THE LOCUS OF PLANE CURVES IN THE MODULI STACK OF CURVES
AARON LANDESMAN
ABSTRACT. Let d ≥ 4 and let Ud denote the locus of smooth curves in the Hilbert scheme of degree d plane curves. If the members of Ud have genus g, let Mg denote the moduli stack of genus g curves. We show that the natural map [Ud/ PGL3] Mg is a locally closed embedding. Along the way, we show that [Ud/ PGL3] represents a functor parameterizing families of plane→ curves.
1. INTRODUCTION One of the most fundamental objects of study in algebraic geom- etry is the study of the moduli curves. Centuries ago, when people were first exploring curves, they first considered plane curves: those cut out from P2 by a single equation. The goal of this note is to define a stack parameterizing families of plane curves. Perhaps the most natural candidate definition for such a stack would be families of curves so that all geometric fibers are isomorphic to plane curves. While it is not to difficult to see this is a substack of Mg, it is not at all clear whether this is algebraic. In order to fix this issue, we add the extra condition that the first de- rived pushforward of the family is locally free of the appropriate de- gree, or equivalently that the family commutes with base change (see Lemma 3.5). We define the functor in Definition 3.1. Recall that Mg is a smooth Deligne-Mumford stack for g ≥ 2. Since plane curves of degree < 4 have genus ≤ 1, for the purposes of examining these sub- stacks of Mg parameterizing plane curves, it is natural to restrict to the case d ≥ 4. Given this definition, we have two natural questions:
Question 1.1. For d ≥ 4, is the stack of plane curves as defined in Definition 3.1 a locally closed algebraic substack of Mg (i.e., is the natural map to Mg representable by locally closed embeddings)? Question 1.2. For d ≥ 1, is the substack of degree d plane curves smooth? 1 2 AARON LANDESMAN
We answer both these questions in the affirmative. To do so, we first recall classical facts line bundles on plane curves in section 2. Next, we show in section 3 (specifically in Theorem 3.7) that the stack of plane curves defined in Definition 3.1 is isomorphic to the quotient stack [Ud/ PGL3]. Here, Ud is the locus in the Hilbert scheme of smooth degree d plane curves and PGL3 acts on Ud via its action on 2 2 the universal family Cd ⊂ Ud × P by automorphisms of P . This implies Pd is smooth, as mentioned in Corollary 3.11. We show that the natural map Pd Mg is a locally closed embedding in section 4 (specifically in Theorem 4.5). Indeed, it is claimed→ in many places that [Ud/ PGL3] is the locus of plane curves. For example, it is done in [SB88, p. 51], [H+04, p. 1], and (implicitly in) [BGvB10]. The main goal of this document it to provide a rigorous stack-theoretic proof of this claim. All stacks, unless otherwise specified are defined over Spec Z. We work with stacks in the etale´ topology and in general follow the conventions used in [Ols16].
2. CLASSICAL FACTSABOUTPLANECURVES In this section, we recall some classical facts regarding plane curves over an algebraically closed field. The main result of this section is Proposition 2.11, which states that a smooth plane curve has a 2 2 unique gd, and that gd corresponds to a reduced point in the scheme 2 parameterizing gd’s. We will need this later to test a certain map of stacks is an isomorphism, by testing it on geometric points. Many of the results of this section can be found in the exercises [ACGH85, Appendix A, Exercises 17 and 18], and we include proofs for completeness. We note that the results of this section hold over fields of arbitrary characteristic (as we prove) even though [ACGH85, Appendix A, Exercises 17 and 18] typically has the hypothesis that the field has characteristic 0. This independence of characteristic is crucial for defining our stacks over Spec Z (instead of over a field of characteristic 0). We begin with some standard definitions. Definition 2.1. Let k be an algebraically closed field. A 0-dimensional 2 subscheme S ⊂ Pk is said to impose independent conditions on curves of degree n if 0 2 0 2 h (P , IS(n)) = h (P , O 2 (n)) − d, k k Pk where IS ⊂ O 2 is the ideal sheaf of S. Pk THE LOCUS OF PLANE CURVES IN THE MODULI STACK OF CURVES 3
r Definition 2.2. A gd on a smooth curve C is a line bundle L of degree d on C with h0(C, L) ≥ r + 1. 2 2.1. Showing there is a single gd. In this section, we show that a 2 smooth plane curve has only one gd in Proposition 2.6. We start with a lemma characterizing when finite reduced sub- schemes of P2 supported on at most n + 2 points impose indepen- dent conditions on curves of degree n. Lemma 2.3. Let S be any reduced subscheme of P2 whose support consists of n + 1 points. Then, S imposes independent conditions on curves of degree n. Further, if S ⊂ P2 is a reduced subscheme supported on n + 2 points, then S fails to impose independent conditions on curves of degree n if and only if S is contained in some line. Proof. Since S has degree d, it follows from the exact sequence
(2.1) IS(n) O 2 (n) OS Pk that 0 2 0 2 h (P , IS(n)) ≥ h (P , O 2 (n)) − d. k k Pk So, we only need verify the reverse inequality. By induction on the degree of S, it suffices to show that for any d ≤ n + 1 we can find some plane curve passing of degree n through all but one point of S, but not passing through the last point of S. Further in the case d = n + 2, it suffices to show we can find such a curve passing through all but one point of S, provided the n + 2 points do not lie on a line. In the case d ≤ n + 1, to see this, let pd denote a particular point of S and for each point pi ∈ S, pi 6= pd, choose a line `i passing through pi but not through pd. In the case d ≤ n + 1, taking C to be the union of the lines ∪i6=d`i provides a curve of degree ≤ n passing through all but one point of S. Taking the union of this with a curve of de- gree n − d − 1 not passing through pd provides the desired curve of degree n. To conclude, we only need verify that if S is supported on n + 2 non collinear points, there is some curve passing through all but one of these points. Choose three noncollinear points p1, p2, and p3 in the support of S. Upon reordering points, it suffices to show there is a curve passing through all points except p3. Then, let `1 be the line joining p1 and p2. For 2 ≤ i ≤ n, let `i be a line passing through pi+2 n but not p3. Then, ∪i=1`i provides the desired curve of degree n not passing through p3. 4 AARON LANDESMAN
Using Lemma 2.3 we can compute the cohomology of invertible sheaves of low degree on smooth plane curves.
Lemma 2.4. Let C be a smooth plane curve of degree d over an algebraically closed field k. Let p1, ... , pm be distinct points and L := OC(p1, ... , pm). Then, if m ≤ d − 2, we have H0(C, L) = 1. If m = d − 1, then 0 0 h (C, L) = 1 unless p1, ... , pm lie in a line `, in which case h (C, L) = 2 0 and h (C, OC(` ∩ C)) = 3.
m Proof. Let S := ∪i=1V(pi). By Lemma 2.3 applied in the case n = d − 3, observe that we have an exact sequence
0 2 0 2 0 2 0 H (P , O 2 (d − 3) ⊗ IS) H (P , O 2 (d − 3)) H (P , OS) 0. k Pk k Pk k
We obtain a corresponding map of exact sequences (2.2) 0 2 0 2 0 2 0 H (P , O 2 (d − 3) ⊗ IS) H (P , O 2 (d − 3)) H (P , OS) k Pk k Pk k
0 ∨ 0 0 0 H (C, OC(d − 3) ⊗ L ) H (C, OC(d − 3)) H (C, OS) coming from the natural restriction of sheaves to C and using that ∨ IS|C ' L . Observe that the latter two maps of (2.2) are isomorphisms, as fol- lows from the exact sequence on cohomology associated to
(2.3) 0 O 2 (α − d) O 2 (α) OC(α) 0 Pk Pk applied in the cases α = 0 and α = d − 3. Therefore, the first vertical map of (2.2) is also an isomorphism by the five lemma. We next claim that h0(C, L) = 1 if S is not contained in a line. Indeed, if S is not contained in a line, by Lemma 2.3 and the isomor- 0 0 phisms from (2.2), h (C, OC(d − 3) ⊗ L) = h (OC(d − 3)) − m. Using geometric Riemann-Roch and the fact that the canonical bundle of C is OC(d − 3), (since OC(d − 3) is a degree 2g − 2 bundle with g global THE LOCUS OF PLANE CURVES IN THE MODULI STACK OF CURVES 5 sections,) we have 0 0 ∨ h (C, L) = h (C, KC ⊗ L ) + m − g + 1 0 ∨ = h (C, OC(d − 3) ⊗ L ) + m − g + 1 0 = h (C, OC(d − 3)) − m + m − g + 1 d − 1 = − g + 1 2 = g − g + 1 = 1. To conclude, we prove the second statement of the lemma. If 0 m = d − 1 and p1, ... , pm are collinear, we know H (C, OC(p1 + ··· + 0 pm−1)) = 1 by the above. Hence, H (C, OC(p1 + ··· + pm)) ≤ 2. But, if the points lie on a line `, then we know H0(C, C ∩ `) ≥ 3. Since C ∩ ` − (p1 + ··· + pm) is an effective divisor of degree 1, we obtain 0 the two inequalities must be equalities, so H (C, OC(p1 + ··· + pm) = 0 2 and H (C, C ∩ `) = 3. Using the prior cohomological calculations, in preparation for prov- 1 ing Proposition 2.6, we show smooth plane curves have no gd−2’s 1 and characterize the gd−1’s. 2 Lemma 2.5. Let C ⊂ Pk be a smooth plane curve of degree d ≥ 4, with k an algebraically closed field. Then, 1 (1) C has no gm for m ≤ d − 2 and 1 (2) any gd−1 is of the form D − p for p ∈ C and D in the linear system 0 H (C, OC(1)). 1 Proof. We first show that C has no gm for m ≤ d − 2. Suppose that 1 C has a gm for m ≤ d − 2. Such a line bundle determines a map C P1 of degree at most m (after possibly removing basepoints by twisting the line bundle down). Therefore, it suffices to show that 1 C →has no dominant degree m maps to Pk for m ≤ d − 2. That is, it suffices to show C has no basepoint free line bundles of degree m for m ≤ d − 2. So, suppose C has some basepoint free line bundle of degree m ≤ 1 d − 2 corresponding to a dominant map C Pk. We next reduce to showing that C has no line bundles corresponding to generically 1 separable dominant maps C Pk. This is automatic→ if k has char- 1 acteristic 0. If k has characteristic p, and C Pk is generically in- 1 separable and dominant, then→C Pk factors as the composition of → → 6 AARON LANDESMAN
1 some generically separable dominant map C Pk with some power of Frobenius. Therefore, it suffices to show C has no generically sep- 1 arable dominant map to Pk. → So, we now show that there are no generically separable maps 1 C Pk. Such a map has general fiber given by a collection of m distinct points on C. Thus, it remains to show there are no line bun- 0 dles→L = OC(p1 + ··· + pn) with h (C, L) ≥ 2. (The reason for the re- duction to generically separable morphisms is that we may assume the points defining the line bundle are distinct, so we may apply 1 Lemma 2.4.) Therefore, by Lemma 2.4, C has no gm’s. 1 We next verify the second claim that only gd−1’s on C are given by divisors of the form D − p with p ∈ C and D in the linear sys- 0 tem H (C, OC(1)). The proof is quite similar to the previous case. Let 1 L be some gd−1. Note that L must be basepoint free, as otherwise, 1 twisting down by the basepoints, we would obtain some gm for m ≤ d − 2, contradicting the previous part. Thus L determines a map 1 C Pk. This map is necessarily separable, as otherwise it would factor as the composition of a generically separable map of lower de- gree→ with Frobenius. But, there are no generically separable maps to 1 1 Pk of lower degree because there are no gm’s for m ≤ d − 2. Hence, we know L determines a generically separable dominant morphism 1 ∼ C Pk. Therefore, we may assume that L = OC(p1 + ··· + pd−1) 0 for p1, ... , pd−1 distinct. Then, by Lemma 2.5, we have h (C, L) = 1 unless→ the points lie on a line. In the case that the points p1, ... , pd−1 lie on a line, taking D to be the intersection of C with that line in 2 Pk, we obtain that p1 + ··· + pd−1 = D − q, where q is by definition D − (p1 + ··· + pd−1).
Using Lemma 2.5, we can now deduce the main result of this sec- tion.
2 Proposition 2.6. Let C ⊂ Pk be a smooth plane curve of degree d ≥ 4, 2 with k an algebraically closed field. Then, C curve has at most one gd and 2 that gd is a complete linear series.
2 Proof. First, suppose C is a smooth curve with a gd. That is, C has an invertible sheaf L with h0(C, L) ≥ 3. We claim h0(C, L) = 3. To see this, let D be an effective divisor in the linear system H0(C, L). 0 1 Let p1, p2 be two points. If h (C, L) > 3, then D − p1 − p2 is a gd−2, which does not exist by Lemma 2.5. THE LOCUS OF PLANE CURVES IN THE MODULI STACK OF CURVES 7
To complete the proof, we only need to check C has a unique 2 2 gd. For this, let M be any gd. Then, for any D in the linear sys- 0 1 tem H (C, M) and p ∈ Supp(D), we have OC(D − p) is a gd−1. By Lemma 2.5, D − p consists of d − 1 collinear points. Therefore, there ∼ is some point q so that M(−p) = OC(1)(−q). Consider the invertible sheaf, ∨ ∼ L := M ⊗ OC(1) = OC(−q + p). To conclude the proof, it suffices to show p = q. We know M⊗(d−3) =∼ ∼ ⊗(d−3) ⊗(d−3) ⊗(d−3) KC = L , since M and L are degree 2g − 2 bundles with a g dimensional space of global sections. Therefore, L⊗(d−3) =∼ ∼ OC. This implies OC((d − 3)p) ⊗ OC(−(d − 3)q) = OC. Using Lemma 2.5, 0 we see H (C, OC((d − 3)p)) = 1. This implies that the only section of OC((d − 3)p) is the trivial section, and so OC((d − 3)p) ⊗ OC(−(d − ∼ 3)q) has no sections unless p = q. Therefore, M = OC(1), as de- sired. 2 2.2. Showing that gd is reduced. In this section, we show that the 2 unique gd on a smooth plane curves (whose uniqueness was estab- lished in Proposition 2.6) corresponds to a reduced point of a param- 2 eter space for gd’s (which we shall define in Definition 2.9). We do so in Proposition 2.11. In order to do so, we first recall a standard fact that plane curves are projectively normal. 2 Lemma 2.7. Let k be a field. Any smooth plane curve C ⊂ Pk is projec- 0 2 tively normal, meaning that for all n > 0, the map H (P , O 2 (n)) k Pk 0 H (C, OC(n)) is surjective. → Proof. By the long exact sequence associated to
(2.4) 0 IC O 2 OC 0 Pk
1 2 to verify projective normality, we only need verify H (P , IC(n)) = ∼ 0 for all n ≥ 0. Say C has degree d. Then, IC(n) = O 2 (n − d). Pk 1 2 1 2 Therefore, H (P , IC(n)) = H (P , O 2 (n − d)) = 0. Pk r We next define the scheme Gd(p) for p : C S a family of smooth genus g curves. For the moment, we will only need it in the case S is a field, in which case the functor parameterizes→ invertible sheaves on C and a space of global sections of dimension at most r + 1. However, later in section 4 we will need this functor for arbitrary families p : C S, so we define it in general now.
→ 8 AARON LANDESMAN
Definition 2.8. We say a morphism C S is a family of smooth genus g curves if it is a projective flat morphism so that each fiber is a geometrically connected smooth curve→ of genus g. Definition 2.9 ( [ACG11, Chapter XXI, Definition 3.12]). Suppose p : C S is a family of smooth genus g curves. Define the fibered r category Gd(p) sending a map f : T S to the set of equivalence classes→ of pairs (L, H) / ∼ defined as follows: Let ι : t T be a point and define the corresponding fiber square→
ιC → Ct CT
(2.5) pt pT ι t T.
A pair (L, H) consists of a line bundle L on C ×S T whose restriction to each fiber of pT : CT T has degree d and a locally free sheaf H which is a subsheaf of pT∗L of rank r + 1 so that for each fiber ι : t T, the natural composition→ ∗ ∗ ∗ (2.6) ι H ι pT∗L pt∗ιCL → is injective. The equivalence relation ∼ defined on pairs (L, H) dic- tates that two pairs (L, H)→and (L0, H→0) are equivalent if there is an 0 ∼ ∗ invertible sheaf Q on T and an isomorphism L = L ⊗ pT Q which induces an isomorphism H 0 ' H ⊗ Q. Theorem 2.10 ( [ACG11, Chapter XXI, Theorem 3.13]). For p : C r S a family of smooth genus g curves admitting a section, the functor Gd defined in Definition 2.9 is represented by an S scheme. → We are now ready to state and prove the main result of this section.
Proposition 2.11. Let C be a degree d ≥ 4 smooth plane curve over an 2 algebraically closed field k. Then, Gd(p : C Spec k) is isomorphic to a reduced point. 2 → 2 Proof. Since the underlying set of Gd(p) is the set of gd’s on C, to- gether with a 3-dimensional space of global sections, by Proposi- 2 tion 2.6, Gd(p) is supported on a point. Here we are using that the 2 3 unique gd is not a gd, as was shown in Proposition 2.6. It only remains to show this point is reduced. Indeed, using [ACGH85, Chapter IV, Proposition 4.1(iii)], (whose proof holds equally well in positive characteristic,) the tangent spaces will be 0-dimensional if THE LOCUS OF PLANE CURVES IN THE MODULI STACK OF CURVES 9 the multiplication map 0 0 0 H (C, OC(1)) ⊗ H (C, KC ⊗ OC(−1)) H (C, KC) ∼ is surjective. Under the identification KC = OC(d − 3), we want to show the map → 0 0 0 H (C, OC(1)) ⊗ H (C, OC(d − 4)) H (C, OC(d − 3)) is surjective. To verify this, note that we have a commutative→ square (2.7) 0 2 0 2 0 2 H (P , O 2 (1)) ⊗ H (P , O 2 (d − 4)) H (P , O 2 (d − 3)) k Pk k Pk k Pk
0 0 0 H (C, OC(1)) ⊗ H (C, OC(d − 4)) H (C, OC(d − 3)).
We know that the vertical maps are surjective by projective normal- ity of C, as shown in Lemma 2.7. Further, the top horizontal map is surjective, since every degree d − 3 polynomial is a linear com- bination of products of degree 1 and degree d − 3 polynomials in 3 variables. Therefore, the bottom horizontal map is also surjective, as desired.
3. THEMODULISTACKOFPLANECURVES In subsection 3.1, we define the moduli stack of plane curves of de- gree d, which we will denote Pd, and verify it is an algebraic stack. Then, in subsection 3.2, we review a standard application of coho- mology and base change. Finally, in subsection 3.3, we show that Pd ' [Ud/ PGL3], where Ud denotes the open subscheme of the Hilbert scheme parameterizing smooth degree d plane curves. In particular, this implies that Pd is a smooth algebraic stack of finite type.
3.1. Defining the stack of plane curves. To begin, we define the stack of plane curves. d−1 Definition 3.1. Let d ≥ 1 be an integer. Let g := 2 . Define the moduli stack of degree d plane curves denoted Pd to be the fibered category of pairs (f, L) where f : C S are projective flat morphisms such that for every geometric point s ∈ S, the fiber Cs is a geometrically proper smooth curve of genus→ g, (i.e., f is a family as defined in Definition 2.8,) and L is a degree d invertible sheaf on 10 AARON LANDESMAN
1 C so that R f∗L is locally free of rank 2 − d + g such that for every geometric point s S, L|s is a very ample invertible sheaf. A morphism of two families (f0 : C0 S0, L0) (f : C S, L) is a fiber square → g0 → → → C0 C
(3.1) f0 f g S0 S so that there is some line bundle Q on S0 with g0∗L ⊗ f0∗Q =∼ L0. This makes Pd into a fibered category over the category of schemes by sending a family C S to S.
We note that Pd is a stack. → Lemma 3.2. The functor Pd is a stack in the ´etaletopology.
Proof. First, observe that Pd is indeed a fibered category as all mor- phisms are Cartesian arrows. Full faithfulness follows from a general fact for fppf coverings (in particular for etale´ coverings) as written in [Ols16, Corollary 4.2.13]. Now, suppose we have an etale´ cover g : S0 S and are given an object (f0 : C0 S0, L0) together with descent data σ. (see [Ols16, 4.2.1] for precise details of what is meant by descent data, but it is essentially an isomorphism of the two pull- → 0 0 → backs of f S ×S S which satisfies the cocycle condition). By descent for polarized schemes, see [Ols16, Proposition 4.4.12], it follows that the line bundle L0 and family C0 S0 over S0 is the pullback of some line bundle L on a family f : C S. It only remains to verify that 1 R f∗L is locally free. Since by assumption→ g is flat, it follows from flat base change that the base change→ map is an isomorphism. Hence, if we consider the diagram g0 C0 C
(3.2) f0 f g S0 S, 1 0 0 1 0∗ 0∗ ∗ 1 0 0 we see R f∗(L ) = R f (g L) ' g R f∗(L). Since f∗L is locally free ∗ 1 1 by assumption, g R f∗L is locally free, which implies R f∗L is also locally free by [Ols16, Exercise 4.C(a)]. 3.2. Applications of cohomology and base change. We next aim to prove that Pd is the quotient [Ud/ PGL3]. Before doing so, we will need some preparatory results using cohomology and base change, THE LOCUS OF PLANE CURVES IN THE MODULI STACK OF CURVES 11 which we prove in this section. The main result is Proposition 3.6. To start, we introduce notation for the base change map. Definition 3.3. Suppose we have a Cartesian diagram
ψ0 W X
(3.3) π0 π ψ Z Y. p Then, for F a sheaf on X, we denote by φZ the natural map ∗ p p 0 0∗ ψ R π∗(F) R π∗(ψ F). We next note a variant of cohomology and base change, showing it can be verified on geometric points,→ as opposed to points. Lemma 3.4 (Cohomology and base change for maps from points). Suppose π: X Y is proper, Y is locally Noetherian, F is coherent and flat over Y and for each point q ∈ Y there is some ψ : Spec L Y with image q L φp , for a field,→ so that Spec L is surjective. Then, the following hold: p → (i) For any Z Y, the base change map φZ is an isomorphism. (ii) Furthermore, for any map Spec L0 Y with L0 a field with image q φp−1 , we have→Spec L0 is surjective (hence an isomorphism) if and only p if R π∗F is locally free in some neighborhood→ of q. Proof. We will reduce this version to the version [?, 28.1.6] (which replaces our maps Spec L Y with inclusions of points of Y). We have a Cartesian diagram
→ iq XSpec L Xq X (3.4) t Spec L q Y. p To complete the proof, it suffices to show φq is surjective (respec- p tively, an isomorphism) if and only if φSpec L is surjective (respec- tively, an isomorphism). We just show the surjectivity statement, as the proof of the isomorphism is nearly the same, mutatis mutandis. By flat base change ([?, Theorem 24.2.8]) the base change map for ∗ iqF applied to the left square is an isomorphism. Therefore, the base change map applied to F for the outer rectangle is surjective if and p only if the map φq pulled back along t is surjective. Since pulling back along t is merely a base change of a map of finite dimensional 12 AARON LANDESMAN vector spaces along a field extension, it follows that the pullback of p p φq along t is surjective if and only if φq is surjective. 1 We next verify the equivalence of R f∗(L) being locally free and f commuting with base change. Lemma 3.5. Suppose f : C S is a family of curves of genus g over S, d−1 L is a locally free sheaf of degree d on C, with g = 2 , and S is locally 1 0 Noetherian. Then, R f∗L is locally→ free if and only if φq is an isomorphism 1 for all points q. Further, if R f∗L is locally free of rank 2 − d + g, then for any map g : S0 S with corresponding fiber square
0 0 g → C C (3.5) f0 f g0 S0 S. p the base change maps φS0 are isomorphisms and f∗L is locally free of rank 3. 1 0 Proof. First, let us show that R f∗L is locally free if and only if φq is p an isomorphism for all points q. Observe that for all p ≥ 2, φq is an isomorphism for all points q in S because all cohomologies in degree p ≥ 2 vanish for relative curves. Further, for p ≥ 2, we have R f∗L = 0, hence it is locally free. Applying these two statements above in 1 the case p = 2, by cohomology and base change, we obtain that φq 1 is an isomorphism for all q. Then, since φq is an isomorphism for all 1 points q, cohomology and base change implies R f∗L is locally free 0 of rank 1 if and only if φq is an isomorphism for all q ∈ S. It remains to prove the second statement. Indeed, note that since 0 φq is an isomorphism for all points q, and we automatically have −1 φq is an isomorphism (as negative cohomology groups vanish), we obtain f∗L is locally free. By cohomology and base change, we know 0 p that for all maps S S the base change maps φS0 are isomorphisms. It remains only to show that the rank of f∗L is 3. For this it suf- fices to show that for→ every point q that (f∗L)|q has rank 3. Because 0 the map φq is an isomorphism, this is equivalent to showing that 0 f|q∗(L|q) has rank 3. That is, we want to show h (C|q, L|q) = 3. By 0 1 Riemann-Roch, we know h (C|q, L|q) − h (C|q, L|q) = d − g + 1. So, 0 1 showing h (C|q, L|q) = 3 is equivalent to showing h (C|q, L|q) = 1 2 − d + g. This is equivalent to showing R f|q∗L|q has rank 2 − d + g. 1 By cohomology and base change, we know φq is an isomorphism, THE LOCUS OF PLANE CURVES IN THE MODULI STACK OF CURVES 13
1 so this is equivalent to showing (R f∗L)|q has rank 2 − d + g, which 1 follows from the assumption that R f∗L is locally free of rank 2 − d + g. Using Lemma 3.5, we can immediately deduce the first two parts of the following proposition.
Proposition 3.6. Let (f : S C, L) ∈ Pd be an object. Then, (1) For any map g : S0 S we may construct the fiber square → 0 0 g → C C (3.6) f0 f g0 S0 S 0 and corresponding map of objects of Pd (f : C S, L) (f : C0 S0, g∗L). Then, the base change map ∗ 0 0∗ g f∗L f∗g L → → → is an isomorphism. (2) We have that f∗L is locally free→ sheaf of rank 3 on S. ∗ (3) The map f f∗L L is surjective and the resulting map C P(f∗L) is a closed embedding. Proof. Note that since f→is flat and L is locally free, hence flat on →C, it follows that L is flat over S. By spreading out, writing S as a limit of its finite type Z-algebras, it suffices to verify the special case that S is Noetherian. The first two parts then follow immediately from Lemma 3.5. To conclude, we verify the third part. We start by reducing to the case that S is the spectrum of an algebraically closed field. As above, by spreading out, we may assume S and S0 are Noetherian. To check ∗ surjectivity of f f∗L L, it suffices to check surjectivity at all stalks. ∗ Further, since L and f∗f L are finite type, it suffices to check sur- jectivity at all geometric→ fibers. This explains why we it suffices to check the surjectivity statement when S = Spec k. Next, we explain why it suffices to check the closed embedding statement for S the spectrum of an algebraically closed field. Since a morphism is a closed embedding if and only if it is a proper monomor- phism (and being a monomorphism is equivalent to the fiber over each point having degree 0 or 1, by [Gro67, Proposition 17.2.6]) it suffices to check the map is a closed embedding over each point. Summing up what we have just shown, taking the map g : q S 0∗ ∗ 0∗ to be the inclusion of a point, we only need verify g f f∗L g L → → 14 AARON LANDESMAN is surjective and determines a closed embedding. We can write the preceding map as the composition
0∗ ∗ 0∗ ∗ 0∗ 0 0∗ 0∗ g f f∗L ' f g f∗L ' f f∗g L g L, where the middle map is an isomorphism from cohomology and → base change and the last map is the natural adjunction map. Hence, it suffices to verify that the adjunction map is surjective and deter- mines a closed embedding, which completes the reduction to the case that S = Spec k, for k an algebraically closed field. So, we now assume that S = Spec k, and complete the proof. In this case, by the second part, f∗L is locally free of rank 3. Since we are assuming C is a plane curve by definition of Pd, it follows form Proposition 2.6 that since h0(C, L) ≥ 3, L must be the unique in- vertible sheaf on C which determines a closed embedding C P2. ∗ Further, the resulting map f f∗L L is then a surjection because the 2 ∗ map C P is base point free. (In more detail, since f f∗L at→ some point p can be identified with non-projectivized→ local coordinates for 2 p in P →while L can be identified with the non-projectivized point p. The statement that the map is surjective corresponds to not all coor- dinate functions vanishing at p, which means the map is basepoint free.)
3.3. The stack of plane curves as a quotient stack. In this section, we prove the map [Ud/ PGL3] Pd is an isomorphism. The proof consists of constructing an inverse map and doing a fairly routine verification that the two maps→ are mutually inverse. However, the proof is fairly lengthy as there are a number of details to verify.
Theorem 3.7. For all d ≥ 1, we have an isomorphism
Pd [Ud/ PGL3] .
Proof. We prove this in four steps: In subsubsection 3.3.1 we define → the map above. In subsubsection 3.3.2 we define an inverse map. In subsubsection 3.3.3 we show one composition of these maps is naturally isomorphic to the identity. In subsubsection 3.3.4 we show the other composition is naturally isomorphic to the identity.
3.3.1. Defining a map from Pd. We now define a map Pd [Ud/ PGL3]. We define this map on T points, for T a scheme. Say we have some family of degree d plane curves (f : C T, L) in Pd(T)→. We want to
→ THE LOCUS OF PLANE CURVES IN THE MODULI STACK OF CURVES 15 obtain a corresponding point [Ud/ PGL3]. By definition of the quo- tient stack, this corresponds to the datum
h E Ud (3.7) g T with g : E T a PGL3 torsor and E U a PGL3 equivariant map. We construct such an E and show it is a PGL3 torsor in Lemma 3.8. We then construct→ the map h and show→ it is PGL3 equivariant in Lemma 3.9. 3 g Lemma 3.8. The scheme E := IsomT (Pf∗L, POT ) − T is a principal PGL3-torsor, where g is the natural projection.
Proof. From the definition for principal PGL3-torsor→ given in [Ols16, Definition 4.5.4], it is apparent that conditions (i) and (ii) are satisfied, so we only need check condition (iii), that the map
(ρ, π2): PGL3 ×T E E ×T E (g, p) 7 (gp, p) → is an isomorphism. To verify this, we may do so Zariski locally. → Choose a cover {Ti}i∈I of T so that f∗L is trivial on each Ti. This is possible since f∗L is locally free of rank 3 by Proposition 3.6, hence O3 isomorphic to Ti . Since the construction of proj commutes with base E| =∼ (P(f L| ) PO3 ) change, we see that Ti IsomTi ∗ Ti , Ti . Therefore, upon f L| ' O3 E| fixing an isomorphism ∗ Ti Ti the above Ti can be identi- (PO3 PO3 ) =∼ fied with IsomTi Ti , Ti PGL3 in which case it is clear that the action map (ρ, π2) is an isomorphism. Therefore, we obtain that g : E T is a PGL3 torsor.
Next, we define a map h : E Ud, which we will verify is PGL3 equivariant→ in Lemma 3.9 Define the fiber square
→g0 CE C
(3.8) f0 f g E T 0 0∗ 0 0 Define L := g L. We claim f∗L is a locally free rank 3 sheaf on E. To see this, observe that by Proposition 3.6(1), we know the ∗ 0 0 map g f∗L f∗L is an isomorphism. Since f∗L is locally free by
→ 16 AARON LANDESMAN
0 0 Proposition 3.6, it follows that f∗gL is as well. Therefore, the fam- 0 0 ily (f : CE E, L ) defines an element of Pd(E). Further, applying 0 0 Proposition 3.6(3) to the element (f : CE E, L ) ∈ Pd(E), we ob- 0 0 2 tain a resulting→ closed embedding CE P(f∗L ) ' PE. The latter isomorphism essentially follows from the→ universal property of E as 0 0 ∗ an Isom scheme, using that f∗L ' g f∗→L. This is a flat family whose geometric fibers are smooth plane curves of degree d by construc- tion. Therefore, the universal property of the Hilbert scheme deter- mines an map from E to the Hilbert scheme of plane curves, which factors through Ud as all geometric fibers are smooth. Call this map h : E Ud.
Lemma 3.9. Further, the resulting map defined above h : E Ud is PGL3 equivariant.→ Proof. We want to check that the diagram →
id ×ρ PGL3 ×Spec ZE PGL3 ×Spec ZUd (3.9) h E Ud commutes, where the vertical maps are the multiplication maps. To see this, observe that we have a universal family over Ud, call it fd : Cd Ud. This comes with a universal line bundle Ld on Cd and an (d+2)−1 embedding C P 2 . Let h0 : C C be the resulting map. d Ud E d Observe→ that from the definition of the Hilbert scheme, L0 (defined 0∗ 0 0∗ as g L, with g →as in (3.8),) is isomorphic→ to h Ld. We next claim that cohomology and base change commutes for Ld, as we prove in the following sublemma.
Lemma 3.10. Let Ud denote the open subscheme of the Hilbert scheme of degree d plane curves corresponding to smooth curves, let Cd Ud be the universal family and let Ld denote the universal line bundle. Then, cohomology and base change commutes for Ld. → d+2 ( 2 )−1 Proof. Observe that for any point q ∈ P with fiber Cq, we Ud know Ld|Cq is a locally free sheaf of degree d on Cq and so by Propo- 0 sition 2.6 we know that h (Cq, Ld|Cq ) = 3. By Riemann-Roch, we ob- 0 tain h (Cq, Ld|Cq ) = 2 − d + g for all points q. Since we know Ud is d+2 −1 explicitly an open subscheme of P( 2 ) , hence reduced, it follows 1 from Grauert’s theorem that R h∗(Ld) is locally free of rank 2 − d + g. THE LOCUS OF PLANE CURVES IN THE MODULI STACK OF CURVES 17
Therefore, by Proposition 3.6, cohomology and base change com- mutes for Ld. 0 3 By construction of E, we know E = IsomT (PL , POT ). The multi- plication maps PGL3 ×Spec ZUd Ud and PGL3 ×Spec ZE E are (P(f L ) PO3 ) given by the respective actions of PGL3 on IsomUd d∗ d , Ud and → →
(Pf0 L0 PO3 ) ' (P(h∗f L ) P(h∗O3 )) (3.10) IsomE ∗ , E IsomE d∗ d , Ud ' E × (P(f L ) PO3 ) Ud IsomUd d∗ d , Ud .(3.11) where the first isomorphism is obtained via Lemma 3.10. Using these identifications, we claim the diagram (in which the top and bottom squares are separately Cartesian)
2 2 PGL3 ×Spec ZPE PE
(3.12) PGL3 ×Spec ZCE CE
PGL3 ×E E is the pullback along h : E Ud of the diagram
2 2 PGL3 × P P →Spec Z Ud Ud
(3.13) PGL3 ×Spec ZCd Cd
PGL3 ×Ud Ud.
In more detail, our identifications in (3.10) show that the top row of (3.12) is the pullback along h of the top row of (3.13). Restricting C ⊂ P2 C ⊂ P2 this to E E and d Ud tells us the middle row of (3.12) is the pullback along h of the middle row of (3.13). This finally implies from the universal property of the Hilbert scheme that the bottom row of (3.12) is the pullback along h of the bottom row of (3.13). It follows that that (3.9) is in fact a fiber square. In particular, it com- mutes, as desired. 18 AARON LANDESMAN
By definition of the quotient [Ud/ PGL3], the PGL3 bundle g : E T and PGL3 equivariant map h : E Ud determines a map T [Ud/ PGL3]. All in all, we have determined a map → α : Pd [Ud/→ PGL3] → defined by sending an element of Pd(T) to the element of [Ud/ PGL3] (T) → 3 obtained from the PGL3 bundle E = IsomT (Pf∗L, POT ).
3.3.2. Defining an inverse map to Pd. We next define a map β inverse to α,
β : [Ud/ PGL3] Pd.
First, as shown in Lemma 3.10, the universal line bundle Ld on the universal family Cd over Ud satisfies cohomology→ and base change, and therefore determines an element (fd : Cd Ud, Ld) ∈ Pd(Ud). Now, for any scheme T, a T point of [Ud/ PGL3] is a principal PGL3-bundle ν : P T together with a PGL→ 3-equivariant map ε : P Ud and corresponding fiber square → ε C C C → P d
(3.14) fP fd ε P Ud. To construct our map β, we will describe how to obtain an element (fT : CT T, LT ) ∈ Pd(T). (Here, we are using that PGL3 is affine, so principal PGL3-bundles are the same as PGL3 torsors, by [Ols16, Proposition→ 4.5.6].) Since Ld satisfies cohomology and base change, 1 the bundle LP := Ld|CP has R fP∗LP locally free of rank 2 − d + g, be- 1 ing the pullback along ε of R fd∗Ld. Note that the map P T is fppf by definition of principal PGL3-bundle, and the two pullbacks of LP ∼ to P ×T P = PGL3 ×P (the isomorphism holding because P→is a PGL3 torsor over T) are isomorphic, as follows from PGL3 equivariance of the map ε. Further this descent data satisfies the cocycle condition because the PGL3 action is associative. Hence, by fppf descent for polarized families, we know that the family (CP P, LP) descends to a polarized family of curves (CT T, LT ). We have a fiber square
ν0 → CP →CT
(3.15) fP fT P ν T THE LOCUS OF PLANE CURVES IN THE MODULI STACK OF CURVES 19
To complete our definition of β, we only need verify that (fT : 1 CT T, LT ) ∈ Pd(T). To see this, we must show that (R fT )∗LT is lo- cally free of rank 2 − d + g. Indeed, this follows from flat base change 1 ∗ ∼ applied→ to the map ν: Flat base change implies that R ν (fT )∗LT = 1 0∗ 1 R fP∗(ν LT ) is an isomorphism. Therefore, R fT∗LT has pullback along ν which is locally free, and hence it is locally free.
3.3.3. Showing β ◦ α ' id. Finally, we note that the two maps we have constructed in both directions between [Ud/ PGL3] and Pd are mutually inverse. First, we claim β ◦ α ' id. To verify this, we check it on T points. If we start with a T point of Pd, this is a family C T. Following the construction of α, we obtain a particular bundle E T, and pullback CE E, which determines a map E Ud. Now,→ we wish to show that β recovers the family C T. However, by construction→ of the map→E Ud, we realized CE E as→ the pullback of the universal family Cd Ud, and so we indeed→ recover CE E. Then, we also recover→C T from the full faithfulness→ of descent (as the resulting object CT →T was constructed via descent in the→ map β). → 3.3.4. Showing α ◦ β ' id. To finish the proof of Theorem 3.7, we → only need verify α ◦ β ' id. This will follow if we show the map β is fully faithful. To check this, by [Ols16, Proposition 3.1.10], we can restrict to the case that both morphisms are from the same test scheme T. That is, for x, y ∈ Pd(T), it suffices to show that
homPd(T)(x, y) ' hom[Ud/ PGL3](T)(β(x), β(y)). We first show β is faithful. To this end, suppose we have two distinct maps f and g in homPd(T)(x, y). We want to show β(f) 6= β(g). We can think of x, y as the datum of (fx : Cx T, Lx) and (fy : Cy T, Ly) with a map T T so that Cy pulls back to Cx and Ly pulls back to some line bundle of the form Lx ⊗ (fx)∗Q for x y → Q an invertible→ sheaf on T. If the→ two maps f , f are distinct, the induced map on PGL3 bundles will also be distinct. Therefore, we may assume the two maps fx, fy agree. So, we may assume T T is the identity. Then, two such maps simply correspond to a choice y ∼ x x∗ of isomorphism between L = L ⊗ f Q, in which case the→ two bundles maps were already deemed equivalent in Pd(T). So, β is faithful. We next verify β is full. To this end, suppose we have two objects x, y given by (fx : Cx T, Lx) and (fy : Cy T, Ly) so that β(x) agrees with β(y). We want to show x agrees with y. If β(x) agrees → → 20 AARON LANDESMAN with β(y), this means the resulting map T T is the identity. In this x y case, we suppose that the two PGL3 bundles associated to L and L are isomorphic, meaning → x x 3 ∼ x x 3 IsomT (Pf∗L , POT ) = IsomT (Pf∗L , POT ). We want to show that Ly ' fx∗(Q) ⊗ Lx for Q some invertible sheaf x x ∼ y x on T. For this, it suffices to show P(f∗L ) = P(f∗L ). To show this, taking a Zariski cover {Ui} T which simultaneously triv- ializes both line bundles, we see that on each such Zariski open, upon choosing trivializations for the→ two line bundles, we obtain an x x ∼ y y isomorphism Pf∗L = Pf∗L given by some gi in PGL3 with re- spect to the chosen trivializations. These gi restrict compatibly and hence determine some element of PGL3 defining an isomorphism x x ∼ y y P(f∗L ) = P(f∗L ), as desired. ∼ We have therefore produced an isomorphism [Pd/ PGL3] = Pd, completing the proof of Theorem 3.7. Corollary 3.11. For all d ≥ 1, Pd is a smooth algebraic stack.
Proof. From Theorem 3.7, we have produced an isomorphism Pd ' (d+2)−1 [Ud/ PGL3] where Ud is the open subscheme of P 2 which is the locus of smooth curves the Hilbert scheme of plane curves of degree d. It follows that Pd is an algebraic stack. To show it is smooth, we only need show [Ud/ PGL3] has a smooth cover by a smooth scheme. Indeed, this follows as Ud [Ud/ PGL3] is such a smooth cover. Remark 3.12. At this point, if we wished, we could prove that for d ≥ 3, the stack Pd is in→ fact Deligne-Mumford. Essentially, this would follow because for d ≥ 4 there are only finitely many auto- morphisms of curves and for d = 3, there are only finitely many automorphisms of curves which preserve a degree 3 line bundle. However, there is the delicate issue of verifying the isotropy groups at points are actually etale.´ Instead, since we will show the map Pd Mg is a locally closed embedding for d ≥ 4, it will follow that for d ≥ 4, Pd is Deligne-Mumford because Mg is. → 4. VERIFYING THAT Pd Mg ISALOCALLYCLOSEDEMBEDDING d−1 Let g := Our next goal is to verify that the natural map Pd 2 → Mg, sending a pair (f : C T, L) 7 (f : C T) is a locally closed embedding. → We do this in several steps,→ completing→ the→ proof in Theorem 4.5. (1) We show that in the case f : C T has a section, the resulting
fiber product T ×Mg Pd is in fact a scheme (in Proposition 4.2). → THE LOCUS OF PLANE CURVES IN THE MODULI STACK OF CURVES 21
(2) We show that the map Pd Mg is a monomorphism (in Proposition 4.4). (3) We show that Pd Mg is a→ locally closed embedding, us- ing the (little known!) valuative criterion for locally closed embedding (in Theorem→ 4.5). r Recall the definition of the stack Gd(p) for p : C S a family of smooth genus g curves given in Definition 2.9 and recall that it is representable when p has a section, by Theorem 2.10.→ 2 2 Definition 4.1. First, we define Fd(p) to be the subfunctor of Gd(p) which associates to any S scheme T pairs (L, H) / ∼ as in Defini- tion 2.9 with the additional condition that pT∗L is locally free of rank ∗ 3 and pT pT∗L L is surjective. 2 2 Next, we define Kd(p) to be the subfunctor of Fd(p) which asso- ciates to any S→scheme T pairs (L, H) / ∼ as in Definition 2.9 with the additional conditions that pT∗L is locally free of rank 3, that and that the resulting morphism C PpT∗L is a closed embedding. Proposition 4.2. Suppose p : C S is a family of smooth genus g curves with a section, corresponding→ to a map S Mg Then, we have morphisms 2 2 2 S ×Mg Pd Kd(p) Fd(p)→ Gd(p) with the first map being an isomorphism and the latter two maps being→ open embeddings. In particular, S P the fiber product→ ×Mg→d is in fact→ a scheme. 2 Proof. We will verify that there are maps S ×Mg Pd Kd(p) 2 2 Fd(p) Gd(p) with the first morphism being an isomorphism and the latter two morphisms being open embeddings. The→ last claim→ 2 would→ then follow from Theorem 2.10, since Gd(p) is representable as p has a section. 2 2 First, we show that Fd(k) is an open subfunctor of Gd(p). To see 2 this, we know Gd(p) is representable by some scheme X with a uni- versal invertible sheaf LX on the universal curve pX : CX X. It 2 follows that Fd(p) is then represented by the open subscheme of X ∗ on which pX∗LX is locally free of rank 3 and the map pXpX∗L→X LX is surjective (which is open because it is the complement of the sup- port of the cokernel, and the support of a sheaf is a closed locus).→ 2 2 Next, we show Kd(p) Fd(p) is an open embedding. Note that because pT∗L is locally free of rank 3, we may construct the rela- 2 tive proj P(pT∗L) which→ is isomorphic to PT . The condition that ∗ pT pT∗L L is surjective implies that the sheaf L is basepoint free, and hence we obtain a resulting projective morphism C P(pT∗L). 2 Now, say→ the functor Fd(p) is represented by some scheme Y with → 22 AARON LANDESMAN a universal invertible sheaf LY on the universal curve pY : CY Y, and let φ : CY PpY∗LY be the resulting closed embedding. 2 2 To show Kd(p) Fd(p) is an open embedding, it suffices to show→ there is an open→ locus over which the map C P(pY∗L) is a closed embedding. This→ is intuitively clear because the locus on which it is not a closed embedding should be the one→ in which the degree of the fiber jumps. We formally codify this as follows: Then, let V denote the cokernel of the resulting map OPpY∗LY φ∗OCY . Let W := Supp V ⊂ P(pY∗LY) denote the support of V and let Z := −1 φ (W). Then, it follows that Z is a closed subscheme→ of CY and 2 pY(CY \ Z) represents the functor Kd(p). 2 Finally, we define a map S ×Mg Pd Kd(p) and verify it is an isomorphism of stacks. First define the map
→ 2 S ×Mg Pd Gd(p) (f : C T, L) ∈ Pd(T) 7 (L, f∗L) / ∼ . → 2 Observe this factors through Kd(p) by Proposition 3.6. → → 2 We want to verify the resulting map S ×Mg Pd Kd(p) is an equivalence of stacks. We wish to show the map is fully faithful and essentially surjective. Full faithfulness follows tautologically→ from the definition, as it sends a pair (f : C T, L) to essentially the same data, with the same equivalence relation. That is, the equivalence relation ∼ in Definition 2.9 is the same→ as the isomorphism relation given in Definition 3.1. So, to conclude, we only need verify essential surjectivity. For this, we will start by showing that if we have some family pT : C T and an invertible sheaf L on C whose pushforward is locally free of rank 3, then pT∗L has no subsheaves H which are locally free→ of rank 3 and satisfy (2.6), other than pT∗L itself. Along the way, we 0 will also see that for all points q ∈ T, the base change map φq is an isomorphism. To verify this, suppose we have such datum. Let ∗ 0 Ct := t ×T CT . Note that pt∗ιCH H (Ct, L|κ(t)) is an injective map between vector spaces. Observe that since we are assuming pT∗L is an embedding, it follows pt∗L is→ an embedding. This means that in 2 fact Ct is a plane curve, as L determines an embedding Ct Pt . By 0 Proposition 2.6, if Ct is a plane curve, h (Ct, L|κ(t)) ≤ 3 and so if (2.6) is an injection from a 3-dimensional vector space, it must be→ an iso- morphism. Now, recall the map (2.6) was defined as the composition ∗ ∗ ∗ ι H ι pT∗L pt∗ιCL. Since the composition is surjective, the lat- ter map must be surjective, hence an isomorphism. This implies, by → → THE LOCUS OF PLANE CURVES IN THE MODULI STACK OF CURVES 23 cohomology and base change that the map H pT∗L is an isomor- phism when restricted to any fiber, hence an isomorphism. 1 It only remains to show that if pT∗L is locally→ free then so is R pT∗L. This will imply the map is essentially surjective. However, since 0 the base change map φq is an isomorphism for all points q ∈ T, 1 it follows from Lemma 3.5 that R pT∗L is locally free. Since the base change maps commute by Lemma 3.5, it follows from Riemann-Roch 1 that R pT∗L is locally free of rank 2 − d + g.
Now, since Mg is Deligne-Mumford, there is an etale´ cover T Mg. This corresponds to a family C T. After choosing a further etale´ cover of T, we may assume that p : C T has a section.→ In 2 this case, we know from Theorem 2.10→ that the functor Gd(p) is rep- resentable by a scheme. That is, the fiber product→ τ T ×Mg Pd T (4.1)
Pd Mg is a scheme, using Proposition 4.2. Hence, in order to verify our map Pd Mg is a locally closed embedding, we only need check τ is a locally closed embedding. For this, we will use the following valuative→ criterion for locally closed embeddings. Lemma 4.3 (Valuative criterion for locally closed embeddings, [Moc14, Chapter 1, Corollary 2.13]). Suppose S is a Noetherian scheme and X and Y are S-schemes of finite type. A morphism f : X Y is a locally closed embedding if and only if f is a monomorphism and the following condition holds: For all discrete valuation rings R with fraction→ field K and residue field κ, and all maps g : Spec R Y with commutative diagrams
Spec K X Spec κ X → (4.2) f f g g Spec R Y Spec R Y, there exists a unique morphism h : Spec R X making the diagrams
Spec K X Spec κ X h → h (4.3) f f g g Spec R Y Spec R Y 24 AARON LANDESMAN commute. So, we need to verify the map τ of (4.1) is a monomorphism and that it satisfies the valuative criterion of (4.3). First, we show it is a monomorphism. Proposition 4.4. The map τ defined in (4.1) is a monomorphism. Proof. Using [Gro67, Proposition 17.2.6] in order to verify the nat- ural map τ is a monomorphism, it suffices to verify each fiber ei- ther has degree 0 or 1. This can be verified on geometric points. So, let Spec k T be some geometric point. Let Ck be the corre- sponding curve over k. We wish to show the fiber over Spec k has degree 1. For→ this, we will check the fiber is reduced and is sup- 2 ported on a single point. From the definition of the stack Gd(p) with p : Ck Spec k, which has a section as k is algebraically closed, we 2 2 see Gd(p) parameterizes the underlying set of gd’s on C. By Propo- 2 2 2 sition 2.11,→ Gd(p) is a degree one scheme. Since Pd(p) ⊂ Gd(p) is 2 an open subscheme by Proposition 4.2, it follows that Pd(p) also has degree at most 1, completing the proof. We can now conclude prove our main theorem, by verifying the valuative criterion for locally closed embeddings holds.
Theorem 4.5. The map Pd Mg is a locally closed embedding. Proof. First, it suffices to check after pull back to an etale´ cover of Mg, and hence it suffices to→ check the map τ defined in (4.1) is a lo- cally closed embedding. For this, by Lemma 4.3, it suffices to verify the map is a monomorphism (which follows from Proposition 4.4) and the valuative criterion for locally closed embeddings. To apply Lemma 4.3 we are using that Pd and Mg are both finite type over Spec Z. It is well known that Mg has finite type: this follows from the construction of Mg as a quotient of a Hilbert scheme of canoni- cally embedded curves modulo a PGL action. The statement for Pd follows from Theorem 3.7. It only remains to verify the valuative criterion for being a locally closed immersion. Retaining the notation of Lemma 4.3, the valua- tive criterion can be rephrased in the following way: Let Cκ and CK denote the restriction of C to the closed and generic fibers of Spec R. We may suppose we have a family of curves f : CR Spec R whose 2 closed and generic fibers are plane curves with the maps to PK and 2 Pκ given by invertible sheaves LK and Lκ. We want→ to show there ex- ists a unique invertible sheaf L of degree d on CR so that L|CK = LK THE LOCUS OF PLANE CURVES IN THE MODULI STACK OF CURVES 25
1 and LCκ = Lκ with R f∗L locally free of rank 2 − d + g. First, unique- ness follows the valuative criterion for separatedness, since Pd and Mg are both separated, so the map Pd Mg is separated. Therefore, it suffices to show there exists a morphism Spec R Pd, restricting to the given maps from Spec→ K and Spec κ, which we now construct. Since R is regular, it follows CR is regular. Therefore,→ the natural map from Weil divisors to Cartier divisors is an isomor- ∼ phism, and hence LK = OCk (DK) for some Weil divisor DK ⊂ CK. Let DR denote the closure of DK inside CR. Let L := OC (DR). By ∼ R construction, we know L|C = LK. ∼ K Next, we verify L|Cκ = Lκ. Indeed, since f : CR R is a proper morphism of Noetherian schemes and L is flat over R, we obtain that 0 the map q 7 h (Cq, L|Cq ) is upper semicontinuous.→ In particular, 0 0 since h (CK, L|CK ) = 3, we obtain h (Cκ, L|Cκ ) ≥ 3. It follows from 0 Lemma 3.5→ that h (Cκ, L|Cκ ) = 3. Then, again using Lemma 3.5, 0 there is a unique invertible sheaf M on Cκ with h (Cκ, M) = 3, which ∼ ∼ ∼ implies that both L|Cκ = M and Lκ = M, so L|Cκ = Lκ. 1 To conclude, we only need verify that R f∗L is locally free of rank 0 2 − d + g. But indeed, we have already shown that h (CK, L|CK ) = 0 1 1 h (Cκ, L|Cκ ) = 3, which implies h (CK, L|CK ) = h (Cκ, L|Cκ ) = 2 − d + g. Since Spec R is reduced, it follows from Grauert’s theorem that f∗L is locally free.
5. ACKNOWLEDGEMENTS I thank David Zureick-Brown for running the REU project which originally prompted this question. I thanks Maksym Fedorchuk for suggesting the method of showing Pd Mg is a locally closed em- 2 bedding by showing Pd Gd is an open embedding and then show- 2 ing Gd Mg is a locally closed embedding.→ I thank Anand Patel and 2 Joe Harris for explaining→ why smooth plane curves have a unique gd and suggesting→ other methods of approaching this question. I thank Brian Conrad for pointing out the useful valuative criterion for being a locally closed embedding. I thank Ravi Vakil and Michael Kemeny for listening to my argument in detail. I also thank Tony Feng, Ben Lim, Arpon Raksit, Zev Rosengarten, Bogdan Zavyalov, and Yang Zhou for helpful discussions. This material is based upon work sup- ported by the National Science Foundation Graduate Research Fel- lowship Program under Grant No. DGE-1656518. 26 AARON LANDESMAN
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