MODULI SPACES - IMPA

MODULI SPACES
EDUARDO ESTEVES
These are draft notes for a course on moduli spaces. They are not in final form: they may contain
many misprints or even more fundamental errors. They are in no way suitable for publication, not
even to the standard of the author!
1

2 EDUARDO ESTEVES
1. The moduli space of curves and stacks
We will construct two parameter spaces in the next chapters, that of subvarieties
of a given variety and that of smooth curves. The first will represent a functor,
while the second will only corepresent it.
Consider a closed subvariety X of P n . It corresponds to a homogeneous saturated
ideal I ⊆ K[x 0 , . . . , x n ], the ideal generated by the forms that vanish on X.
For each integer d ≥ 0, let I d be the space of forms of degree d that belong to I;
it is a finite-dimensional space. As we will see, since I is finitely generated, the
function d ↦→ dim K I d is polynomial in d, for large values of d. Denote by P I (t)
this polynomial. So P I (t) is the unique polynomial of degree at most n such that
P I (t) = dim K I d for all d larger than a certain integer. Of course, the difference
( ) t + n
P X (t) := − P I (t)
n
is also polynomial, and
P X (d) = dim K
K[X 0 , . . . , X n ] d
I d
for large values of d. We call P X (d) the Hilbert polynomial of X.
Important numerical invariants of X can be read from its Hilbert polynomial.
For instance, the dimension dim X, the maximum dimension of one of its irreducible
components, is the degree of P X (t). Also, the degree of X, the number of points
X intersects a sufficiently general linear subspace of P n of codimension dim X, is
(dim X)! the leading coefficient of P (t).
We will see that there is a projective parameter space for subvarieties of a fixed
Hilbert polynomial. It represents a functor, but to define it now we would need to
introduce a few concepts that we prefer to leave out. To at least have a statement,
let us just consider the open subspace parameterizing smooth subvarieties.
Fix now the Hilbert polynomial P (t) of a smooth subvariety of P n . Consider
the functor Hilb P (t) that associates to each variety S the set of closed subvarieties
X ⊆ P n × S for which the second projection p 2 : X → S is smooth, and its fibers
p −1
2 (s) have Hilbert polynomial P (t). Given any morphism t: T → S, define the
function Hilb P (t) (S) → Hilb P (t) by letting the image of a subvariety X ⊆ P n × S be
Y := (1, t) −1 (X) ⊆ P n × T.
Of course, the fibers of the second projection q 2 : Y → T are also fibers of X over
S, and thus are smooth and have the same Hilbert polynomial P (t). We can also
show that the map q 2 is smooth.
We have thus a contravariant functor from (Varieties) to (Sets). We will show
in the following chapters that it is representable by a quasiprojective variety.

MODULI SPACES 3
Another parameter space we will be interested in is that of smooth curves.
Given a smooth projective curve C there is a natural line bundle associated to it,
called the canonical or cotangent bundle ω C := Ω 1 C . Its degree is 2g − 2, where g
is the genus of C. It follows from Riemann–Roch that, if g ≥ 2, then the global
sections of the tricanonical bundle ω ⊗3
C
embed C in PN , where N := 5g − 6. The
embedding is almost canonical: we need to choose a basis of the global sections.
Hence, the image of C in P N is defined up to linear transformations. The image has
dimension 1 and degree 6g − 6. It turns out that its Hilbert polynomial is
(1) P (t) := (6g − 6)t + 1 − g.
Conversely, consider the variety H representing the functor Hilb P (t) defined
above, with P (t) given in (1). Consider the subset K of H of points h ∈ H such
that the corresponding curve C h ⊆ P N spans P N and its hyperplane section is tricanonical.
Then K is a closed subvariety of H. Furthermore, every tri-canonical
image of every smooth curve of genus g is C h for some h ∈ K, and vice-versa. We
would then like to construct the quotient of K by the action of PGL(N + 1), the
group of linear transformations of P N or, what amounts to the same, the induced
action σ : SL(N + 1) × K → K.
The categorical quotient exists: it is the so-called moduli space of genus-g
smooth curves, M g . Does it represent a natural functor
We call a map f : C → S a family of curves of genus g if f is projective,
smooth and the all the fibers f −1 (s) have genus g. We say that f is parameterized
by S or over S. (A map f : C → S is projective if there is a closed embedding
h: C → P r × S for some r such that f = p 2 h, where p 2 is the second projection.)
The natural functor to consider would be the functor F defined for every
variety S as the set of families of curves of genus g parameterized by S modulo
isomorphisms. More precisely, we identify two families of curves f 1 : C 1 → S and
f 2 : C 2 → S over the given S if there is an isomorphism ι: C 1 → C 2 such that
f 2 ι = f 1 . This establishes an equivalence relation; the set of equivalence classes is
F (S) by definition.
If t: T → S is any map, we consider the fibered product:
C × S T −−−→ C
⏐ f ⏐↓ ⏐
T f↓
t
T −−−→ S.
The second projection C × S T → T , denoted f T above, is also a family of curves, of
the same genus as the original f. Isomorphic families over S are taken to isomorphic
families over T under this construction. In other words, there is a corresponding
function F (S) → F (T ) taking the class of f to that of f T . It is not difficult to show
that this function is compatible with taking compositions. (Essentially, if u: U → T

4 EDUARDO ESTEVES
is another map, you need only observe that there is a natural isomorphism
(C × S T ) × T U ∼ = C × S U.)
Does M g represent this functor The answer is no: M g only corepresents F .
The reason is the curves with automorphisms: they prevent M g , the categorical
quotient of K by SL(N + 1), from representing F . There are two ways we can make
the last statement more meaningful. We can argue that the functor paramaterizing
families of curves as above but whose fibers have no automorphism but the identity
is representable. And we can argue that M g cannot represent F by using a family
of curves with nontrivial automorphisms.
Let us consider the following example: Pick a curve C of genus g ≥ 2 with a
nontrivial automorphism σ. Let n be its order. Pick any elliptic curve E, in other
words, a nonsingular cubic curve. Choosing a point O ∈ E we have a natural group
structure on E for which O is the neutral element. There is a point P ∈ E of order
n. (Think of the torus; otherwise see Mumford, Abelian varieties.) Consider the
translation-by-P map τ, sending each Q ∈ E to Q + P . Then τ is an element of
order n of the group Aut(E). Identify the subgroup of Aut(C) generated by σ and
the subgroup of Aut(E) generated by τ with G := Z/nZ, by taking τ and σ to
ξ := 1. Then G acts on the product C × E, by letting
ξ(Q, R) := (σ(Q), τ(R)).
Of course, G acts also on E and on C, and the projections p 1 : C × E → C and
p 2 : C × E → E are G-invariant. The action of G on E and thus on C × E is free,
that is the stabilizers are trivial. This means the quotient X := C × E//G is a
smooth connected projective surface. Since p 2 is G-invariant, it induces a morphism
f : X → S, where S is the quotient of E by the subgroup generated by P , thus
another elliptic surface. This map is smooth and all of its fibers are isomorphic to
C: in fact, the diagram
C × E −−−→ X
⏐ p 2
⏐↓ ⏐
f↓
E −−−→ S
is a fiber product, where the horizontal maps are the quotient maps. Another such
family is simply the second projection q 2 : C × S → S. If M g corepresented the
functor F , then these families would correspond to maps to M g , but these two maps
would be the same map t: S → M g . Moreover, if M g represented F then there would
be a family of curves over M g inducing families isomorphic to f and q 2 by means of
t. More directly, f and q 2 would be isomorphic, but there is no such isomorphism.
This is usually the example presented to argue that M g does not represent
F . However, we see from the above diagram that the family f is trivial after an
étale base change. So we may still ask whether F ét is representable. Elements of
F ét (S) are not anymore families over S but noneffective patching data of families

MODULI SPACES 5
over S. But this kind of situation happens for other functors, for example, the one
associated with relative Jacobians, and is not a big issue.
In fact, the étale associated sheaf Gét of the quotient functor G for the action
σ is isomorphic to F ét as we will argue soon. (This shows in particular that M g
corepresents F .) So, in principle, it could be that M g represents F ét . But this fails
as well.
In fact, F ét is not even an algebraic space. The argument given in the last
chapter to show that Gét is an algebraic space requires that the stabilizers of the
action be trivial. This is not the case: a nontrivial automorphism of C induces
an isomorphism of the space of sections of ω ⊗3
C , and thus an automorphism of PN ,
where C is tri-canonically embedded. This automorphism of P N induces that of C
by restriction, so is nontrivial. Since it sends C to itself, it fixes the point in H
corresponding to C.
Let us argue that Gét is isomorphic to F ét . Since H represents Hilb P (t) , there
is a “universal” subscheme X ⊆ P N × K, whose projection µ: X → K is a family
of curves of genus g. Given an element of G(S), thus the equivalence class of a map
f : S → K, consider the fibered product
X × K S −−−→ X
⏐ µ S
⏐↓ ⏐
µ ↓
S
f
−−−→ K.
Then µ S is a family of curves of genus g over S, thus defining an element of F (S).
Another representative of the equivalence class of f would yield an isomorphic family.
Also, it is clear by the above construction that this construction is compatible with
the functions associated to maps of varieties by F and G, thus defining a natural
transformation G → F . Buy their very definitions, it is clear that this defines a
natural transformation Gét → F ét .
Conversely, let S be a variety, S ′ → S an étale surjection and f : C ′ → S ′ a
family of curves of genus g yielding an element of F ét (S). Up to changing S ′ by a
Zariski covering, we may assume that the bundle f ∗ (ω ⊗3
C ′ /S
) of sections of the relative
′
tri-canonical bundle is trivial, that is, isomorphic to the trivial bundle of rank 5g−5.
Choosing a trivialization, we obtain a closed embedding ι: C ′ → P N × S ′ such that
p 2 ι = f, where p 2 : P N × S ′ → S ′ is the second projection. We thus have a map
S ′ → K, giving rise to an element of G(S ′ ). However, the fact that f yields an
element of F ét (S), and not simply F (S ′ ), means that there is an étale surjection
S ′′ → S ′ × S S ′ such that the images of f in F (S ′′ ) under the two projections
q 1 : S ′′ → S ′ and q 2 : S ′′ → S ′ are the same. Call f 1 : C 1 × S ′′ and f 2 : C 2 × S ′′ the

6 EDUARDO ESTEVES
induced families. In other words, we have fibered products:
C 1 −−−→ C ′
⏐ f ⏐↓ ⏐
1 f↓
S ′′ q 1
−−−→ S
′
and
C 2 −−−→ C ′
⏐ f ⏐↓ ⏐
2 f↓
S ′′ q 2
−−−→ S ′ .
The fact that they are equal in F (S ′′ ) means there is an isomorphism between
them. This isomorphism induces isomorphisms between the bundles of sections of
the tricanonical relative bundles of the two families, which are trivial, thus giving
rise to a map S ′′ → PGL(N + 1) establishing an automorphism of P N × S ′′ that
restricts to an isomorphism between the embedded C ! and C 2 . This means that the
element of G(S ′ ) constructed above induces tyhe same element (equivalence class)
in G(S ′′ ), whether we take the element induced by q 1 or by q 2 . Thus the element of
G(S ′ ) gives rise to an element of Gét (S).
It remains to see that this element of Gét (S) is independent of the choices
made. So, we would have to consider another étale surjection T → S and another
family of curves h: Y → T of genus g yielding the same element of F ét (S). This
means there would be an étale surjection U → T × S S ′ such that the induced families
of curves over U through the first and second projections are isomorphic. We would
have to replace T by a Zariski covering to get a map T → K. In the end we will get
that this map and S ′ → K constructed above give the same element of Gét (S).
Finally, we would have to see that the natural transformations constructed,
Gét → F ét and F ét → Gét are inverse to each other. This takes some work but is
rather simple.
If our functor F parameterizes only curves without nontrivial automorphisms,
we can show that F ét is an algebraic space. (As a matter of fact, F ét is indeed a
scheme.) In fact, we need a little more, as the triviality of the stabilizers shows only
that the natural map
(σ, p 2 ): G × K −→ K × K
is injective. We need to observe as well that the automorphisms of a smooth curve
do not deform, not even infinitesimally. This is simply the fact that C has no global
vector field for g ≥ 2.
Finally, a word about stacks. We want the diagram
G × K
⏐
p 2
⏐↓
K
σ
−−−→ K
⏐ ⏐↓
−−−→ F ét
to be Cartesian. This would allow us to study F ét by means of K and its action, as
if K were an atlas of F ét . The idea is to replace F ét by something else. We would
enlarge the category of schemes by another category, a category in which we can

MODULI SPACES 7
in a similar way construct a quotient object and in which the diagram would be
Cartesian. This is the category of stacks. A stack is not a functor but a pseudofunctor,
or a 2-functor, taking objects in a category, in our case (Varieties), to the
category of grupoids, where grupoids are categories themselves.

8 EDUARDO ESTEVES
References
[1] S. Aronhold. Zur Theorie der homogenen Functionen dritten Grades von drei Variablen. Crelle
39 (1850), 140–159.
[2] J. Dieudonné and J. Carrell. Invariant Theory, old and new. Academic Press, New York, 1971.
(Also, Adv. Math. 4 (1970), 1–80.
[3] Gordan. Beweiss dass jede Covariante und Invariante einer binären Form eine ganze Function
mit Coefficienten einer endlicher Anzahl solcher Formen ist. Crelle 69 (1868), 323–354.
[4] A. Grothendieck with J. Dieudonné. Éleménts de Géometrie Algébrique IV-4. Inst. Hautes
Études Sci. Publ. Math. 32 (1967).
[5] W. J. Haboush. Reductive groups are geometrically reductive. Ann. of Math. 102 (1975), 67–83.
[6] D. Hilbert. Über die Theorie der algebraischen Formen. Math. Annalen 36 (1890) 473–534.
[7] H. Kraft and C. Procesi. Classical Invariant Theory, a primer.
[8] S. Mukai. An introduction to Invariants and Moduli. Cambridge studies in advanced mathematics
vol. 81, Cambridge University Press, Cambridge, 2003.
[9] D. Mumford. The red book of varieties and schemes. Lecture Notes in Math., vol. 1358,
Springer, Berlin Heidelberg, 1999.
[10] P. E. Newstead. Introduction to moduli problems and orbit spaces. Tata Lectures on Math.
and Physics, vol. 51, Springer, Heidelberg, 1978.
[11] G. Salmon. Higher plane curves. W. Metcalfe and Son, Cambridge, 1879.
[12] T. Shioda. On the graded ring of invariants of binary octavics. Amer. J. Math. 89 (1967)
1022–1046.
[13] B. Sturmfels. Algorithms in invariant theory. Springer, Wien, 2nd ed., 2008.
[14] E. Weisstein. Cubic Curve. From MathWorld – A Wolfram Web Resource. Available at
http://mathworld.wolfram.com/CubicCurve.html.
[15] H. Weyl. The classical groups. Princeton University Press, Princeton, 1939.