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Universit`

a degli Studi di Pisa

Facolta di Scienze Matematiche, Fisiche e Naturali

Corso di Laurea in Matematica

Tesi di Laurea Triennale

The stack of admissible double covers

12 maggio 2017

Candidato

Federico Scavia

Relatore

Prof. Angelo Vistoli

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3

Introduction

Fix an integer g ≥ 2. Then there exists a 3g −3-dimensional moduli space Mg of smooth projective

curves of genus g, and a more refined object, the stack Mg.

In what follows we will assume that we are working on a field of characteristic not equal to 2. Recall that a smooth projective curve C is hyperelliptic if there exists an involution σ of C such that C/hσi is the projective line. In this case σ is well known to be unique. One can define a family of hyperelleptic curves of genus g as a family C → S with an involution σ, such that the quotient C/hσi → S is a family of curves of genus 0. This is a good definition, which yields a smooth algebraic stack1 H

g of dimension 2g − 1. The forgetful map Hg → Mg turns out to be a

closed embedding.

If Hg denotes the moduli space of Hg, this implies that the resulting map Hg → Mg is a closed

embedding, in characteristic 0 (to the best of our knowledge, this is an open question in positive characteristic).

Recall that Mg has a compactification Mg, the stack of stable curves of genus g. An obvious

question is: what does the closure Hg of Hg in Mg look like? The naive guess, that is, taking

families of stable curves C → S with an involution σ such that C/hσi → S is a family of curves of geometric genus 0 turns out to work, and gives a smooth stack Hg containing Hg as open

substack, such that the forgetful map Hg → Mg is closed embedding.

However, there is another possible compactification eHg of Hg defined as a stack of admissible

covers.

To explain it, we start from the problem of compactifying some of the most classical moduli spaces: the Hurwitz schemes Hd,g. These are fine moduli spaces for branched covers of P1, that

is, pairs (C, f ) where C is a smooth curve of genus g and f is a simply branched non-constant morphism of degree d2. In order to construct these schemes, one may first try to parametrize the ramification divisors R ⊆ P1 of the covers. By the Riemann-Hurwitz formula, every such R has

degree b := 2d + 2g − 2. Divisors of degree b in P1 are parametrized by (P1)b/S

b = Pb, and one

might hope to construct Hd,g as a cover of this parameter space. However, this is impossible for

very simple reasons:

• if a divisor admits a point with multiplicity greater than d, it cannot come from a cover of degree d,

• more generally, the number of covers with a given ramification divisor depends on the multiplicities of the points in the divisor.

However, these problems do not arise if we restrict ourselves to divisors consisting of b distinct points. These are parametrized by the dense open subscheme

Ub := ((P1)b\ ∆)/Sb = Pb\ ∆0

where ∆ is the union of all the diagonals in (P1)b and ∆0 is the discriminant locus. One may prove that there exist a quasi-projective variety Hd,g parametrizing ramified covers of P1 of degree

1Defining a family of hyperelliptic curves simply as a family in M

g whose fibers are all hyperelliptic is not

satisfactory, because being a hyperelliptic curve is not an open condition for any g ≥ 3. In particular, any infinitesimal deformation of a hyperelliptic curve would be a family of hyperelliptic curves, and this prevents this definition to give rise to an algebraic stack.

2There are many variants of this definition. For example, one of the most commonly used parametrizes covers

of P1with ordered ramification values. Here we allow the base curve to vary and we do not choose an ordering of the ramification values.

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b with simple branch points (the Hurwitz scheme), and an ´etale cover Hd,g → Ub. In particular

Hd,g has dimension of dimension b. The problem has now become to compactify Hd,g. Again, one

possible idea is to first try to compactify Ub and then try to somehow lift the compactification.

As we have discussed, the obvious compactification Ub ⊆ Pb, i.e. allowing the ramification points

to coincide, does not work.

One possible solution is to compactify Ub allowing the base to degenerate while keeping simple

ramification. More precisely, we first compactify Ub ⊆ M0,b inside M0,b. Then, we compactify

Hg,d by adding the so called admissible covers. The idea is that one lets C → P1 degenerate,

by simultaneously degenerating C to a nodal curve of arithmetic genus g, P1 to a nodal curve of

arithmetic genus 0, stable when marked by its ramification divisor, and f to a map which is an ´etale cover of degree b away from the singular locus and with a prescribed local behaviour around each node. In this way, one obtains a moduli space eHd,g, which is a projective variety, and that

contains Hd,g as a dense open subscheme.

Admissible covers of degree 2 date back at least to the article of Beauville [6], and their definition is generalized to any degree in the article of Harris and Mumford [10].

Two problems now arise. First, the definition in [10] is about families of admissible covers. In other words, not every map whose geometric fibers are admissible covers is a family of admissible covers. This is of course a bit dissatisfying. A first solution to this issue came in [19], using logarithmic structures. A second solution arrived in [1], through twisted covers.

Second, the description of the deformation spaces shows that the Hurwitz stack Hd,g is in

general not normal, but its normalization is always smooth. In [1], a modular description of its normalization is given, as a stack of twisted covers.

We now go back to the case of degree 2. We give two definitions of admissible double covers. The first one (3.1.1 and 3.3.4) is taken from [1], and (by removing the request that the degree be 2) immediately generalizes to that of admissible covers. The second one is used in [6], and it involves an automorphism of order 2 of the family (3.1.5). This extends the notion of the hyperelliptic involution for smooth curves, and allows us to view the stack of these covers (which we will denote by eHg) as a compactification of the stack Hg of smooth hyperelliptic curves of

genus g. By letting a more general group G act on the family, instead of the cyclic group of order 2, one obtains the notion of an admissible G-cover. As we have said, these two notions coincide in the case of double covers.

The question is now to understand how these two stacks are related. There is a natural map from eHg to Hg, given by associating to a family X → S its stable model (4.1). This morphism is

bijective on geometric points, so one might try to prove that it gives an isomorphism of eHg with

Hg, which is to say that eHg → Mg is also a closed embedding. This is not true, because the map

in fact admits some ramification. However, the ramification may be computed explicitly, and this allows us to prove that, over a field of characteristic 0, the induced map between coarse moduli spaces eHg → Mg is a closed embedding with image Hg, giving an isomorphism eHg ∼= Hg.

We now describe the content of the thesis in greater detail. In the first chapter we start from the definition and the basic properties of a nodal curve over an algebraically closed field. We discuss their ´etale-local structure and their line bundles, especially the dualizing sheaf. We then turn to families of nodal curves, using the properties of their fibers to obtain information on the whole family, locally in the ´etale topology. For example, ´etale-locally on the base, every such family is projective. We will spend time constructing quotients and studying the completions and the strict henselizations of the local rings at the nodes. We will describe how the dualizing sheaves of the various fibers glue together to a globally defined invertible sheaf, called the dualizing sheaf of the family.

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5

that the two definitions we give are equivalent. We will then be in the position to define the stack of admissible double covers eHg. The chapter ends with the study of the deformation theory of

e

Hg. This will be useful in the third chapter, and will immediately prove that eHg is a smooth

compactification of Hg.

In the third chapter, we study the operation of passing to the stable model. This gives a morphism eHg → Hg, called the collapsing map. A combinatorial argument shows that the stable

model determines the admissible cover. Combined with the result that every stable hyperelliptic curve comes from an admissible cover, this gives bijectivity on geometric points. The map is proper because eHg is proper. We then turn to the study of the deformation theory of the map

to prove (in characteristic 0) that the induced map between coarse moduli spaces eHg → Hg is

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Contents

1 Nodal curves 9

1.1 Basic properties . . . 9

1.2 Local structure . . . 13

1.3 Families of nodal curves . . . 15

1.4 Dualizing sheaf . . . 19

1.5 Quotients by involutions . . . 20

2 Hyperelliptic curves 23 2.1 Smooth hyperelliptic curves . . . 23

2.2 Stable curves . . . 24

2.3 Hyperelliptic stable curves . . . 25

3 Admissible covers 27 3.1 The admissible involution . . . 27

3.2 Ramification . . . 31

3.3 The stack of admissible covers of hyperelliptic type . . . 34

3.4 Deformation theory . . . 38

4 The collapsing map 45 4.1 The stable model . . . 45

4.2 Coarse moduli spaces . . . 49

4.3 Injectivity on geometric points . . . 51

4.4 Ramification . . . 53

A The stack of nodal curves 57 B Algebraic results 61 B.0.1 Power series . . . 61

B.0.2 Quotients . . . 62

B.0.3 Noetherian approximation . . . 65

B.0.4 Trace and involutions . . . 65

B.0.5 Henselian rings . . . 66

B.0.6 Grothendieck’s Existence Theorem . . . 68

B.0.7 Other results . . . 68

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Chapter 1

Nodal curves

1.1

Basic properties

Definition 1.1.1. Let k be an algebraically closed field. A curve over k is a purely one dimen-sional scheme of finite type over k.

Definition 1.1.2. Let C be a curve over an algebraically closed field k. A closed point p ∈ C(k) is called a node if the complete local ring bOC,p is isomorphic to k[[x, y]]/(xy). The curve C is

called a nodal curve if every closed point of C is either a smooth point or a node.

Definition 1.1.3. Let C be a nodal curve over the algebraically closed field k, and let p ∈ C be a node. The two minimal primes of bOC,p are called the branches of the curve C at the point p.

Proposition 1.1.4. Let C be a nodal curve over the algebraically closed field k, π : N → C be the normalization morphism, and let L be a line bundle on C. There is an exact sequence of coherent sheaves

0 → L → π∗π∗L → ⊕p nodek → 0.

Proof. Since π is finite, the pushforward of a coherent sheaf is coherent, so π∗π∗L is coherent.

Start by considering the natural map L → π∗π∗L, and let Q be the cokernel (which is also a

coherent sheaf, being a quotient of coherent sheaves). This is an isomorphism in the smooth locus of C. Hence Q is supported at the nodes of C. Fix a node p ∈ C and choose an isomorphism

b

OC,p ∼= k[[x, y]]/(xy). After formal completion the exact sequence becomes

0 → k[[x, y]]/(xy) → k[[x]] ⊕ k[[y]] → bQp → 0

which implies that the map L → π∗π∗L is injective and that bQp ∼= k, so that Qp ∼= k.

Lemma 1.1.5 ([21] 13.2.6). Let k be an algebraically closed field and let C be a curve over k. Then for any node p ∈ C(k) there exists an ´etale morphism U → C whose image contains p and an ´etale morphism

U → Spec k[x, y]/(xy) sending a point over p to the point x = y = 0.

Proof. Let π : N → C be the normalization. By B.0.25 there exists an ´etale neighbourhood U → C such that N ×CU = V1` V2 with Vi connected and such that there exist pi ∈ Vi mapping

to p. We can choose U = Spec R to be affine, so that Vi = Spec Ai are also affine. The proposition

above with L = OC yields

0 → R → A1× A2 → k → 0

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where the last map is given by the difference of the evaluation maps at p1 and p2. We may also

assume (by further restriction, if necessary) that there exists an element u ∈ R whose image in A1 is a uniformizer at p1 and mapping to 0 in A2. Symmetrically, we can also find v mapping to

a parameter in A1 and to 0 in A2. Then uv = 0, so we get a map

k[u, v]/(uv) → R.

This ´etale in a neighbourhood of p since it induces an isomorphism of completions (B.0.32). Corollary 1.1.6. Let C be a nodal curve over the algebraically closed field k, and let p ∈ C be a node. Then the strict henselization of C at p (see B.0.5) is isomorphic to that of k[x, y]/(xy) at the origin.

Proposition 1.1.7. Let C be a nodal curve over the algebraically closed field k, with δ nodes and irreducible components of geometric genera g1, . . . , gν. Then the arithmetic genus of C is

pa(C) =

X

gi+ 1 − ν + δ.

Proof. Call π : N → C the normalization and consider the exact sequence

0 → OC → π∗ON → kδ→ 0.

This gives a long exact sequence

0 → H0(C, OC) → H0(C, π∗ON) → kδ → H1(C, OC) → H1(C, π∗ON) → 0.

Since π is finite, for any coherent sheaf F on N , the higher direct images Riπ

∗F vanish, and

hence, by the Leray spectral sequence, Hi(N, F ) = Hi(C, π∗F ) for any i. Thus Hi(N, ON) =

Hi(C, π

∗ON) for i = 0, 1. This gives:

pa(C) = pa(N ) + δ.

Now N is a disjoint union of ν curves of geometric genera gi, so

pa(N ) =

X

(gi− 1) + 1 =

X

gi+ 1 − ν.

Proposition 1.1.8. Let C be a nodal curve over the algebraically closed field k of arithmetic genus pa(C) = g, and let L be a line bundle on C. Then

χ(L) = deg L + 1 − g

Proof. Consider the normalization morphism π : N → C. The exact sequence of 1.1.4 yields

χ(L) = χ(π∗L) − δ

where δ is the number of nodes of C. Consider the irreducible components of N N1, . . . , Nν of

geometric genera g1, . . . , gν. These are also the connected components of N . Apply standard

Riemann-Roch to each component to get

χ(π∗L|Ni) = deg π ∗

L|Ni+ 1 − gi.

Take the sum of the above equations to get

χ(π∗L) = deg π∗L + ν −Xgi = deg L + ν −

X gi

so that, applying the genus formula for C

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1.1. BASIC PROPERTIES 11

We now describe the dualizing sheaf of a nodal curve.

Theorem 1.1.9. Let k be a field and X a projective k-scheme of pure dimension r. Then there exist uniquely (up to a unique isomorphism) a coherent OX-module ωX and an isomorphism

ηX : Hr(X, ωX) → k such that, for any coherent OX-module F and integer p, there exists a

canonical pairing

Hp(X, F ) × Extr−pO

X(F, ωX) → H r

(X, ωX) → k

which is always non-degenerate for p = r and is non-degenerate if and only if X is Cohen-Macaulay. Furthermore, if X is a closed subscheme of Pnk, then

ωX = ExtOn−k Pnk

(OX, OPnk(−n − 1)).

If X is smooth, then ωX = det ΩX/k. If X is a smooth curve, ηX is defined by the classical residue

symbol.

Proof. [2] 1.3.

Definition 1.1.10. The sheaf ωX is called the dualizing sheaf of X.

Let C be a nodal curve over the algebraically closed k, with normalization π : N → C. Call Z the divisor of the nodes of C, and let D ⊆ N its inverse image in the normalization. Set U = N \ D and call j : U → N the inclusion. We start by definining

ΩN(log D) ⊆ j∗ΩU

to be the image of the natural map

ΩN ⊗ ON(D) → j∗ΩU.

Concretely, ΩN(log D) is the sheaf of meromorphic differentials that have at most simple poles at

D. To see this, pick a point q ∈ D and call z ∈ ON,q a uniformizer. Then the stalk of ΩN(log D)

at q is generated by dzz . For every point q ∈ D, we have the residue map

resq : ΩN(log D) → k(q)

given by setting resq(dzz) = 1. We obtain a map

π∗ΩN(log D) → ⊕p nodek(p)

by choosing for each node p ∈ C an ordering of the two points of the fiber and taking the difference of the the residues. Let L the kernel of this map (which of course is independent of the orderings). This sheaf is invertible, because this can be verified ´etale locally.

Proposition 1.1.11. The sheaf L represents the contravariant functor

H1(C, −)∨ : (line bundles on C) → (Set).

In particular L ∼= ωC and π∗ωC ∼= ΩC(log D).

Proof. [21] 13.2.9.

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1. The degree of ωC is 2g − 2.

2. The vector space H0(C, ω

C) has dimension g.

Proof. By the dualizing property h0

C) = h1(OC) = g and h1(ωC) = h0(OC) = 1, so by 1.1.8

g − 1 = deg ωC + 1 − g ⇒ deg ωC = 2g − 2.

Proposition 1.1.13. Let C be a nodal curve over the algebraically closed field k and let L be an invertible sheaf on C. Then L is ample if and only if its pullback to each irreducible component of the normalization eC has positive degree.

Proof. The normalization eC → C is finite and surjective. Hence by [11] III.5.3 L is ample if and only if its pullback to the normalization is ample. We conclude since for smooth curves being ample is equivalent to having positive degree, by [11] IV.3.1.

We conclude this section by proving that one may glue smooth points of a nodal curve into a node.

Proposition 1.1.14. Let C be a (non necessarily connected) nodal curve over the algebraically closed field k. Consider two distinct smooth points p1 and p2 of C. There exist a nodal curve C0

and a map π : C → C0 with the following properties:

1. as a topological space, C0 is the quotient of C under the equivalence relation p1 ∼ p2, and π

is the projection map;

2. if V is an open subset of C0, then OC0(V ) = {h ∈ OC(π−1(V )) : h(p1) = h(p2)},

3. the map π is an isomorphism away from p1 and p2.

Remark 1.1.15. We say that C0 is obtained from C by glueing of the points p1 and p2.

Proof. We define C0 as a ringed space using 1 and 2 and we show that it is covered by open subspaces that are affine schemes. This is obvious away from p = π(p1) = π(p2). Consider an

affine open subscheme U = Spec A of C containing p1 and p2, and let U0 be its image in C0 (so

that U = π−1(U0)). After restricting U , we may assume that A is a domain. Let f1 and f2 ∈ A

be the generators of the ideals of p1 and p2. Since p1 and p2 are distinct we have (f1, f2) = A,

hence (f1) ∩ (f2) = (f1f2). Let A0 = k ⊕ (f1f2)A ⊆ A as a k-vector space, with the ring structure

defined by

(λ, f ) · (µ, g) = (λµ, λg + µf ).

We study the map ϕ : Spec A → Spec A0 induced by the injective map (λ, f ) 7→ λ + f . If q is a prime of A different from p1 and p2, then

ϕ(q) = (0) ⊕ f1f2q.

We also have

ϕ(p1) = ϕ(p2) = (0) ⊕ (f1f2).

This implies that Spec A0 is the quotient of Spec A by the equivalence relation p1 ∼ p2 and that

ϕ is the projection map (the injectivity follows because A is a domain).

In order to prove that U0 ∼= Spec A0 we still have to show that the sheaves of functions are the same. If h ∈ OC(U ), then h(p1) = h(p2) if and only if for some λ ∈ k we have

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1.2. LOCAL STRUCTURE 13

This is equivalent to h − λ ∈ (f1) ∩ (f2) = (f1f2), that is, h belongs to the image of A0 → A. The

same reasoning works on every principal open subset of Spec A0, and these subsets form a basis of both Spec A0 and U0 (under the identification of the previous paragraph), hence U0 ∼= Spec A0 as schemes. The fact that π is an isomorphism away from p1 and p2 is obvious.

We now prove that C0 is a nodal curve. The only non obvious point is to show that p is a node of C0. Denote by bA the completion of A along the ideal (f1f2). Then the map

b

A → k[[x]] ⊕ k[[y]]

given by x 7→ f1 and y 7→ f2 is an isomorphism. The completion of of A0 with respect to the

maximal ideal p = (0) ⊕ (f1f2) is then given by the kernel of the map

k[[x]] ⊕ k[[y]] → k

sending (f, g) to f (0) − g(0), which is just k[[x, y]]/(xy).

1.2

Local structure

Definition 1.2.1. Let k be a field and X → Spec k be a one dimensional scheme over k. We say that a closed point x ∈ X is a node of X if it is the image of a node of Xk along the natural morphism Xk → X.

Remark 1.2.2. Of course, if k is algebraically closed, this notion of node coincides with the old one.

Lemma 1.2.3. Let k be a field and X a one dimensional scheme over k. Let p ∈ X be a closed point. The following are equivalent:

1. p is a node;

2. the complete stalk at p bOp is reduced, and given a presentation

b

Op ∼= k[[x1, . . . , xn]]/(f1, . . . , fr),

the (n − 1) × (n − 1) minors of the Jacobian matrix (δfi/δxj) generate the maximal ideal

mp ⊆ bOp.

3. the field extension k(p)/k is separable, bOX,p∼= k[[x, y]]/(ax2+bxy+cy2), where ax2+bxy+cy2

is a non degenerate quadratic form over k;

4. any p ∈ Xk mapping to p defines a nodal singularity.

Proof. First of all, let us see that if p is a node then k(p)/k is separable. Call p is a node of Xk mapping to p, and Z the closed subscheme defined by the first Fitting ideal of ΩX/k (see [24] 07Z6

for more about Fitting ideals). Then Zk is the subscheme defined by the first Fitting ideal of ΩXk/k. It is supported at the nodes and is reduced at every node: if bOp ∼= k[[u, v]]/(uv), its ideal is

(u, v). This implies that Z is geometrically reduced at p. In a neighbourhood of x, Z = Spec k(p), so k(p)/k is geometrically reduced, and this implies that k(p)/k is separable.

Next recall the following algebraic fact:

Lemma 1.2.4. If A is a noetherian ring, M is a finitely generated A-module, and I is an ideal of A, then the natural map M ⊗AA → cb M is an isomorphism, where completions are along I.

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Proof. [17] 8.7

From this it is clear that 3 implies 4, since every quadratic form on k can be put into the form xy with a linear change of variables. It is also obvious that 4 implies 1. We now prove that 1 implies 2.

Let p ∈ X be a node. We fix an isomorphism

b

Op ∼= k[[x1, . . . , xn]]/(f1, . . . , fr),

and we also know that

b

Op⊗kk ∼= k[[x, y]]/(xy).

Let J be the ideal generated by the determinants of the (n − 1) × (n − 1) minors of the Jacobian matrix (∂fi/∂xj). It is the first Fitting ideal of ΩX/k. We start by proving that J = mp. In fact,

since Spec k → Spec k is faithfully flat and the Fitting ideals behave well with respect to base change, it suffices to see this after having made base change to k. In this case the thesis is easily verified by passing to the presentation of bOp⊗kk as k[[x, y]]/(xy), where the first Fitting ideal is

(x, y) and so equals the maximal ideal. Since the complete stalk is obviously reduced, this proves 2

We are left with proving that 2 implies 3. We keep the same notation for bOp. Suppose that

there exist i and j such that ∂fi/∂xj is invertible in k[[x1, . . . , xn]]. Then we can substitute fi to

xj as a variable: by B.0.3 we still have a system of variables, and by B.0.7 the ideal generated by

the minors may be computed in the new variables too. By repeating this operation until we can, we may assume that f1 = x1, . . . , fm = xm and ∂fi/∂xj ∈ m for i, j ≥ m + 1. Now the Jacobian

matrix has this shape

Im×m ∗ 0 M



where the coefficients of M are all in m. Hence the determinants of the minors belong to mn−m−1. This implies that m = n − 2 because the determinants must generate m. So we can assume that r = n − 1 f1 = x1, . . . fn−2 = xn−2, so that

b

Op ∼= k[[x, y]]/I

for some ideal I. The next lemma, applied to A = k[[x, y]]/(xy), proves that I is principal.

Lemma 1.2.5. Let (A, m) be a regular local ring of dimension 2 and let I ⊆ m be a radical ideal. Then I = m or I is principal.

Proof. Write I = (a1, . . . , ar). Since A is regular, it is a unique factorization domain ([17] 20.3)

and we can write ai = abi where a is a greatest common divisor of a1, . . . , ar. Since A is a

unique factorization domain, every prime ideal of height 1 is principal ([24] 0AFT). There are no height 1 primes over b1, . . . , br, so since dim A = 2 we get m = p(b1, . . . , br). Now if a is a unit

(b1, . . . , br) = I is radical and so equals m. On the other hand if a ∈ m then an ∈ (b1, . . . , br) ⊆ I,

so a ∈ I and this implies I = (a).

Since I is not equal to the maximal ideal and I 6= (0), we can then write I = (f ) for some nonzero f . The condition on the minors implies that the quadratic part Q(x, y) = ax2+ bxy + y2

of f is non degenerate. We want to find a change of variables x, y so that f is mapped to ax2+ bxy + cy2. Consider x1, y1 ∈ m2 and write

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1.3. FAMILIES OF NODAL CURVES 15

By nondegeneracy of Q we can choose x1 and y1 such that Q(x + x1, y + y1) ∈ m4. Inductively,

we find for each n xn, yn ∈ mn+1 such that

Q(x + · · · + xn, y + · · · + yn) ∈ mn+3.

To see this, just expand in xn, yn to get to the same equations of the first step. Now the required

change of basis is x 7→ x +P

n≥1xn, y 7→ y +

P

n≥1yn.

Corollary 1.2.6. Let L/k be a field extension, where both k and L are algebraically closed. Let X be a scheme over k, and assume that XL is a nodal curve over L. Then X is a nodal curve

over k.

Proof. Since Spec L → Spec k is faithfully flat and XL → Spec L is finitely presented, flat and

proper, X → Spec k is too (see for example [24] 02KN). Since Spec L → Spec k is flat, XL→ X is

too, so by the dimension formula (see [16] 4.3.12), given a closed point of XL its image is a closed

point too, and X → Spec k is purely one-dimensional. Now fix p ∈ X and take q ∈ XL mapping

to it. We may write

b

OX,p∼= k[[x1, . . . , xn]]/(f1, . . . , fr),

and

b

OXL,q ∼= L[[x1, . . . , xn]]/(f1, . . . , fr).

The Jacobian criterion ([16] 4.2.19) tells us immediately that if q is a smooth point than p is smooth too. On the other hand, we have seen in 1.2.3 a characterization of a node through an analogous of the Jacobian criterion. This proves instantly that if q is a node, then p is a node too, since being geometrically reduced may be verified on an arbitrary extension ([24] 0384). Proposition 1.2.7. Let S = Spec A be an affine scheme, and let X → S be a family of nodal curves, where X is a scheme. Then there exist a subring A0 ⊆ A that is a Z-algebra of finite

type and a scheme X0 → Spec A0 such that X0 → Spec A0 is a family of nodal curves and the

following diagram is cartesian:

X X0

Spec A Spec A0

Proof. Combine the general approximation result given in B.0.14 with 1.2.6. Definition 1.2.8. A node x of X is split if there are two minimal primes in bOx.

Let x be a node of X and choose a presentation of bOX,x as k[[x, y]]/(ax2+ bxy + cy2). The node

is split exactly when the quadratic form ax2+ bxy + cy2 splits, in which case bOX,x ∼= k[[x, y]]/(xy).

1.3

Families of nodal curves

Definition 1.3.1. Let S be a scheme. A family of nodal curves over S is a morphism from an algebraic space X to S that is flat, proper and of finite presentation, and such that for every geometric point s : Spec k → S (where k is algebraically closed), the geometric fiber

Xs := Spec k ×SX

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Proposition 1.3.2 ([21] 13.2.5). Let f : X → S be a family of nodal curves over a scheme S. Then, ´etale locally on S, f is projective.

Proof. Let X0 ⊆ X be the maximal open subset over which f is smooth. Since f is flat, x ∈ X is

in X0 if and only if x is a smooth point of the fiber Xf (x). In particular, X0 is dense in every fiber.

Fix a point s ∈ S, and let p1, . . . , pr be smooth points in Xs, one for each irreducible component

of Xs.

Lemma 1.3.3. Let f : X → Y be a smooth morphism, x ∈ X be a point and call y = f (x) ∈ Y . Assume that the field extension k(x)/k(y) is separable. Then there exists an ´etale morphism π : Y0 → Y with image containing y and a morphism s : Y0 → X such that f ◦ s = π.

Proof. [9] 17.16.3 (ii).

We combine this with [9] 17.16.2 (iii), which tells us that for every s ∈ S the points x ∈ Xs

such that k(x)/k(s) is separable form a dense subset of Xs. Hence, we may assume after an ´etale

base change that there exist sections q1, . . . , qr : S → X0 of f (not necessarily ´etale) such that

the image of qi contains pi. Since X0 → S is smooth of relative dimension 1, the image of each qi

is a Cartier divisor on X. This is because by [9] 17.12.1 any section of a smooth morphism is a regular embedding, and a regular embedding of codimension 1 is by definition a Cartier divisor. Tensoring the corresponding invertible sheaves, we get an invertible sheaf L on X. Since Ls is

ample by 1.1.13, there exists a Zariski neighbourhood of s ∈ S such that L is relatively ample in that neighbourhood [8] 4.1.7.

Definition 1.3.4. A closed embedding X → Y of schemes is called a regular embedding of codimension d if for every point x ∈ X, there exists an affine neighbourhood V = Spec A of x in Y such that X ∩ V ⊆ V is defined by a regular sequence in A of length d.

A morphism X → Y is called a local complete intersection morphism of codimension d if for every x ∈ X there exists a neighbhourhood U ⊆ X of x and a factorization

U −→ Pi −→ Yg

where i is a regular embedding of codimension e and g is smooth of relative dimension d + e. See one of [24] 06C3, [16] 6.3.2, [21] Exercise 13.A.

Proposition 1.3.5. A family of nodal curves X → S is a local complete intersection morphism.

Proof. The property of being a local complete intersection is ´etale local on domain and codomain. Hence we may assume that X is a scheme and S = Spec R. After noetherian approximation (1.2.7), we can even assume that R noetherian. In this case the proposition follows because for flat maps being a locally complete intersection morphism may be checked on the geometric fibers ([16] 6.3.15).

Remark 1.3.6. Let U ⊆ X be a schematically dense open subset (this is equivalent to U being topologically dense and containing all the associated points of X). Let F be a locally free sheaf on X and let s ∈ F (X) be a global section such that s|U = 0. Then, since the support of s is

closed and contains a schematically dense open subset, s = 0.

Now consider two locally free sheaves F , G and a morphism F |U → G|U. By the previous

reasoning applied to the sheaf Hom(F , G), we obtain that the morphism extends at most uniquely to X.

Proposition 1.3.7. Let X → S be a family of nodal curves. Then the arithmetic genus of the geometric fibers is locally constant (in the Zariski topology) on the base S.

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1.3. FAMILIES OF NODAL CURVES 17

Proof. Fix a geometric fiber s : Spec k → S. Assume that the thesis holds for an ´etale neighbour-hood of s. This means that there exists a commutative diagram

Spec k S S0

s

such that S0 → S is an ´etale morphism of schemes, and such that the fibers of X ×SS0 → S0 have

locally constant genus. Since an ´etale map is open, we obtain a Zariski open subset of S where all geometric fibers have the same genus as that of s. We can then assume that S = Spec R affine and X is a scheme. By noetherian approximation (1.2.7), there exists a ring map R0 → R and a scheme

X0 on Spec R0 such that X = Spec R ×Spec R0 X0 and X0 → Spec R0 is a family of nodal curves

whose fibers have locally constant genus. Then, for every geometric point s : Spec k → Spec R, we obtain the following cartesian squares:

Xs X X0

Spec k Spec R Spec R0

s

From this and from 1.2.3 it follows that we may assume R noetherian. The result then follows from flat base change along proper morphisms, since the fibers are all one dimensional, see B.0.33.

Now we discuss the smooth locus. Recall the following definition ([24] 0831 for more).

Definition 1.3.8. Let X → S be an algebraic space, U ⊆ X an open subspace. We say that U is schematically dense in X if one of the following two equivalent conditions is satisfied ([24] 0833):

• For every scheme V and every ´etale morphism V → X, the open subscheme U ×X V is

schematically dense.

• There exist a scheme V and a surjective ´etale morphism V → X such that U ×X V is

schematically dense.

Proposition 1.3.9. Let X → S be a flat, separated, finitely presented morphism whose fibers are nodal curves (for example an open subset of a family of nodal curves). Then the smooth locus is open, schematically dense on every fiber and schematically dense.

Proof. By picking an ´etale cover of X which is still a nodal curve, we may assume that X is as scheme. Also, we may assume that S = Spec R is the spectrum of a ring and that X is a scheme. Now, by noetherian approximation (1.2.7), we may assume that R is a noetherian ring, hence X, being of finite type over S, is a noetherian scheme. Then it is well known that the smooth locus of X → S is an open subscheme of X. Now by [9] 11.10.9 we see that being schematically dense is equivalent to being schematically dense on geometric fibers, which is true in our case.

Proposition 1.3.10. Let X → S be a family of nodal curves, s : Spec k → S be a geometric point and t : Spec k → Xs be a nodal point of the geometric fiber. Assume that 2 is invertible in

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X U W

S V

where the three horizontal maps are ´etale, U is an ´etale neighbourhood of t, V = Spec A is an affine ´etale neighbourhood of s, W = Spec A[x, y]/(xy − a) for some a ∈ A.

Proof. The question is ´etale local on X and S, so we can assume that S is an affine scheme Spec R and X is a scheme. By noetherian approximation 1.2.7, we can then assume that R is a finitely generated Z-algebra.

We now want to apply the following theorem:

Theorem 1.3.11. Suppose that X → S is a flat morphism of finite type, the fiber X0 over a

point s0 ∈ S is a curve over k(s0) with isolated singularities, and p ∈ X0 is a split node. Then

there exists u ∈ bOS,s0 and an isomorphism of bOS,s0-algebras

b

OX,p ∼= bOS,s0[[x, y]]/(xy − u).

Proof. [25] 7.5

The problem here is that the image t ∈ X of t is not necessarily split, that is, we do not have an isomorphism of the completed stalk with k[[x, y]]/(xy). We want to reduce to this case. Call s ∈ S the point in the image of s. The finite field extension k(t)/k(s) is separable, because of 1.2.3. Also, we know from the same lemma that, in the non split case, bOX,t∼= k[[x, y]]/(ax2+ bxy + cy2)

for some non degenerate form with, say, a 6= 0. Call L = k(t)[x]/(ax2 + bx + c) and notice that

L/k(t) is a separable field extension (because char k 6= 2), so L/k(s) is separable too.

Lemma 1.3.12. Let A be a ring and p be a prime of A. Let L/k(p) be a finite separable field extension. There exists an ´etale map A → A0 together with a prime p0 lying over p such that k(p0) ∼= L over k(p).

Proof. By the primitive element theorem, L = k(p)(α), with g a minimal polynomial for α over k(p). We may assume that g lifts to a polynomial f in R[x]: it is enough to replace α with cα for some c ∈ R, so that the coefficients of its minimal polynomial all come from R. Notice that g0 is invertible in L, so we can define a map of R-algebras, so the map R[x] → L given by x 7→ f induces a map R0 = (R[x]/(f ))f0 → L. Now R0 is an ´etale R-algebra, and if we call p0 the kernel

of the previous map, the pair (R0, p0) satisfies all requirements.

Applying this lemma to L/k(s), we obtain an ´etale neighbourhood V → S containing s for which X ×SV has a split node above x. In other words, we can assume that x is a split node of

X.

Now we can apply the previous theorem. Then we conclude by an application of Artin’s Approximation Theorem [5] 2.5 (which holds because the base ring is the strict henselization of the localization of a finitely generated Z-algebra, see 0CAW).

Corollary 1.3.13. Let X → S be a family of nodal curves, let t : Spec k → X be a node of X, s its image on S, and t, s the respective images on X and S. Then:

Osh

X,x ∼= OS,s[x, y]sh/(xy − a)

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1.4. DUALIZING SHEAF 19

Proposition 1.3.14. Let A be a local henselian ring with maximal ideal m, a ∈ m, and set R = A[x, y]sh/(xy − a). Let x0 and y0 ∈ R such that x0y0 ∈ A and the ideals (x, y, m), (x0, y0, m)

are equal. Then, after switching x0 and y0 if necessary, there exist units ux, uy ∈ R∗ such that

uxvx ∈ A∗ and x0 = uxx, y0 = uyy.

Proof. [19] 3.7.

1.4

Dualizing sheaf

Definition 1.4.1. Let f : X → S be a local complete intersection morphism of codimension d, and suppose there exists a global factorization

X −→ Pi −→ Sg

of f with i a regular immersion of codimension e and g smooth of relative dimension d + e. We define the relative dualizing sheaf of f as:

ωX/S := i∗ExteOP(i∗OX, ∧ d+e

Ω1P /S) = det(I/I2)∨⊗OX i ∗

(det ΩP /S)

where I is the ideal sheaf of X inside P .

Proposition 1.4.2. The sheaf ωX/S is an invertible sheaf. It is independent of the choice of

factorization of f .

Proof. [16] 6.4.5.

Let f : X → S be a proper morphism which is a local complete intersection morphism, and which ´etale locally on S is projective. Then there exists a sheaf ωX/S obtained by ´etale locally on

S choosing projective immersions of X over S, using the independence of the choice of immersion to glue.

Definition 1.4.3. For such a morphism f : X → S, the invertible sheaf ωX/S is called the

relative dualizing sheaf.

Remark 1.4.4. The independence of the choice of factorization tells us that this definition agrees with the previous one, when a global factorization exists (it may be verified ´etale locally).

Proposition 1.4.5. Let f : X → S be a proper morphism which is a local complete intersection morphism, and which ´etale locally on S is projective, so that we have defined ωX/S. Let S0 → S be

a morphism. Set X0 = X ×S S0 and call p : X0 → X the projection. If either S0 → S or X → S

is flat, then X0 → S0

is also a local complete intersection and the natural map

ωX0/S0 → p∗ωX/S

is an isomorphism.

Proof. [16] 4.9.

Let X → S be a family of nodal curves. The previous discussion implies that there exists a dualizing sheaf ωX/S and that it is the correct dualizing sheaf on each geometric fiber.

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1.5

Quotients by involutions

We will consider proper nodal curves. Recall that a scheme X of dimension 1 proper over a field is automatically projective (see for example [24] 0A26).

Proposition 1.5.1. Let X → S be a projective morphism of schemes, and let G be a finite group of automorphisms of X over S. Then the quotient π : X → X/G exists in the category of schemes. Proof. By [16] 3.3.36 (b), we see that every G-orbit is contained in some G-invariant affine open subset of X. By [21] 6.2.2, this property is equivalent to the existence of the quotient.

This implies that if G is a finite group acting on X through automorphisms, then the geometric quotient X/G exists (as a scheme) and is of dimension 1. We will be exclusively interested in the case G =<σ > where σ is an involution with isolated fixed points.

Lemma 1.5.2. Let C be a nodal curve, σ an involution with isolated fixed points, q a point of D := C/ < σ > whose fiber consists of two points p1 and p2. Then there exists an ´etale

neighbourhood U → D of p such that U ×D C = V1` V2, where Vi are connected and the natural

map Vi → U is an isomorphism for i = 1, 2.

Proof. By B.0.25 there exists an ´etale neighbourhood U → D of p such that U ×D C = V1` V2,

the Vi are connected and the fiber of pi is contained in Vi. Consider on U ×D C the natural

involution given by pullback of σ. We still call this involution σ. Since V1 and V2 are the

connected components of U ×DC, either σ switches them or sends them to themselves. However,

the fiber of pi is contained in Vi for i = 1, 2, so σ(V1) = V2 and σ(V2) = V1. Since quotient

commutes with flat maps (B.0.11), U is the quotient of this action, and so the projection restricts to an isomorphism Vi ∼= U .

Proposition 1.5.3. Let k be an algebraically closed field of characteristic different from 2 and let C be a nodal curve over k. Consider an involution with isolated fixed points σ acting on C. Then C/ <σ > is a nodal curve.

More precisely, if p ∈ C is a node fixed by σ, we can choose an isomorphism bOp ∼= k[[x, y]]/(xy)

in such a way that the involution acts by x 7→ −x and y 7→ −y if the branches at p are σ-invariant, or by x 7→ y and y 7→ x if they are switched. If p ∈ C is a smooth point fixed by the involution, we can choose an isomorphism bOp ∼= k[[x]]/(x) in such a way that the involution acts by x 7→ −x.

Proof. It is well known that the quotient X/G is still a proper one-dimensional ([21] 6.2.11) connected k-scheme of finite type (B.0.10).

Let p ∈ C be a nodal point, mapping to q ∈ C/ <σ >. The fiber of q consists of one or two points, since < σ > acts transitively on the fibers by [21] 6.2.11. If the fiber of q consists of two points, then by 1.5.2 we have bOp ∼= bOq.

Assume that p is a fixed point of σ and fix an isomorphim bOp ∼= k[[x, y]]/(xy). Then, by B.0.11,

b

Oq is isomorphic to the subring of invariants of the induced involution of k[[x, y]]/(xy) (which we

still call σ). The ideals (x) and (y) are either sent to themselves or switched, since there are no other minimal primes.

Let us assume that the branches are σ-invariant. In this case, by quotienting k[[x, y]]/(xy) by (y) or by (x), we get involutions σx and σy on k[[x]] and k[[y]]. To begin with, we want to show

that both of these are nontrivial.

Consider π : N → C the normalization. By the universal property, the involution σ lifts to a unique involution τ of N , and we have σπ = πτ . Over p there are two points p1 and p2 and we

have the exact sequence

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1.5. QUOTIENTS BY INVOLUTIONS 21

If we let the cyclic group C2 of order 2 act as <σ > on the first term, as <τ > on the second and

trivially on the third, the sequence is C2-equivariant, since σπ = πτ . The condition that σ does

not switch the branches tells us that τ sends bOpi to itself, for i = 1, 2. Now if σ acts trivially on a

branch, then τ acts trivially on one of the bOpi, say bOp1. By B.0.13 this implies that τ acts trivially

on Op1 too, which is a domain, so it acts trivially on a non empty open subset of N containing

p1. This gives that τ acts trivially on an irreducible component of N , but then σ acts trivially on

its image, absurd.

Since the involution of bOp comes from an automorphism of Op, it is continuous, hence the

involution of k[[x]] is determined by its value on x. By B.0.3, it is determined by its value on any f (x) =P

n≥1aix

i with a

1 6= 0, (since k[[x]] = k[[f (x)]]).

Our objective is now to find an element u such that k[[u]] = k[[x]] and σ(u) = −u. Decompose x as x = x++ x−, where x+= x + σ(x) 2 , x − = x − σ(x) 2

(recall that char k 6= 2). Clearly σ(x+) = x+ and σ(x) = −x. We may write σ(x) =P

j≥0cjx j

where c0 = 0 and c1 6= 0, since σ(x) ∈ (x). We obtain c1 = −1, otherwise

x+= c1 + 1

2 x + . . .

would be such that k[[x+]] = k[[x]] and so the involution would be equal to the identity, which we

have already excluded. This implies that x− = x + . . . and so u = x− satisfies σx(u) = −u and

k[[u]] = k[[x]], by B.0.3. Since (x) ∩ (y) = 0 in k[[x, y]]/(xy), we also have σ(u) = −u.

By applying the same reasoning to k[[y]] and changing variables, we may assume that the action on k[[x, y]]/(xy) is given by σ(x) = −x and σ(y) = −y. Now the ring of invariants is generated by x2 and y2, so it is of the form k[[s, t]]/(st).

Let us now assume that the involution switches branches at the node p. We may write b

Op = k[[x, y]]/(xy) where x 7→ uy and y 7→ vx, for some units u and v. Now x = σ(σ(x)) =

σ(uy) = σ(u)σ(y) = σ(u)v, so v = σ(u)−1 and if we let z = uy, we have found a presentation b

Op = k[[x, z]]/(xz) such that x 7→ z and z 7→ x. It is now clear that the ring of invariants is

k[[x + z]].

If p ∈ C is a smooth point that is fixed by the involution, we fix an isomorphism bOp ∼= k[[x]],

and call σ the induced action of the involution on bOp. We may again decompose x = x++ x−,

where σ(x+) = x+ and σ(x) = x. We must have x= x + . . . , otherwise x+ would be of the

form cx + . . . where c 6= 0 and the action would be trivial (it is not, because this would imply triviality on Op, hence on an open neighbourhood of p). By B.0.3, setting u = x− gives the

conclusion.

If A is a local ring, we denote by Ash its strict henselization.

Proposition 1.5.4. With the notation of the previous proposition, if p is a node fixed by σ and the branches at p are σ-invariant, fixed an isomorphism Oshp ∼= k[x, y]sh/(xy), we may find w, z such that (x) = (w), (y) = (z) and w 7→ −w, z 7→ −z.

Proof. We start by proving the following lemma.

Lemma 1.5.5. Let A → R be a faithfully flat map, and let a, b ∈ A. If a = br for some r ∈ R, then a = bc for some c ∈ A.

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Proof. The map f : A/bA → R/bR is also faithfully flat, hence injective by B.0.22. The condition a = br means f (a) = 0, hence a = 0, that is, a = bc for some c ∈ A.

We fix some isomorphism Osh

p ∼= k[x, y]sh/(xy), using 1.1.5. We write x = x+ + x

, where

σ(x+) = x+ and σ(x) = x. We must have x= x + . . . , otherwise (x) = (x+) in bO

p and

the action would be trivial (by B.0.3), but we know from 1.5.3 that the action is nontrivial on each branch. Hence (x) = (x−) in bOp, so by B.0.21 and the previous lemma we conclude that

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Chapter 2

Hyperelliptic curves

2.1

Smooth hyperelliptic curves

Definition 2.1.1. A family of smooth curves of genus g is a morphism X → S of schemes that is flat, proper and finitely presented whose fibers are smooth curves of genus g.

Remark 2.1.2. Consider a morphism X → S, where S is a scheme and X is an algebraic space, that is flat, proper and finitely presented and whose fibers are smooth curves of genus g. By [21] 8.4.6, any such morphism is projective. In particular X is a scheme.

Definition 2.1.3. A family of smooth hyperelliptic curves of genus g is a commutative triangle

X Y

S

where X → S is a family of smooth curves of genus g, Y → S is a family of smooth curves of genus 0 and X → Y is a finite ´etale morphism of degree 2 (a double cover).

Definition 2.1.4. The category Hg of hyperelliptic curves of genus g is the category defined as

follows:

• its objects are families (X → Y → S) of hyperelliptic curves of genus g;

• an arrow from two objects (X0 → Y0 → S0) to (X → Y → S) is a triple of morphisms

a : X0 → X, b : Y0 → Y and c : S0 → S of schemes such that the following squares are

cartesian: X0 X Y0 Y S0 S f

Composition of arrows is defined in the obvious way.

The category Hg is a fibered category in groupoids over Spec Z[1/2].

Proposition 2.1.5. The fibered category Hg is equivalent to the fibered category A over Z[1/2]

defined as follows:

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• its objects are pairs (X → S, σ), where X → S is a family of smooth curves of genus g (where 2 ∈ OS is invertible) and σ is an automorphism of X such that σ2 = idX, σ 6= idX

and X/ <σ > is a family of smooth curves of genus 0;

• an arrow from (X0 → S0) to (X → S) is a pair of morphisms of schemes f : X0 → X and

S0 → S such that the diagram

X0 S0

X S

f

is cartesian and σ ◦ f = f ◦ σ0. Composition of arrows is defined in the obvious way.

Proof. Let us first describe a functor Hg → A. An object (X → S, σ) is sent to the object

(X → X/ <σ >→ S). One can define the image of a morphism

X0 X

S0 S

f

in the obvious way, after having noticed that, if π : X → X/ <σ > and π0 : X0 → X0/ <σ0>

are the two projections, we have

πf σ0 = πσf = πf

so that X0 → X → X/ <σ> factors uniquely through X0/ <σ0> by universal property.

On the other direction, an object X → Y → S may be sent to (X → S, σ) where σ is the hyperelliptic involution of the cover (B.0.18). The map on morphisms

X0 X Y0 Y S0 S f

is defined by forgetting the middle horizontal arrow, but it must be verified that f σ0 = σf . This follows from the local description of the involution, see B.0.4.

It is obvious that the two functors are quasi-inverse to each other.

Theorem 2.1.6. The fibered category Hg is an irreducible smooth Deligne-Mumford stack of finite

type over Spec Z[1/2], of relative dimension 2g − 1. Proof. [4] 4.2.

2.2

Stable curves

Definition 2.2.1. Let k be an algebraically closed field. A nodal curve C of genus g ≥ 2 over k is called stable if Aut C is a finite group.

Proposition 2.2.2. A nodal curve C of genus g ≥ 2 over k is stable if and only if every smooth rational component of C contains at least three nodes of C.

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2.3. HYPERELLIPTIC STABLE CURVES 25

A smooth rational component of C containing at most two nodes of C is called unstable.

Proof. If D ⊆ C is an unstable component, then there are infinitely many automorphisms of D that fix the nodes of C lying on D. These can be extended to automorphisms of C by setting them equal to the identity on every irreducible component different from D. Therefore C has infinitely many automorphisms.

For the other implication, consider the subgroup G < Aut(C) of those automorphisms of G that send every irreducible component to itself. Then G is a normal subgroup of Aut(C) and the quotient is a finite group: if C1, . . . , Cν are the irreducible components of C, Aut(C)/G embeds

in the symmetric group Sν. So it suffices to prove that every irreducible component D of C has

finitely many automorphisms respecting the divisor of nodes of C lying on D. By passing to the normalization of D we are reduced to the following lemma.

Lemma 2.2.3. Let C be a smooth curve of genus g, and let p1, . . . , pn be distinct points on C.

Assume that there are infinitely many automorphisms of C that send the divisor D = p1+ · · · + pn

to itself. Then C belongs to one of the following classes:

• g = 0, n ≤ 2, • g = 1, n = 0.

Proof. If g = 0, C ∼= P1 and it is easy to see that an automorphism of P1 is determined by the images of three points. If g = 1, fixing a point makes C into an elliptic curve, for which translations give an infinite family of automorphisms. It is then clear that the cases in the above list all have infinite Aut C.

On the other hand, Hurwitz’s theorem ([12]) tells us that when g ≥ 2 the group of automor-phisms of C has cardinality at most 84(g−1). It is also well known that the group of automorphism of an elliptic curve fixing the origin is finite, and in fact its order divides 24 ([23] III.10.1).

Since g ≥ 2 and C is connected, every component of genus 1 of C is automatically stable. This proves the claim.

Proposition 2.2.4. Let C be a nodal curve over the algebraically closed field k. Then C is stable if and only if the dualizing sheaf ωC is ample.

Proof. [21] 13.2.14.

Definition 2.2.5. Let S be a scheme, X → S be a family of nodal curves over S. We say that X is a family of stable curves if every geometric fiber is stable.

We will denote by Mg the stack of smooth curves of genus g ([21] 8.4.3), and by Mg the

stack of stable curves of genus g. The second one was first defined by Deligne and Mumford in [20] (see also [21] 13.2).

2.3

Hyperelliptic stable curves

Definition 2.3.1. A stable curve C of genus g ≥ 2 is called hyperelliptic if there exists an involution σ ∈ Aut(C) with isolated fixed points such that the quotient C/ < σ > is a nodal curve of arithmetic genus 0. We will also say that (C, σ) is a hyperelliptic stable curve.

Proposition 2.3.2. Let C be a stable curve of genus g ≥ 2. There exists at most one involution σ of C which has isolated fixed points and is such that C/ <σ > is a genus zero nodal curve.

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Proof. [3] X.3.5.

Proposition 2.3.3. Let (C, σ) be a hyperelliptic stable curve of genus g ≥ 2. Then σ is in the center of Aut(C).

Proof. Let ϕ be another automorphism of C and call π : C → C/ < σ > the projection to the quotient. The involution associated to πϕ is ϕσϕ−1. By uniqueness, ϕσϕ−1 = σ.

Definition 2.3.4. Define Hg to be the full subcategory of the inertia stack IMg whose objects

over a scheme S are pairs (X → S, σ) such that for every geometric point s of S the pair (Xs, σs)

is a hyperelliptic stable curve. It is clearly a substack of the inertia stack of Mg, called the stack

of hyperelliptic stable curves.

Proposition 2.3.5. The stack Hg is an open and closed substack of IMg. In particular, it is a

proper smooth stack over Spec Z[1/2].

Proof. We start by proving that Hg is open. Consider an object (X → S, σ) of Hg(S). The

condition that the quotients of the fibers be of arithmetic genus 0 is an open condition by 1.3.7. To see that the condition of having only isolated points as singularities is also open, consider the embedding j = (idX, σ) : X → X ×S X. The fixed points of the involution are exactly

Z := j−1(∆), where ∆ ⊆ X ×SX is the diagonal. Then the condition is open by an application

of Chevalley’s theorem ([9] 13.1.5) to the proper morphism Z → S.

To prove that Hg is closed, it suffices to prove that the inclusion map is representable, formally

unramified, proper and injective on geometric points, by [24] 04XV. Representability and unram-ifiedness follow from [24] 04ZZ, since by definition the inclusion is fully faithful. Injectivity on geometric points follows from 2.3.2. To prove properness, we use the valuative criterion. Suppose we have a commutative diagram:

Spec K Hg

Spec R IMg

α

where R is a discrete valuation ring and K is its quotient field. The morphism α corresponds to a pair (X Spec R, σ) where X → Spec R is a stable curve of genus g and σ : X → X is an automorphism over Spec R. Let η be the generic point of Spec R. The commutativity of the diagram implies that the pullback (Xη → Spec K, ση) to η is an element of Hg(Spec K). This

means that σ2

η = idXη and Xη/ < ση>→ Spec K is a smooth curve of genus 0. Since σ

2 and

idX coincide on the generic fiber Xη and X is reduced and separated over Spec R, σ2 = idX. By

1.5.3, since char k 6= 2, X/ < σ >→ Spec R is a family of nodal curves. Since the arithmetic genus is locally constant (1.3.7), the closed fiber of X/ < σ >→ Spec R has arithmetic genus 0. This proves that (X → Spec R, σ) is an object of Hg(Spec R), this object correspond to a morphism

Spec R → Hg fitting as a diagonal in the square. The uniqueness of such an arrow is clear, because

Hg(Spec R) injects into IMg(Spec R).

We will prove in 4.3.6 that the natural inclusion Hg ⊆ Hg gives a compactification of Hg.

Proposition 2.3.6. The natural map Hg → Mg is a closed embedding.

Proof. For every Deligne-Mumford stack X , the natural map IX → X is representable and

un-ramified ([15] 8.1). As a consequence, the map Hg → Mg is also representable and unramified.

It is proper by 2.3.5. By the uniqueness of the hyperelliptic involution (2.3.2) it is also injective on geometric points. Hence by [24] 04XV it is a closed embedding.

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Chapter 3

Admissible covers

3.1

The admissible involution

Definition 3.1.1. Let S be a scheme such that 2 is invertible in OS. An admissible double

cover f : X → P over S is a finite morphism such that:

1. X → S and P → S are families of nodal curves;

2. On geometric fibers, every node of X maps to a node of P , and conversely the inverse image of every node of P only consists of nodes of X;

3. The restriction of X → P to Pgen is ´etale and finite of constant degree 2. Here Pgen is the

complement of the locus of nodes and ramification points (it is an open subscheme of P ), and is called the generic locus;

4. If p ∈ X is a node over s ∈ S, mapping to q ∈ Y , there is a commutative diagram Osh p Os[x, y]sh/(xy − a) Osh q Os[u, v]sh/(uv − ae) Osh s

where a ∈ Oshs , u 7→ xe, v 7→ ye, for e = 1, 2, and the two horizontal arrows are isomorphisms.

Remark 3.1.2. Notice that the condition 1 above implies that f : X → P is finitely presented.

Definition 3.1.3. Let C be a nodal curve over an algebraically closed field k. An admissible involution of C is an automorphism of C over k satisfying the following conditions:

• σ is an involution, that is, σ2 = id C;

• σ has only isolated fixed points, that is, it does not fix any irreducible component of C; • assume that σ fixes a node p. Then σ acts on OC,p, hence on bOC,p, and we assume that the

action does not switch the two branches at p.

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Remark 3.1.4. By 1.5.3, we see that an involution σ of C with isolated fixed points is admissible if and only if there are no nodes of C mapping to a smooth point of the quotient curve.

Definition 3.1.5. Let X → S be a family of nodal curves over S. An admissible involution of X is an automorphism of X over S such that on every geometric fiber s : Spec k → S, the induced automorphism σs of the nodal curve Spec k ×SX over k is an admissible involution.

Proposition 3.1.6. Let X → S be a family of nodal curves over a scheme S, and let σ be an admissible involution. Assume that 2 is invertible in OS. Then the quotient X/ <σ > exists and

is a family of nodal curves over the same base S.

We will in fact find a geometric quotient for the action of G =< σ >, that is, an algebraic space X/G together with a map π : X → X/G such that:

• the map π is G-invariant and finite;

• the map of sheaves OX/G→ (π∗OX)G is an isomorphism.

This is equivalent to asking that the functor Hom(X, ·)Gis corepresentable, and that the universal object π : X → X/G is finite.

Proof. The condition that the sheaf HomS(X, ·)G on the category of algebraic spaces over S be

representable by an algebraic space is ´etale local on the base S ([21] 04U0). Hence we may assume that S = Spec R is an affine scheme and, by 1.3.2, that X is scheme. By noetherian approximation (1.2.7), there exists a cartesian diagram

X X0

Spec R Spec R0

such that R0 is a noetherian ring and X0 → Spec R0 is a family of nodal curves. We are reduced

to proving the thesis for X0 → Spec R0. In other words, we may assume that R is a noetherian

ring.

The existence of the algebraic space X/ < σ > is now guaranteed by ([14] Proposition 1.5 page 180). The same result gives that the projection is finite. The fact that the fibers of X/ < σ > are nodal curves follows from B.0.8 and 1.5.3. Flatness also follows from B.0.8. The finite presentation is consequence of B.0.9.

Theorem 3.1.7. Let X → S be a family of nodal curves and let σ be an admissible involution of X, with quotient Y . Then the projection morphism X → Y is an admissible cover of degree 2.

Proof. We have already seen that Y → S is a family of nodal curves, so property 1 is satisfied. We have also proved that nodes map to nodes. On the other hand, if a point of Y has a smooth point in the inverse image, then it is also smooth. Thus property 2 holds. We already know that the projection map is finite. The ´etaleness on the generic locus follows from 1.5.2, so we have proved property 3 too.

Before we prove property 4, let us introduce a notation that we have already used once. Let M be an R-module on which an involution σ acts, and assume that 2 is invertible in R. Then we can write m = m++ m, where

m+= m + σ(m) 2 , m

= m − σ(m) 2 .

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3.1. THE ADMISSIBLE INVOLUTION 29

Note that σ(m+) = m+, σ(m) = −m, so this gives a decomposition M = M+⊕ Min two

σ-invariant submodules, on which σ = ± id.

Let us now prove 4. The case of a non-fixed node follows from a combination of 1.5.2 and 1.3.13. Consider the nontrivial case of a fixed node p ∈ X mapping to a node q ∈ Y and to s ∈ S. Write

Osh

p ∼= Os[x, y]sh/(xy − a), Oqsh∼= Os[u, v]sh/(uv − b),

where a, b ∈ m = mshs (the maximal ideal of Ossh). The ring homomorphism ϕ : Oshq → Osh p

between stalks identifies the Oqsh with the ring of invariants of Opsh (B.0.11 and B.0.21).

We now want to prove that (ϕ(u)2, ϕ(v)2, m) = (x, y, m). This is equivalent to showing that,

called

ϕ : k[u, v]sh/(uv) → k[x, y]sh/(xy)

which results from taking the quotient by m, I = (ϕ(u), ϕ(v)) equals (x2, y2) (which is the same as (x, y)2). By 1.5.4, there exist w, z ∈ k[x, y]sh/(xy) such that (w) = (x) and (z) = (y) and such

that w, z 7→ −w, −z. It is clear that w2, z2 ∈ I, since they are invariant. On the other hand, 0 is

the only element that belongs to I of the form λw + µz, where λ, µ ∈ k, hence I = (x2, y2). By 1.3.14 there exist invertible elements αx, αy ∈ Oshp with product in Oshs such that

u 7→ αxx2, v 7→ αyy2.

Since Osh

p and Oshs are strictly henselian, there exist βx, βy ∈ Opsh such that βx2 = αx, βy2 = αy, and

their product is in Oshs too. If we set x0 = βxx, y0 = βyy0, and a0 = x0y0 ∈ Ossh, we have obtained a

presentation

Osh

p = Os[x0, y0]sh/(x0y0− a0)

where u 7→ x02 and v 7→ y02, so property 4 is satisfied as well.

Theorem 3.1.8. Let X → Y be an admissible cover of degree 2 over S. Then there exists a unique admissible involution σ of X such that Y = X/ <σ >.

Proof. We may immediately reduce to the case in which X is a scheme by 1.3.2. Let σ0 : X0 → X0

be the unique involution of the ´etale part of X → Y , given by B.0.18. We want to prove that σ0 can be extended uniquely to an admissible involution σ of X. To do this, we can of course

assume that S = Spec R is an affine scheme. By noetherian approximation, we may assume that R is noetherian. The uniqueness (even locally) is then clear, because X0 is schematically dense

in X. We now want to extend the involution locally (in the ´etale topology) at every node and ramification point. We start by extending on the strict henselizations.

Nodes. Let us begin with the local situation at a fixed node p ∈ X above q ∈ Y and s ∈ S. The hypotheses give us a presentation of the map between strict henselizations of p and q:

Os[u, v]sh/(uv − a2) → Os[x, y]sh/(xy − a)

with u 7→ x2, y 7→ v2. Call A = O

s[x, y]sh/(xy − a). By hypothesis we have an involution on the

schematically dense open subscheme Spec A \ V(x, y, a). Notice that the element x − y ∈ A is not a zero divisor and

V(x, y, a) ⊆ V(xy − a, x − y) = V(xy − a, x2− a, y2− a)

in Os[x, y]sh (for the more difficult inclusion, note that x(x − y), y(x − y) ∈ (xy − a, x2− a, y2− a),

so (x − y)2 belongs there too). Hence Spec A

x−y is invariant for the involution, and we get an

involution of Ax−y. Now, since A → Ax−y is injective, we can view A ⊆ Ax−y and ask ourselves

what the images of x and y may be. Since x2 7→ x2, by 1.3.14 (applied to x, y and σ(x), σ(y))

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Lemma 3.1.9. Let (R, m) be a local ring such that 2 is invertible in R. Let c ∈ R such that c2 = 1. Then c = ±1.

Proof. Quotienting by m, we see that c = ±1 + m where m ∈ m. Now from 1 = (±1 + m)2 = 1 ± 2m + m2 we obtain m(m ± 2) = 0. Since 2 is invertible in R, m ± 2 is invertible too. This

implies that m = 0.

We know that x is not invariant, because it is not invariant in the geometric fiber, so necessarily x 7→ −x. In the same way we see that y 7→ −y in the ´etale locus. To extend this involution it then suffices to set x 7→ −x and y 7→ −y as elements of A. Notice that by construction Oshq → Osh

p <σ>

is an isomorphism.

The same reasoning works for nodes y with two nodes of X in the fiber. Note that however the map there is already ´etale by 1.5.2, so the involution is already defined.

Ramification points. The local situation at a ramification point is similar. Let p ∈ X be a ramification point over some q ∈ Y . We have isomorphisms Osh

p ∼= Os[x]sh, Oqsh ∼= Os[u]sh, such

that the natural map Oshq → Osh

p is given by u 7→ x2. If we set Os[x]sh = A, we have an involution

of Ax which fixes x2 and does not fix x, so it must be x 7→ −x by 3.1.9. The extension is then

obviously defined by x 7→ −x. In this case too, we have an isomorphism Osh

q ∼= Oshp <σ>

.

Global extension. We now want to provide an extension over an ´etale neighbourhood of every p ∈ X where the involution is not yet defined. Let q and s be the images of p in Y and S, respectively. We have already remarked that an extension around p must be given only if p is a ramification point or if p is a node and the unique point in the fiber of q.

We fix an affine neighbourhoods Spec R ⊆ S of s and Spec A ⊆ X of p, such that Spec A maps to Spec R. Then we may write Osh

q = lim−→Aλ for some filtered family of ´etale A-algebras Aλ. The

involution of Oqsh gives a map A → Aλ for some large enough index λ. To see this, consider the

composition f of the natural map A → Osh

p followed by the involution of Oshp , which of course is

not A-linear, but only R-linear. Since A is a finitely generated R-algebra, we may pick a set of generators x1, . . . , xr of A over R. The element f (xi) ∈ Oshp comes from some A-algebra Aλi, and

by choosing an index λ such that λ ≥ λi for every i we obtain the desired map.

We have obtained an ´etale neighbourhood U = Spec Aλ of p in X and an ´etale map g : U → X

whose restriction U ×X X0 → X0 to X0 coincides with the involution σ0. Letting p vary and

following the same procedure for all such p, we construct an ´etale cover {ϕi : Ui → X} of X (we

can of course assume that X0 → X also belongs to the cover) such that for every i there exists a

morphism gi : Ui → X making the following diagram commute:

Ui×X X0

X0 X0 ϕ|X0 gi|X0

σ0

For every i and j the maps Ui ×X Uj → Ui → X and Ui×X Uj → Uj → X coincide, since

their restrictions to X0 are equal and X0 is schematically dense in X. Hence they glue to a map σ : X → X which restricts to σ0 on X0. It is an involution, since σ2 and idX coincide on X0. Let

us show that it is admissible. First, the irreducible components of the geometric fibers cannot be fixed, because they are not fixed by the standard involution on X0, which is dense in every geometric fiber of X → S. Second, the two branches at a fixed node are not switched by σ, as we have proved at the end of the paraphraph on nodes.

Finally, let us show that the quotient of X by the involution is Y . The morphism X → Y is σ-invariant, hence induces a morphism X/ <σ >→ Y which is an isomorphism on the smooth

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3.2. RAMIFICATION 31

part. To conclude, it is enough to check that the map induces isomorphisms at the henselization of every node and every ramification point. This was already remarked at the end of the previous paragraphs.

3.2

Ramification

Assume that 2 is invertible in OS. In the following let f : X → Y be an admissible cover of degree

2 over S. We will assume throughout that the fibers of X have constant genus g and those of Y have constant genus g0.

Lemma 3.2.1. Call f0 : X0 → Y0 the restriction to the smooth locus of X and Y . Then the

canonical injection

0 → f0∗Ω1Y0/S → Ω1X0/S

extends uniquely to an injection between dualizing sheaves

0 → f∗ωY /S → ωX/S

and it is an isomorphism near the nodes.

Proof. Let us begin with the ´etale-local situation at a node which is fixed by the involution. So let X = Spec A[x, y]/(xy − a), Y = Spec A[u, v]/(uv − a2) and S = Spec A. The quotient map is given by u 7→ x2, v 7→ y2. In this case, Ω

X/S is the sheaf associated to the A-module

< dx, dy > < ydx + xdy >.

The dualizing sheaf ωX/S is then generated by the element

dx ∧ dy xy − a

as an A[x, y]/(xy −a)-module. Here we are making use of the regular embedding X ⊆ Spec A[x, y], with ideal (xy − a), and of the formula in 1.4.1. In the same way, the sheaf of differentials and the dualizing sheaf for Y /S are given by sheafifying

< du, dv > < udv + vdu >, <

du ∧ dv uv − a2 >,

respectively. The map on differentials is given by du 7→ 2xdx, dv 7→ 2ydy. The extension is then given by setting du ∧ dv uv − a2 7→ 2xdx ∧ 2ydy x2y2− a2 = 4xy xy + a · dx ∧ dy xy − a = 2 dx ∧ dy xy − a .

It is injective since in all our schemes 2 is invertible.

Now let us assume that X is a scheme. The general case of X algebraic space immediately follows, by the uniqueness statement. The problem is clearly Zariski-local in S, so we may assume that S is affine. In this case, by noetherian approximation we may assume that S = Spec R is the spectrum of a noetherian ring (in fact, we can take to be a Z-algebra of finite type). The uniqueness then follows from the schematic density of the smooth locus (see 1.3.6 and 1.3.9). In fact, even every partial extension of the map is unique, for the same reasons.

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