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The locus of plane curves in the moduli stack of curves

Aaron Landesman

TL;DR

The paper identifies the locus of smooth degree $d$ plane curves, with genus $g=\binom{d-1}{2}$, as a locally closed substack of the moduli stack ${\mathscr M}_g$ for $d\ge4$ by equating the quotient stack $[U_d/\mathrm{PGL}_3]$ with a moduli stack ${\mathscr P}_d$ of embedded plane curves and showing ${\mathscr P}_d \to {\mathscr M}_g$ is a locally closed embedding. The authors introduce an intermediate moduli functor ${\mathscr G}^2_d(p)$ parameterizing curves with a $g^2_d$ and prove that ${\mathscr P}_d$ embeds openly into this functor, which is itself representable by a scheme; they then verify a monomorphism and a valuative criterion to conclude the locally closed embedding. A key technical step is proving that a smooth plane curve has a unique, complete $g^2_d$, and that this $g^2_d$ corresponds to a reduced point in the associated parameter space, leveraging projective normality to control deformations. The results unify different approaches to the moduli of plane curves and provide a robust framework for studying the locus within ${\mathscr M}_g$, with potential applications to Chow rings, arithmetic statistics, and related moduli problems.

Abstract

Let $d \geq 4$ and let $U_d$ denote the locus of smooth curves in the Hilbert scheme of degree $d$ plane curves. If the members of $U_d$ have genus $g$, let $\mathscr{M}_g$ denote the moduli stack of genus $g$ curves. We show that the natural map $[U_d/\operatorname{PGL}_3] \to \mathscr{M}_g$ is a locally closed embedding.

The locus of plane curves in the moduli stack of curves

TL;DR

The paper identifies the locus of smooth degree plane curves, with genus , as a locally closed substack of the moduli stack for by equating the quotient stack with a moduli stack of embedded plane curves and showing is a locally closed embedding. The authors introduce an intermediate moduli functor parameterizing curves with a and prove that embeds openly into this functor, which is itself representable by a scheme; they then verify a monomorphism and a valuative criterion to conclude the locally closed embedding. A key technical step is proving that a smooth plane curve has a unique, complete , and that this corresponds to a reduced point in the associated parameter space, leveraging projective normality to control deformations. The results unify different approaches to the moduli of plane curves and provide a robust framework for studying the locus within , with potential applications to Chow rings, arithmetic statistics, and related moduli problems.

Abstract

Let and let denote the locus of smooth curves in the Hilbert scheme of degree plane curves. If the members of have genus , let denote the moduli stack of genus curves. We show that the natural map is a locally closed embedding.

Paper Structure

This paper contains 11 sections, 15 theorems, 8 equations.

Table of Contents

  1. Introduction
  2. Idea of the proof
  3. Outline
  4. Classical Facts about plane curves
  5. Showing there is a single $g^2_d$
  6. Showing that $g^2_d$ is reduced
  7. The moduli stack of plane curves
  8. Defining the stack of plane curves
  9. Relating the stack of plane curves to $\mathscr G^2_d(p)$
  10. Verifying that $\mathscr P_d \rightarrow {\mathscr M_g}$ is a locally closed embedding
  11. Acknowledgements

Key Result

Theorem 1.1

For $d \geq 4$ and $g = \binom{d-1}{2}$, the map $[U_d/\mathop{\mathrm{PGL}}\nolimits_3] \to {\mathscr M_g}$ is a locally closed embedding of stacks.

Theorems & Definitions (35)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.6
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 25 more