from having a non-zero residue over the cusp. We have a
double pole but no higher order. An example of a suitable differential is dt/t2 = dx/2y.
Example 25.5. Consider a tacnode locally of the form V (y 2 − x4 ). The normalization has two branches
t 7→ (t, t2 ) and t 7→ (t, −t2 ). A function downstairs consists of two power series in t which agree up to first
order. The differentials of the dualizing sheaf over the tacnode can have double poles but no triple ones.
Such an example is the pair (dt/t2 , −dt/t2 ) which is the same as dx/y up to a scalar factor.
In each of the examples we considered, there are several observations worth mentioning:
(i) the dualizing sheaf ωC is locally free of rank 1,
(ii) deg ωC = deg KC ν + 2δ = 2g(C) − 2,
(iii) h0 (ωC ) = h0 (KC ν ) + δ = g(C).
These facts hold whenever C has planar singularities. More generally, also when we are dealing with locally
complete intersection singularities. They are true even more broadly for Gorenstein singularities.
Example 25.6. The spatial triple point is an example of a non-Gorenstein singularity. A function
upstairs comes from a function downstairs if and only if it assumes the same value at the points over the
triple point.
Then
rational differentials on π −1 (U ) with simple poles
ωC (U ) =
.
at 3 points and sum of residues at these 0
This violates the first two conditions we wrote down above. Since the stalk over the triple point has rank 2,
the description already says ωC is not locally free.
By contrast a planar triple point V (x3 − y 3 ) is Gorenstein. The difference is that ωC allows double poles
at the points in the normalization over the triple point. An example of a suitable differential is dx/y 2 .
25.2. DUALIZING SHEAVES
75
Let us return to the general situation. Consider a singular point p on a curve C, and let π : C ν → C
denote the normalization at p. We have already introduced the short exact sequence
/O
/ π∗ OC ν
/ Fp
/ 0.
0
Here Fp is a skyscraper sheaf supported at p. In our previous discussion, we also computed the rank of Fp
at p.
There is another way to look at the singularity at p. We define the conductor ideal to be I ⊂ OC,p
defined by
I = Ann(Fp ) = {g ∈ OC,p | for all f ∈ OC ν , we have gf ∈ OC }.
Then the Gorenstein condition is equivalent to
δ = rank Fp = rank(OC,p /I).
LECTURE 26
November 7, 2012
26.1. Planar singularities and dualizing sheaves
We would like to tie our discussion of dualizing sheaves with that of curve singularities. One of the basic
features of the canonical bundle is it allows for adjunction to hold. In other words, given a smooth curve C
on a smooth surface S, then
KC = KS (C)|C ,
hence
2g(C) − 2 = C · C + KS · C.
This formula applies more generally for singular C as long as C remains Cartier (e.g., satisfied automatically
for reduced curves).
e the
Suppose p ∈ C is a singular point of multiplicity m. Let Se = Blp S be the blowup of S at p, and C
e
e
e
proper transform of C in S. We would like to know how the (arithmetic) genus g(C) of C relates to that
of C. Let π : Se → S denote the blow down morphism, and E ⊂ Se the exceptional divisor. The canonical
divisors of S and Se are related by
KSe = π ∗ KS + E.
It is a standard fact that Pic Se = π ∗ Pic S ⊕ ZhEi, and we can compute any intersection on Se by using the
equalities
E 2 = −1,
E · π ∗ D = 0,
π∗ D · π∗ D0 = D · D0 .
Putting things together, we get
e −2=C
e·C
e + Ke · C
e = (C · C − m2 ) + (C · KS + m) = 2g(C) − 2 − m(m − 1).
2g(C)
S
In conclusion,
e = g(C) −
g(C)
m
.
2
A singularity as the one we drew above
was called a “triple point with an infinitely near double point”
in the classical literature. It has δ = 32 + 22 = 4, so resolving it drops the genus by 4. In a similar way, we
cam compute the δ-invariant of a cusp with a second component passing through it: δ = 32 = 3.
76
26.2. KONTSEVICH SPACE
77
Back to our first example of a “triple point with an infinitely near double point”, let C 00 denote the
smooth curve obtained after two blow ups, S 00 the surface it sits in, and π : S 00 → S the composite blow
down. Suppose E1 and E2 denote the exceptional divisors of the first and second blow ups respectively.
Then
C 00 = π∗ C − 3E1 − 5E2 ,
KS 00 = π∗ KS + E1 + E2 ,
KC 00 = π∗ KC − 2E1 − 3E2 .
In terms of dualizing sheaves we write
ωC 0 = ωC + 2E1 + 3E2 .
Note that sections of ωC viewed as sections of ωC 00 allow double poles at p and triple poles at q, r.
26.2. Kontsevich space
Let X be a projective variety over C and β ∈ H2 (C, Z). We introduce the Kontsevich space
Mg (X, β) = {f : C → X | C nodal, # Aut(f ) < ∞, and f∗ [C] = β} .
Remark 26.1. By an automorphism of the morphism f we mean an automorphism ϕ : C → C of the
curve C which fits in the following diagram.
/C
ϕ
C
f
X
~
f
78
26. NOVEMBER 7, 2012
Theorem 26.2. There exists a projective Deligne-Mumford moduli space parametrizing the maps f
described above.
Example 26.3. We consider M = M0 (P2 , 2), the space of plane conics viewed through the lens of
Kontsevich space. A conic curve can either be smooth, a union of two lines crossing in one point, or a double
line. The Hilbert scheme H of plane conics is isomorphic to P5 = P H0 (OP2 (2)). The smooth conics form a
dense open, the pairs of lines a (dense open in a) hypersurface, and the double lines a surface which can be
identified with the Veronese surface in P5 . The Kontsevich space M and H agree in the way they handle the
first two types. The equivalent of double lines in the Kontsevich space are degree 2 maps onto a line. The
idea is that the image line is insufficient data to determine such a map. We also need to specify the branch
points. To summarize, we have a morphism
M0 (P2 , 2)
/ P2
which is an isomorphism on the compliment of the Veronese surface S ⊂ P5 of double lines. Actually
M0 (P2 , 2) is the blow up of P5 along S. (The fiber over a point of S is Sym2 P1 = P2 .)
Remark 26.4. When two of the branch points approach each other the domain breaks up into a union
of two lines joined in a single point.
LECTURE 27
November 9, 2012
27.1. The Kontsevich space and the Hilbert scheme
We introduced the Kontsevich space
Mg (X, β) = f : C → X
C nodal curve of genus g, # Aut(f ) < ∞,
.
and f∗ [C] = β ∈ H2 (X, Z)
The equality above is actually a theorem asserting there exists a projective variety Mg (X, β) which is a
Deligne-Mumford moduli space for the associated functor. We will not pursue this technical statement
further.
As with the moduli space of genus g curves Mg , there is an alternative way to interpret the finiteness
of the automorphism group Aut(f ). It means that any rational component C0 ⊂ C contracted by f satisfies
# C0 ∩ (C \ C0 ) ≥ 3.
There is an analogous statement for contracted genus components too.
Remark 27.1. When X is a point, we obtain an equality Mg (X, 0) = Mg . More generally, when β = 0
the Kontsevich space is Mg (X, 0) = Mg × X.
We can think of the Kontsevich space Mg (X, β) as an alternative compactification of the open U ⊂ H(X)
parametrizing smooth curves of genus g and class β. Our next goal would be to explore closely the relation
between the Kontsevich space and the Hilbert schemes in several cases.
Remark 27.2. Suppose X = Pn and β = d` where ` ∈ H2 (Pn , Z) is the class of a line. Then we will
write Mg (Pn , d) for Mg (Pn , d`).
Last time we discussed the space of complete conics M0 (P2 , 2). There are three types of degree 2
curves: smooth conics, two lines, and double lines. The first two are represented identically in both the
Hilbert scheme H2m+1 (P2 ) = P5 and the Kontsevich space. In the case of double lines, the Kontsevich space
additionally remembers the points of ramification of a degree 2 map onto the line. In other words, we have
a regular map
/ H2m+1 (P2 ) = P5
M0 (P2 , 2)
which is the blow up along the Veronese surface S ⊂ P5 corresponding to double lines.
27.2. More examples of Kontsevich spaces
Example 27.3. Let us look at M = Mg (P3 , 2), the space of conics in P3 . Note that any stable map
f : C → P3 is finite (i.e., it doesn’t contract any components). As before, the domain of a stable map
f can either be a single P1 or the union of two joined in a single point. Just as in P2 , it is possible to
specialize an embedded curve to a double line. It is possible to show that any curve C in the Hilbert scheme
H = H2m+1 (P3 ) lies in a unique plane. On the other hand, the curves corresponding to double lines in M do
not carry the data of such a unique plane. For this reason, we do not get a regular map M → H. Two analyze
the relation between these two spaces more closely, we introduce the Chow variety C2,1 parametrizing cycles
of dimension 1 and degree 2 in P3 . Both the Hilbert scheme and Kontsevich space admit regular birational
79
80
27. NOVEMBER 9, 2012
morphisms onto C2,1 as shown in the following diagram. It turns out the birational isomorphism M 99K H
is a flip.
Γ
y
M0 (P3 , 2)
$
flip
%
/ H2m+1 (P3 )
C2,1
y
There is also a space Γ sitting above which can be obtained by blowing up the locus of double lines in
H2m+1 (P3 ). It turns our the Hilbert scheme is a P5 bundle over (P3 )∨ , and by blowing up the locus of
double lines Γ can be identifies with a fibration over the same base whose fibers are isomorphic with the
space of complete conics.
Example 27.4. We have already discussed that the Hilbert scheme often has extraneous components
which get in the way. In fact, Kontsevich spaces are not much better behaved. To illustrate this, let us
consider the space of plane cubics M1 (P2 , 3). It contains an open subset
U = {smooth plane cubics} ⊂ P9 = P H0 (OP2 (3)) = H3m+1 (P2 ).
There is a morphism
M1 (P2 , 3)
/ H3m (P2 ) ∼
= P9
which is an isomorphism over the locus of nodal curves in the codomain. Examples of such are rational
cubics, a union of a conic and a line, and three lines.
We know from stable reduction that a cuspidal curve is replaces by a curve with an elliptic tail. Actually,
there is a P1 worth of such curves given by the j-invariant of the elliptic tail. The situation for triple lines
is even more complicated.
The actual point we would like to stress via this example is that M1 (P2 , 3) has three irreducible components. For example, consider stable maps of the following form such that f is constant on the elliptic
component.
The general such map will map the P1 component onto a rational cubic. Counting parameters, the image
curve depends on 8 parameters. The point of attachment of E is 1 additional parameter, and so is the
j-invariant of the elliptic tail. In total, we have a 10-dimensional family of such curves which forms an
irreducible component of M1 (P2 , 3).
27.2. MORE EXAMPLES OF KONTSEVICH SPACES
81
The third component consists of maps of the following form such that f is constant on E and the two
P1 -s are mapped to a line and a conic.
The image curve depends on 5 + 2 = 7 parameters, j(E) is 1, and so are the two points of attachment on E.
In total, we get a 9-dimensional family of such curves. It turns out this is also an irreducible component of
M1 (P2 , 3).
In general, the Kontsevich space compared to the Hilbert scheme is not always nicer to work with.
There are pros and cons on both sides. For example, for plane cubics H ∼
= P9 but M1 (P2 , 3) has extraneous
3
components. On the other hand, for twisted cubics M0 (P , 3) is irreducible but H3m+1 (P3 ) has extraneous
components. The real issue is we cannot tell when a point in H or Mg (X, β) lies in the closure of the locus
of smooth curves. For example, what stable maps f : C → P2 of degree d and genus g lie in the closure of
the Severi variety?
LECTURE 28
November 12, 2012
28.1. Complete subvarieties of Mg
We would like to study the complete subvarieties contained in Mg . In other words, we are interested in
families of smooth curves over a complete base. We can also think of this as placing Mg on a scale between
an affine space on one end of the spectrum and projective space on the other. There are several ways to
make this precise.
Question 28.1. What is the largest dimension of a complete subvariety contained in Mg ?
Imagine we start with An . The only complete subvarieties it contains are points. If, however, we blow
up the origin, we arrive at a space which contains a copy of Pn−1 as the exceptional divisor of the blow up.
It follows that simple birational modifications to a space can significantly change the answer to the previous
question. On the other hand, the behavior we presented is somewhat pathological since the exceptional locus
is rigid. Here is one way we can refine the problem we proposed above.
Question 28.2. What is the largest dimension such that a general point of Mg contains a complete
subvariety of that dimension?
There is a lot we can say about the state of this problem. For example, there exist complete curves
through a general point in Mg . In fact, given an integer n, we can pass a curve through a general n-tuple
of points in Mg . This follows from the existence of the Satake compactification Msg . As an alternative
compactification of Mg , there are two major drawbacks. The Satake compactification is highly singular
and it is not modular. (In fact we can identify points at the boundary with curves of lower genus, but this
does not allow for a suitable notion of a family.) One of the few virtues of Msg is that its boundary has
codimension 2 (when g ≥ 3). Precisely this is the property which allows us to find complete curves.
Another interesting feature, following a construction of Kodaira, is that if we allow g to vary, then Mg
contains families of arbitrarily large dimension m (to be more precise g ∼ 3m ). Suppose we start with a
curve C0 of genus h, and consider all triple covers of C0 branched over only one point:
{C | there exists f : C → C0 of degree 3 with only 1 branch point} ⊂ Mg
for g = 3h − 1. We claim this forms a complete curve in Mg . Note that this space admits a map to C0
which is a branched cover (the idea here is similar to our discussion of Hurwitz space).
It is possible to repeat this construction in greater generality. Suppose we start with a complete variety
Σ ⊂ Mg of dimension m. Then
Σ0 = {C | there exists f : C → C0 of degree 3 with 1 branch point, C0 ∈ Σ} ⊂ M3g−1
is a complete subvariety of dimension m + 1. On the other hand, the varieties we construct in this fashion
do not go through the general point of Mg , so the method does not shed any light on the second question
we posed.
28.2. Diaz’s Theorem
Theorem 28.3 (Diaz). If Σ ⊂ Mg is a complete subvariety of dimension m, then m ≤ g − 2.
82
28.2. DIAZ’S THEOREM
83
Remark 28.4. Combining this result with Kodaira’s construction, it follows that the largest dimension
m of a complete subvariety in Mg satisfies
log3 g < m ≤ g − 2.
Needless to say, this is a very wide gap and there are many related open questions.
Actually, Diaz’s Theorem and the idea of its proof go back to Arbarello. The original approach had two
problems which were later resolved nicely by Diaz.
Proof of Diaz’s Theorem as proposed by Arbarello. We define a stratification of Mg . For
2 ≤ d ≤ g, define
Γd = {C | there exists p ∈ C with r(dp) = h0 (dp) − 1 ≥ 1}.
Alternatively, we can write
Γd = {C | there exists f : C → P1 of degree d totally ramified over a point}.
e d = Γd \ Γd−1 . The smallest one of these is Γ
e 2 = Γ2 which coincides with the hyperelliptic locus.
Then set Γ
We have inclusions
Γ2 ⊂ Γ3 ⊂ · · · ⊂ Γg−1 ⊂ Γg = Mg .
The last equality follows from the fact every curve admits a Weierstrass point. It is possible to compute the
dimensions of Γd :
dim Γd = #all branch points − dim Aut(P1 )
= (2g + 2d − 2) − (d − 1) + 1 − 3
= 2g + d − 3.
The important point is all inclusions Γd ⊂ Γd+1 are of codimension 1. Given an m-dimensional complete
subvariety Σ of Γd , we claim it must intersect Γd−1 in a complete subvariety of dimension at least m − 1. It
e d = Γd \ Γd−1 the previous statement follows.
we show that Γ
Suppose we have a complete variety Σ of dimension m contained in Mg = Σg . Intersecting the filtration
above with Σ, we obtain a new sequence of inclusions
Σ ∩ Γ2 ⊂ Σ ∩ Γ3 ⊂ · · · ⊂ Σ ∩ Γg−1 ⊂ Σ.
Applying the inequality of dimensions we just explained, we can derive
m = dim Σ ≤ 1 + dim(Σ ∩ Γg−1 ) ≤ 2 + dim(Σ ∩ Γg−2 ) ≤ · · · ≤ g − 2 + dim(Σ ∩ Γ2 ) ≤ g − 2.
e d contains any complete
This completes the proof of Diaz’s Theorem modulo showing that no stratum Γ
curves. In reality there is a gap in this argument, but we will discuss how to remedy it at a later time. LECTURE 29
November 14, 2012
29.1. Arbarello’s proof
Our aim would be to discuss two proofs of the following result. The first of these, due to Arbarello, is
incomplete. Later, Diaz was able to refine the argument and give a rigorous proof.
Theorem 29.1 (Diaz’s Theorem). If Σ ⊂ Mg is a complete variety of dimension m, then m ≤ g − 2.
Arbarello’s proof. Start by defining a filtration
Γd = {C | there exists p ∈ C such that r(dp) = h0 (dp) − 1 ≥ 1}
= {C | there exists f : C → P1 with deg f ≤ d and #f −1 (∞) = 1}.
A dimension count reveals that
dim Γd = 2g − 3 + d.
We have inclusions
Γ2 ⊂ Γ3 ⊂ · · · ⊂ Γg−1 ⊂ Γg = Mg .
Set
e d = Γd \ Γd−1
Γ
e g contains no complete curves. Combining this
in order to obtain a stratification of Mg . We claim that Γ
remark with the dimension count above, we arrive at the statement of the proposed theorem.
Remark 29.2. The subvariety Γ2 ⊂ Mg consists of all hyperelliptic curves.
e d be a complete curve. First, we can change our point of view and think
Proof of claim. Let B ⊂ Γ
e
of it as a map B → Γd . After a base change, we can assume the following.
(a) The curve B is smooth.
(b) There exists a family π : C → B.
(c) There exists a section σ : B → C such that:
(i) r(dσ(b)) ≥ 1 for all b ∈ B,
e d = Γd \ Γd−1 ), and
(ii) r((d − 1)σ(b)) = 0 for all b ∈ B (because of containment in Γ
(iii) r(dσ(b)) = 1.
Consider the vector bundle E = π∗ OC (dσ) of rank 2. The fact that r((d − 1)σ(b)) = 0 means that E has no
basepoints, so we get a map ϕ : C → PE ∨ . We have arrived at the following diagram of spaces.
84
29.2. DIAZ’S PROOF
85
The branch locus of ϕ can be broken into ϕ(σ) and a remainder which we will call D. The key is in
analyzing the points at which D meets one of the points of total ramification in ϕ(σ).
As permutations, we have
(12)(1 · · · d) = (2 · · · d).
The ramification index is d − 2, so we break off into two local irreducible components. This means C has a
singularity, hence leading to a contradiction. The problem is D need not meet ϕ(σ).
29.2. Diaz’s proof
1
Above, we arrived at a P -bundle over B with a section σ and a, potentially disjoint, curve D. Diaz’s
insight was noting that there are no P1 -bundles with two disjoint sections σ, τ and a curve D disjoint from
both (except from the trivial bundle and constant sections).
Let us start by seeing why this is the case. After a base change we can separate D into a collection of
sections, disjoint from σ and τ , though potentially intersecting each other. Then we can use σ, τ and one of
the sections of D to give an isomorphism with the trivial P1 -bundle. In fact, it is possible to show further
that the sections comprising D are disjoint.
Diaz’s considers a different stratification which allows for two sections. We define
∆d = {C | there exists f : C → P1 with deg f ≤ g and #f −1 ({0, ∞}) ≤ d}.
86
29. NOVEMBER 14, 2012
As before, there are inclusions
∆2 ⊂ ∆3 ⊂ · · · ⊂ ∆g−1 ⊂ ∆g .
We claim ∆g = Mg . To see this note that any curve has a Weierstrass point. We can arrange so that
this point is the unique preimage of 0, and there is a simple branch point over ∞ (hence the preimage has
cardinality at most g − 1).
1
Remark 29.3. If g = 2k is even, then any curve C has a gk+1
. We can take f : C → P1 to be the degree
1
k + 1 map associated to the gk+1 . We then need to modify f in order to make some arrangements for 0, ∞.
Unlike Arbarello’s filtration, the parts ∆d need not be irreducible. For example, ∆2 contains all curves
which admit maps to P1 ramified only over two points. This contains the hyperelliptic locus, but may include
other components in general. In general, ∆d will have pure dimension 2g − 3 + d but may be highly reducible.
The dimension count follows by studying the branch locus.
As before, we consider a complete curve
B
e d = ∆d \ ∆d−1 .
/∆
Because of 0 and ∞, we have two disjoint sections σ and τ . The remainder of the ramification locus D must
meet one of the sections by the argument above. The only difficulty lies in analyzing the ramification as D
meets one of the sections. Suppose we have a family of curves ft : Ct → P1 branched over 0 and α(t). Let
the monodromy over 0 be some permutation σ ∈ Sd . For simplicity assume the monodromy around α(t)
is a simple transposition (12). If 1 and 2 belong to different cycles of σ, then C0 is a smooth curve but
#f0−1 (0) < #f0−1 (∞). But then we find ourselves in a lower stratum since the number of preimages has
decreased. This means 1 and 2 must belong to the same cycle in σ, so we are in the singular case, leading
to a contradiction.
LECTURE 30
November 16, 2012
30.1. The birational geometry of Mg
We would like to pose the following problem.
Question 30.1. Can you write down the general curve of genus g?
We need to elaborate on the meaning of this statement. We would like to write a set of equations whose
solutions vary freely in several variables, and the family they correspond to contains the general curve. In
other words, we would like to find a family of curves π : C → B such that:
(i) B is isomorphic to an open subset of an affine (or projective) space,
(ii) the induced map B → Mg is dominant.
If finding a family as above is possible for a given value of g, then Mg is unirational. In fact, the converse
is also true for g ≥ 3.
30.2. Unirationality in low genera
Our goal would be to go through some small genera examples and give a positive answer to the question
we posed.
Example 30.2 (Genus 2). All smooth genus 2 curves are hyperelliptic with 6 branch points. We can
consider the locus
6
Y
y2 =
(x − λi )
i=1
which is a family of smooth curves over an open subset of A6λi (where the λi are distinct). In fact this family
is surjects onto M2 .
Example 30.3 (Genus 3). The general genus 3 curve is not hyperelliptic but it is smoothly embeddable
in P2 as a plane quartic. It follows that we can look at the locus
X
f (x, y) =
aij xi y j = 0.
i+j≤4
There are 15 coefficients aij and the singular quartics form a hypersurface in A15 . Throwing this locus away
we get a map from an open of A15 onto the non-hyperelliptic locus in M3 .
Example 30.4 (Genus 4). The general genus 4 curve in the intersection of a quadric and a cubic in P3 .
Similarly, we can use a family defined by
f2 (x, y, z) = g3 (x, y, z) = 0.
Parametrizing the coefficients of f2 and g3 solves the problem.
Example 30.5 (Genus 5). The general genus 5 curve is a complete intersection of three general quadrics
in P4 , so we get a family over A45 including all non-hyperelliptic and non-trigonal curves.
87
88
30. NOVEMBER 16, 2012
Example 30.6 (Genus 6). Note that our analysis of genus 4 and 5 used their canonical models. It is
possible to carry out a similar program for genus 6 (a general curve is the intersection of a quintic del Pezzo
surface and a quadric in P5 ), but this method does not extend easily beyond g = 6.
To remedy the issue, we will go back to a planar model. Brill-Noether theory says that a general genus
6 curve can be birationally embedded in P2 as a sextic with 4 nodes. Furthermore, the nodes are in general
linear position. (The curves which do not satisfy this condition form the Petri divisor in M6 .) We know
that Aut(P2 ) = PGL(3) acts transitively on 4-tuples of point in general linear positions, so we can take these
to be
p1 = [1, 0, 0],
p2 = [0, 1, 0],
p3 = [0, 0, 1],
p4 = [1, 1, 1].
We can then look at the vector space of degree 6 homogeneous polynomials F (X, Y, Z) vanishing to order 2
at p1 , . . . , p4 . There is an open U ⊂ V such F ∈ U define curves nodal at p1 , . . . , p4 and smooth otherwise.
This gives us a dominant map to M6 and resolves our question.
Example 30.7 (Genus 7). We continue our exploration of the planar model. Brill-Noether theory says
that a general genus 7 curve can be birationally embedded in P2 as a curve of degree 7 with δ = 8 nodes. In
other words, the small Severi variety V7,7 = V 7,8 dominates M7 . Consider the map V 7,8 → Sym8 P2 which
only remembers the positions of the nodes. The general fiber of this map is an open in a projective space of
dimension 11. Birationally V 7,8 is a P11 -bundle over Sym8 P2 . We can pull back along (P2 )8 → Sym2 P2 to
avoid talking about the rationality of the later space. The pullback of V 7,8 is then rational and dominates
M7 .
Example 30.8 (Genus 8). Brill-Noether theory implies that the general genus 8 curve can be birationally
embedded as an octic P2 with δ = 13 nodes. As before, there is a morphism
V8,8 = V 8,13
/ Sym13 P2
whose general fiber is an open in P5 . This line of argument leads to the conclusion M8 is unirational.
Example 30.9 (Genus 9). Similarly to genus 8, the general genus 9 curve can be birationally embedded
as an octic in P2 with δ = 12 nodes. We get a morphism
V8,9 = V 8,12
/ Sym12 P2
whose general fiber is birational to P8 . As before, we conclude the moduli space M9 is unirational.
Example 30.10 (Genus 10). We ca take degree d = 9 curves with δ = 18 nodes. In this case, we get a
morphism
/ Sym18 P2
V9,10 = V 9,18
with general fiber P0 , that is, the map is birational.
Example 30.11 (Genus 11). We cab still take d = 10 and δ = 25 but V10,11 does not dominate Sym25 P2 .
The expected dimension of the fiber is negative. In other words, the nodes of such an embedded curve are
not in general linear position.
While studying planar embeddings fails to imply unirationality from genus 11 onward, it is possible to
use different approaches to the problem. At great cost, the cases g = 11, 12, 13, 14 were completes in the
1970s by Sernesi, Ran and Chang.
LECTURE 31
November 19, 2012
31.1. Unirationality and Mg
We already explained that Severi observed that for g ≤ 10 there exists a family C → B of smooth genus
g curves such that the associated morphism B → Mg is dominant. The details of this construction were
later put on a firm standing by Arbarello and Cornalba. Sernesi, Ran and Chang extended this to g ≤ 114.
These statements relate directly to the geometry of Mg . The existence of such a family implies that Mg is
unirational. In fact, the converse implication is also true for g ≥ 3 where the automorphism locus is dense.
Remark 31.1. One consequence of unirationality is that curved defined over Q form a dense subset for
g ≥ 14. For higher genera, this statement is not known.
In general, there are very few ways to show that a variety X of dimension n is unirational. Other than
exhibiting a dominant map from a rational variety to X, this is a very hard problem. There is however a
way to judge in the negative. If X is smooth and projective, the existence of a dominant rational map
/X
Pn
implies that
m
h0 (KX
)=0
for all m > 0. This is a condition we can test, and we are going to employ this strategy. In order to work
with a projective variety, we switch from working with Mg . While Mg is not smooth, the weaker condition
of having canonical singularities suffices. Our new question is whether
m
6= 0
h0 KM
g
for any m ≥ 0. Alternatively, we are looking for an effective pluricanonical divisor on Mg . Note that
asking questions about the birational geometry of Mg naturally leads us to study the divisor theory of the
projective variety Mg . Even we cannot identify the complete group Pic(Mg ), we would like to get a handle
on a subgroup G such that:
(i) G contains KMg ,
(ii) we understand the effective divisors in G.
31.2. Divisors on Mg
Rather than investigating divisors in isolation, we would like to start by asking to construct subvarieties
of Mg . Geometric conditions naturally give rise to such subvarieties. For example, among all conditions of
the form
{curves C with a special Weierstrass point}
there are two divisors. Another similar condition is
{curves C with a semicanonical pencil} = {curves C a line bundle L with L2 = KC and h0 (L) ≥ 2}.
Fix a Brill-Noether number
ρ = ρ(g, r, d) = g − (r + 1)(g − d + r) < 0.
The associated Brill-Noether locus
{curves C with a gdr }
89
90
31. NOVEMBER 19, 2012
forms a subvariety of codimension −ρ. In particular, when ρ = −1 we obtain a divisor. More generally
P
{curves C such that there exist points p1 , . . . , pk such that h0 ( mi pi ) ≥ r}
is another subvariety. We can find values of k, mi and r for which the above is a divisor. The boundary of
Mg gives another natural divisor in Pic(Mg ).
Our discussions are leaning towards a study of line bundles on Mg . On the other hand, the moduli
space of stable curves Mg is not a regular variety. We defined it as a Deligne-Mumford moduli space, so it
would be best if we can understand line bundles on it in terms of families of curves Let us consider a general
variety X and a line bundle L on it. Given a morphism ϕ : B → X with B a curve, there is an induced line
bundle Lϕ = ϕ∗ L on B. Furthermore, given a morphism α : B 0 → B, we have an equality
Lϕ◦α = α∗ Lϕ .
Of course, the above statement is obvious. The interesting observation is that this package of data (a line
bundle Lϕ for each morphism ϕ : B → X satisfying the naturality condition) gives rise to a line bundle on
X. The task of building line bundles on Mg is equivalent to building natural line bundles on 1-parameter
families of stable curves. In other words, given a 1-parameter family of stable curves C → B of stable curves,
we would like to find line bundles (or divisors) LC on B such that for all α : B 0 → B we have
LC×B B 0 = α∗ LC .
Here are a couple of ways to build such line bundles.
(1) The Hodge bundle of a family π : C → B is defined as
E = π∗ ωC/B .
This is a vector bundle of rank g whose fiber over a point b ∈ B is the vector space of canonical
forms on the fiber π −1 (b). From here we can get a divisor
λC/B = c1 (E)
on B. These satisfy the criteria given above.
(2) The first Chern class c1 (ωC/B ) is a divisor on the surface C. We can then look at its self-intersection
to define another divisor class
κC/B = π∗ (c1 (ωC/B )2 ).
LECTURE 32
November 26, 2012
32.1. Divisor class theory for the moduli space of curves
As we have already seen, there are many ways to introduce divisors on Mg . Firstly, we can do so by
providing codimension 1 loci. We discussed the use of geometric conditions to arrive at such subvarieties.
One such is
∆ = {singular curves}
which corresponds to the divisor δ = [∆]. It is possible to refine this observation and decompose ∆ further. A
general curve in ∆ has a single node. Let ∆0 denote the closure of the locus of curves with a non-separating
node, that is, singular irreducible curves. Similarly, let ∆i for i = 1, . . . , bg/2c denote the closure of the locus
of curves consisting of a genus i and a genus g − i identified at a single point. Then
∆ = ∆0 ∪ · · · ∪ ∆bg/2c ,
hence
δ = δ0 + · · · + δbg/2c
where δi = [∆i ].
Another way of producing divisors is by functorially associating divisor classes on the base B of each
family of stable curves π : C → B. We gave two examples of such divisors. Given a family as above, we can
define its Hodge bundle as
EC/B = π∗ ωC/B .
This is a rank g vector bundle whose fiber over a point b ∈ B can be identified with H0 (ωπ−1 (b) ). This gives
rise to the divisor class
λC/B = c1 (E) ∈ Pic(B).
In a similar fashion, we define
κC/B = π∗ (c1 (ωC/B )2 ) ∈ Pic(B).
There is a hard theorem showing that λ, κ, δ0 , . . . , δbg/2c generate Pic(Mg ).
Given a 1-parameter family of stable curves π : C → B, we would like to be able to compute the degree
of any of these divisors on B. For example, we would like to be able to do this with the divisor δ.
91
92
32. NOVEMBER 26, 2012
Recall that if p is a node of a curve C, we defined the versal family xy − t over Def(C, p) = ∆t . If C is
a stable curve with a single node, we get a map
/ Def(C, p) = ∆t .
Def(C)
The divisor ∆ around [C] ∈ Mg is locally given by (t). If C is stable with a several nodes p1 , . . . ,, we get a
map
Q
Q
/ Def(C, pi ) = ∆ti .
Def(C)
Q
Then ∆ is locally cut by
ti = 0. Note that ∆ is a normal crossing divisor. Suppose we are given a
1-parameter family π : C → B with associated map ϕ : B → Mg → B with b0 ∈ B corresponding to the
i
curve Cb0 = π −1 (b0 ) with nodes pi . Furthermore,
suppose the local equation of C around pi is xy − tm
i . Then
P
∗
the multiplicity of ϕ ∆ at b0 ∈ B is
mi . This allows us to compute the degree of ∆ on a 1-parameter
family of curves from its local equations.
32.2. A pencil of plane quartics
Suppose we consider two general homogeneous quartic polynomials F, G ∈ H0 (OP2 (4)) and we look at
C = V (t0 F + t1 G) ⊂ P1t × P2
as a family of curves parametrized by t = [t0 , t1 ] ∈ P1t . It is possible to show that under the genericity
hypothesis we made all fibers of π : C → P1t are stable. (This is true more generally if the total space C is
smooth.) To see this, node that the locus of quartics is P H0 (OP2 (2)) = P14 , and the locus of curves with a
singularity worse than a node has codimension 2. A general line is disjoint from this locus, so our observation
follows.
We are interested in finding the degrees of δ, λ and κ on this family over P1 . As a mild form of notation
abuse, we will denote these degrees with the same symbols we use for the divisors the come from. Let us
start with δ. We can think of C ⊂ P1 × P2 as a general divisor of type (1, 4). By Bertini’s Theorem the total
space C is smooth, so all nodes have multiplicity 1. Furthermore, the general singular quartic has a single
node, so
δ = #{nodes of singular fibers} = #{singular fibers}.
The best way to carry this computation is by invoking a version of the Riemann-Hurwitz formula.
Digression (The Riemann-Hurwitz formula). Let
X
/B
be a flat family such that X is a smooth n-dimensional projective variety and B is a smooth curve. The
singular fibers of such a family occur over finitely many points
Γ = {b1 , . . . , bδ } ⊂ B,
so we have a decomposition
X = π −1 (B \ Γ) ∪ Xb1 ∪ · · · ∪ Xbδ .
The Euler characteristic is additive, so
χ(X ) = χ(π −1 (B \ Γ)) +
X
χ(Xbi ).
Note that π −1 (B \ Γ) is a fiber bundle over B \ Γ. Let F denote one of its fibers. Then
X
χ(X ) = χ(B \ Γ)χ(F ) +
χ(Xbi )
X
= χ(B)χ(F ) − δχ(F ) +
χ(Xbi )
X
= χ(B)χ(F ) +
(χ(Xb ) − χ(F )) .
b∈Γ
32.2. A PENCIL OF PLANE QUARTICS
93
In the situation we are interested in the general fiber is a genus g curve with δ specializations to a nodal
curve. The Euler characteristic of a singular fiber is the number of nodes it contains more than the Euler
characteristic of the general fiber. Then
δ = #{nodes} = χ(C) − χ(B)χ(F ).
We are now ready to return to out pencil of plane quartics. The general fiber is a smooth genus 3 curve
and the base B is isomorphic to P1 . Furthermore, the total space C is the blowup of P2 at the base locus
which consists of 16 points. It is not hard to show that blowing a smooth point on a surface increases the
Euler characteristic with 1 (the difference χ(P1 ) − χ(point)). Putting this information together, we compute
δ = 3 + 16 − (−4) · 2 = 19 + 8 = 27.
Alternatively, we can also say that the locus of singular points in C is cut out by the three partial derivatives
∂G
∂F
∂G
∂F
∂G
∂F
+ t1
,
t0
+ t1
,
t0
+ t1
.
t0
∂x
∂x
∂y
∂y
∂z
∂z
Each of these is bihomogeneous of degree (1, 3), so their common intersection has 27 points by Bezout’s
Theorem. The second method we presented works only for P2 , while Riemann-Hurwitz is applicable in a
wider variety of cases.
Next, let us compute the degree of κ. By way of notation, given a subvariety Z ⊂ P1 × P2 , we will
abbreviate
OZ (m, n) = π1∗ OP1 (m) ⊗ π2∗ OP2 (n).
Then
KP1 ×P2 = OP1 ×P2 (−1, −3),
KC = KP1 ×P2 (C)|C = OC (−1, 1),
ωC/P1 = KC ⊗ π1∗ KP∨1 = OC (1, 1).
Then the degree of c1 (ωC/P1 )2 counts the triple intersection in P1 × P2 of divisors of bidegree (1, 4), (1, 1)
and (1, 1). Given we have a presentation for the Chow ring
A(P1 × P2 ) = Z[α, β]/(α2 , β 3 ),
we can compute
κ = deg(c1 (ωC/B )2 ) = deg (α + 4β)(α + β)2 = deg(9αβ 2 ) = 9.
As for λ, we note that
⊕3
E = (π1 )∗ OC (1, 1p) = OP1 (1) ⊗ (π1 )∗ OC (0, 1) = OP1 (1) ⊗ OP⊕3
.
1 = OP1 (1)
Then
λ = deg c1 (E) = 3.
LECTURE 33
November 28x, 2012
33.1. Divisor class theory for Mg
We have been working with divisor classes on Mg as gadgets which associate to a family of stable curves
π : C → B a divisor class on B. We can also think of these as tools to measure the non-triviality (actually
the non-isotriviality) of π : C → B. One way to exhibit distinct fibers is by showing the family contains both
smooth and singular fibers. We can use the boundary divisor δ to formalize this. In particular, the degree
of δ on B is the total number of nodes in the family counted with multiplicity. The other two divisors we
introduced were somewhat more abstract:
λ = c1 (π∗ ωC/B ),
κ = π∗ (c1 (ωC/B )2 ).
We also computed the degrees of these divisors on a general pencil of quartic curves in P2 :
δ = 27,
λ = 3,
κ = 9.
33.2. A pencil of hyperplane sections on a quartic surface
We would like to provide another example in which the computations of these divisors is not too difficult.
Let S ⊂ P3 be a smooth quartic surface, and consider a general pencil of sections π : C → B = P1 .
The total space of the family C is the blowup of S along a center Γ consisting of the four basepoints. In
other words, Γ is the intersection of the surface S and the base locus of the pencil of planes which is a line.
There are several spaces and morphisms relevant to the following computation, so we will summarize them
with a diagram.
P1 ×O P3
/ P1 × S
C = BlΓ S
pr1
& 1
P
π
94
pr2
/ P3
O
α
/S
33.2. A PENCIL OF HYPERPLANE SECTIONS ON A QUARTIC SURFACE
95
Given that the pencil we take is general, it follows that π : C → B is s family of stable curves. It is them
immediate to compute
δ = χ(C) − χ(B)χ(general fiber) = χ(S) + 4 − 2(−4) = 24 + 12 = 36.
Remark 33.1. The Euler characteristic of the surface S can either be computed by noting it is a K3,
or via the Hopf Index Theorem.
Alternatively, we can also look at the Gauss map of S ⊂ P3 :
G : C −→ S ∨ ⊂ (P3 )∨ ,
p 7−→ [Tp S].
A pencil of planes in P corresponds to a line in (P3 )∨ , so δ is equal to the degree of deg(S ∨ ). If S = V (F )
for a homogeneous quartic polynomial F ∈ H0 (OP3 (4)), then G is given in coordinates as
∂F ∂F ∂F ∂F
,
,
,
.
∂x ∂y ∂z ∂w
3
Each of these components is a cubic polynomial, so Bezout’s Theorem implies that degree of the image of
the quartic surface S is
δ = deg(S ∨ ) = 4 · 32 = 36.
The first factor 4 comes from the degree of S in the original ambient space P3 . The two factors 3 are derived
from the fact a line in the dual projective (P3 )∨ is the intersection of two hyperplanes each of which is a
cubic surface when pulled back via G.
Let us introduce two divisor classes on C ⊂ P1 × S:
η = π ∗ OP1 (1),
ζ = α∗ OS (1).
As a mild form of notation above, we may use η and ζ to refer to the corresponding classes on P1 × S and
P1 × P3 . Since a pencil of plane sections is the locus of a bihomogeneous polynomial of bidegree (1, 1), it
follows that
[C] ∼ η + ζ
in A(P1 × S). Applying adjunction, we arrive at the equalities
KP1 ×P3 = −2η − 4ζ,
KP1 ×S = −2η − 4ζ + 4ζ = −2η,
KC = −2η + η + ζ = −η + ζ,
ωC/P1 = η + ζ.
The intersections on C among the classes we introduced are
η 2 = 0,
ζ 2 = 4.
η · ζ = 4,
Then, we immediately arrive at
κ = (η + ζ)2 = 12.
Finding the degree of λ requires a little more work:
π∗ ωC/P1 = π∗ (π∗ OP1 (1) ⊗ α∗ OS (1)) = OP1 (1) ⊗ π∗ α∗ OS (1).
The latter of the two factors is a rank 3 bundle whose fiber over a point t ∈ P1 can be identified with
H0 (OC (1)) ∼
= H0 (OH (1)).
t
t
In fact, there is a short exact sequence
0
/ OP1 (1)
/ OP1 (1)⊕4
/ π∗ α∗ OS (1)
We can use this to compute
c1 (π∗ α∗ OS (1)) = 1,
/ 0.
96
33. NOVEMBER 28X, 2012
hence
λ = 4.
LECTURE 34
November 30, 2012
34.1. Basic facts about Pic(Mg )Q
We start by summarizing several results about the rational Picard group Pic(Mg )Q = Pic(Mg ) ⊗ Q.
(A) Pic(Mg )Q is generated by λ, κ, δ0 , . . . , δbg/2c .
(B) The generators above satisfy a relation
12λ = κ + δ,
called the Mumford relation. Furthermore, this relation is unique for g ≥ 3.
(C) A class aλ − bδ is ample if and only if a > 11b > 0.
(D) The canonical class of Mg can be expressed as
KMg = 13λ − 2δ.
Remark 34.1. The expression for the canonical class holds true for the moduli stack of curves. If
working with the coarse moduli space, we need to add a summand δ1 to account for the fact the general
curve C lying in ∆1 has automorphism group Z/2.
Part (A) was conjectured by Mumford and proved much later by Harer. The techniques he used were
topological, and are go well beyond the realm of algebraic geometry. For example, Harer employed Teichmüller theory which uses the fact every curve can be cut using circles into a union of pairs of pants.
It follows that we have a presentation
Mg = Tg /Γg .
Teichmüller space Tg is biholomorphic to a complex disc of dimension 3g−3. We use Γg to denote the mapping
class group in genus g, that is, orientation preserving diffeomorphisms up to isotopy. Harer computed
the group cohomology of Γg in low degrees which lead to finding generators for Pic(Mg )Q and Pic(Mg )Q
subsequently. More recently, Arbarello and Cornalba have a more direct, yet still topological, proof of the
statement (A).
Part (C) is not too difficult to show but it involved the notion of stability which we have not dealt with.
34.2. Grothendieck-Riemann-Roch and its applications
The two parts which are accessible from our point of view are (B) and (D). Demonstrating them involved
the use of Grothendieck-Riemann-Roch formula. Before stating the most general version of this result, we
will go through several historical iterations and describe how each generalizes the previous one.
The original Riemann-Roch formula deals with a degree d divisor D on a curve C of genus g. It says
that
`(D) = d − g + 1 + `(KC − D).
97
98
34. NOVEMBER 30, 2012
It is possible to combine the terms `(D) and `(KC − D) using Serre duality, and also replace the language
of divisors with line bundles. Given a line bundle L on C, we can rewrite the above as
c1 (TC )
.
2
This equation applies more generally. Given a coherent sheaf F on C, we have
χ(L) = c1 (L) +
c1 (TC )
.
2
Next, it is possible this formula to varieties of higher dimension. Given a surface S and a line bundle L on
it, we can write
c1 (L)2 + c1 (TS )c1 (L) c1 (TS )2 + c2 (TS )
χ(L) =
+
.
2
12
There is an analogous extension for coherent sheaves on surfaces.
Hirzebruch was able to give an analogous formula for coherent sheaves on smooth varieties of arbitrary
dimension. To state his version of Riemann-Roch, we need to introduce some notation. Let X be a smooth
variety and F a coherent sheaf on it of rank r. The most basic topological invariant of F is its total Chern
class c(F). The splitting principle says that we can formally factor
Y
c(F) =
(1 + αi ).
χ(F) = c1 (F) + rank(F)
Technically speaking the αi are not well-defined but their symmetric functions are. For example,
X
X
αi = c1 (F),
αi αj = c2 (F).
i<j
The Chern character of F is defined as
X
ch(F) =
eαi
X
α2
=
1 + αi + i + · · ·
2
X
X
= rank(F) +
αi +
αi2 + · · ·
c1 (F)2 − 2c2 (F)
+ ··· .
2
We also introduce another invariant associated to F called the Todd class:
Y
αi
td(F) =
1 − e−αi
c1 (F)2 + c2 (F)
= 1 + c1 (F) +
+ ··· .
12
Given these stipulations, Hirzebruch was able to give a very compact expression embodying all previous
versions:
χ(F) = [ch(F) td(TX )]n .
Grothendieck went even further by extending Hirzebruch’s result to families of sheaves and varieties.
Suppose we have a morphism π : X → B and a sheaf F on X. Treating F as a family of sheaves on the
fibers of π, the Theorem on cohomology and base change relates the cohomology of F fiber by fiber and the
higher direct images Ri π∗ F. Grothendieck’s formula is
X
td(TX )
i
i
(−1) ch(R π∗ F) = π∗ ch(F) ∗
.
π td(TB )
= rank(F) + c1 (F) +
If B is a point, the equality in degree 0 amounts to the Hirzebruch-Riemann-Roch formula. If π : X → B is
a family of smooth curves, we can further simplify
td(X)
∨
= td(TX/B ) = td(ωX/B
).
π ∗ td(B)
34.2. GROTHENDIECK-RIEMANN-ROCH AND ITS APPLICATIONS
99
The Grothendieck-Riemann-Roch formula is designed precisely to handle situations such as the one we
have at hand. Let us consider a 1-parameter family of curves π : C → B and set ω = c1 (ωC/B ). We would
like to compute λ = c1 (π∗ ωC/B ). In this case R1 π∗ ωC/B is not 0 but is trivial, so it only brings a contribution
in degree 0. Assuming all curves in the family are smooth, we can get a simple expression for the fraction of
Todd classes. Then Grothendieck-Riemann-Roch implies the following.
λ = c1 (π∗ ωC/B )
ω ω2
ω2
+ ···
1− +
+ ···
= π∗
1+ω+
2
2
12
2
ω ω
= π∗ 1 + +
2
12
κ
= (g − 1) +
+ ··· .
12
Having singular fibers does not change the first of the two factors in π∗ but it does affect td(C)/π ∗ td(B)
which leads to the additional factor of δ in Mumford’s relation. Proving this relation is unique is a simpler
but laborious computation using test curves.
Our next goal is to attack part (D). To do so we need to get a handle on the tangent bundle of Mg . If
we start with an automorphism free curve [C] ∈ M0g , then deformation theory says that
T[C] Mg = H1 (TC ).
If we use the universal curve C → M0g , we can perform this tangent space computation globally. Then
T Mg = R1 π∗ (TC/M0g ).
Serre vanishing implies
2
T ∨ Mg = π∗ (ωC/B
),
so we can apply Grothendieck-Riemann-Roch:
KMg = π∗ (ch(ω 2 ) td(ω ∨ )).
Going through the computation, we get KMg = 13λ. Accounting for the boundary, we conclude
KMg = 13λ − δ.
LECTURE 35
December 3, 2012
35.1. More about divisors on Mg
We have been building towards making some statements about the global geometry of Mg using divisor
class techniques. Let us start by recalling several facts we already discussed.
(A) Pic(Mg )Q = Qhλ, δ0 , . . . , δbg/2c i.
(B) The class aλ − bδ is ample if and only if a > 11b > 0.
(C) The canonical class of the stack Mg is KMg = 13λ − δ.
We pose the following problem.
Question 35.1. What is the effective cone of Mg ?
Since we are mostly interested in the subspace Qhλ, δi ⊂ Pic(Mg )Q , we will focus on the portion of the
effective cone lying in it. We can show that it contains the quadrant a > 0, b < 0, and one of its edges is
along the negative b-axis. The other edge is along a line in the quadrant a > 0, b > 0 with slope sg .
(i) If sg <
(ii) If sg >
13
2 ,
13
2 ,
then Mg is of general type (i.e., it is not unirational).
k
then h0 (KM
) = 0 for all k > 0. In other words, Mg has negative Kodaira dimension.
g
Note that this translates the question we have previously posed about the birational geometry of Mg to a
statement about the cone of effective divisors on Mg . Our aim will be to find an upper bound for sg by
exhibiting effective divisors on Mg .
For example, we had previously introduced the Weierstrass divisor. One of its components is
W = {curves C with a special Weierstrass point p ∈ C such that h0 (OC ((g − 1)p)) ≥ 2}.
The slope of [W ] is on the order of 9 + O(1/g), so sg ≤ 9 + O(1/g). Another example is the divisor
T = {curves C with a semicanonical pencil}
whose slope is 8 + O(1/g). Neither of these gives enough information to place sg relative to 13/2, the slope
of KMg . The Brill-Noether divisor works for our purposes:
B = {curves C with a linear system gdr with ρ = −1}.
100
35.2. A PENCIL ON A K3 SURFACE
101
To understand the Brill-Noether locus better, it is useful to provide some alternative descriptions. If C is a
curve C of odd genus g = 2k + 1, then it lies in B if and only if it has a pencil of degree k + 1. For example,
B is the hyperelliptic locus in genus 3.
The upshot is that the slope of B is
12
6+
.
g+1
When g ≥ 24, this slope is below 13
2 so Mg is of general type. One glitch is that the Brill-Noether divisor
does not exist in every genus. If
ρ = g − (r + 1)(g − d + r) = −1,
then g + 1 is composite. In the cases g + 1 is prime, we resort to the Petri divisor. We will not go into the
details of this computation.
The basic method for obtaining the slope we quoted above consists of two steps.
(i) Argue that [B] is a linear combination of the form
aλ − b0 δ0 − · · · − bhg/2i δhg/2i .
(ii) Calculate the coefficients a, b0 , . . . , bhg/2i by intersecting with test curves.
Remark 35.2. The first of these steps follows from Harer’s Theorem. It can be deduced independently
and the argument is worth understanding in its own right.
35.2. A pencil on a K3 surface
There is s simple computation which first suggested the slope 6 + 12/(g + 1) we mentioned above. Let us
consider a pencil on a general K3 surface. We consider a K3 surface S with a birational embedding S ,→ Pg
of degree 2g − 2. Given a hyperplane H ⊂ Pg , then S ∩ H is a canonical curve in H ∼
= Pg−1 . Taking a pencil
of hyperplane sections of S, we get a family of stable curves of genus g. Let Γ = S ∩ Pg−2 denote the base
locus of this pencil, consisting of 2g − 2 points. We arrive at the following diagram of morphisms.
Se = BlΓ S
α
/S
π
P1
The total space of the family Se can be identified with the blowup of S along the center Γ. Then
e = χ(S) + 2g − 2 = 24 + 2g − 2,
χ(S)
so
e − 2χ(general fiber) = 24 + 2g − 2 − 2(2 − 2g) = 18 + 6g.
δ = χ(S)
We can also compute λ using the description of Se we have above. The relative dualizing sheaf is
∗
∗
ωS/P
e 1 = π OP1 (1) ⊗ α OPg (1),
so the Hodge bundle can be identified as
∗
E = π∗ ωS/P
e 1 = OP1 (1) ⊗ π∗ α OPg (1).
The last factor is the quotient of a trivial bundle of rank g +1 by OPg (−1). Putting this information together,
we compute
λ = deg E = g + 1.
The remaining question is computing the degree of [B] on the pencil we just constructed. Fortunately
the answer is very simple – zero. To see this we refer to a dichotomy satisfied by K3 surfaces. The following
statements about a K3 surface S are equivalent:
(i) S is the intersection of quadrics,
(ii) S does not contain triples of trigonal points.
102
35. DECEMBER 3, 2012
Surfaces which fail the latter condition are called trigonal. It is immediate that a K3 surface can contain a
trigonal canonical curve if and only if the surface is trigonal. On the other hand, the general K3 surface is
not trigonal, so it is always possible to choose a non-trigonal surface. This implies the degree computation
we just quoted. We deduce the slope of [B] is
6
18 + 6g
= 12 +
.
g+1
g+1
This computation is merely a heuristic and there are many details which need to be filled in.