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THE LOCUS of PLANE CURVES in the MODULI STACK of CURVES One of the Most Fundamental Objects of Study in Algebraic Geom

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THE LOCUS of PLANE CURVES in the MODULI STACK of CURVES One of the Most Fundamental Objects of Study in Algebraic Geom 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 FACTS ABOUT PLANE CURVES 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 2 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 _ ISjC ' 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.
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