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The local geometry of the stack of $A_r$-stable curves

Davide Gori, Ludvig Modin, Michele Pernice

Abstract

In this paper we study the local geometry of the stack of pointed $A_r$-stable curves. In particular, we analyze the deformation theory of $A_r$-stable curves and their automorphism groups in order to study the combinatorics of families of curves over $[\mathbb{A}^1/\mathbb{G}_m]$, and use this to classify all closed points of the stack of $A_r$-stable curves. As a byproduct, we also classify all open substacks of the moduli stack of degree $2$ cyclic covers of $\mathbb{P}^1$ that admit a separated good moduli space. This is the first in a series of three papers aimed at studying obstructions for the existence of good moduli spaces for stacks of curves with $A$-type singularities, and using these to find an open substack of the stack of $A_r$-stable curves that admits a proper non-projective good moduli space when $r=5$.

The local geometry of the stack of $A_r$-stable curves

Abstract

In this paper we study the local geometry of the stack of pointed -stable curves. In particular, we analyze the deformation theory of -stable curves and their automorphism groups in order to study the combinatorics of families of curves over , and use this to classify all closed points of the stack of -stable curves. As a byproduct, we also classify all open substacks of the moduli stack of degree cyclic covers of that admit a separated good moduli space. This is the first in a series of three papers aimed at studying obstructions for the existence of good moduli spaces for stacks of curves with -type singularities, and using these to find an open substack of the stack of -stable curves that admits a proper non-projective good moduli space when .

Paper Structure

This paper contains 13 sections, 49 theorems, 85 equations, 7 figures.

Table of Contents

  1. Good moduli spaces and linearization
  2. Local structure and good moduli spaces
  3. Affine linear GIT
  4. Reduction to the deformation space
  5. A_r-stable curves
  6. A_r-stable curves and crimping
  7. Equivariant properties of normalization
  8. Hyperelliptic A_r-stable curves
  9. Families of curves over [A1/Gm]
  10. Atoms
  11. Curves with positive-dimensional stabilizers
  12. Deformation theory
  13. Isotrivial degenerations

Key Result

Theorem A

Let $\mathcal{M}_{g,n}$ be the stack of smooth curves and $\mathcal{M}_{g,n} \to \operatorname{M}_{g,n}$ its coarse moduli space. Assume $r<2g$ or $n>2$ and $\mathcal{M}_{g,n}\subset \mathcal{U}\subset \mathcal{M}_{g,n}^r$ is a sequence of open embeddings such that $\mathcal{U}$ admits a good moduli

Figures (7)

  • Figure 1: Even atom
  • Figure 2:
  • Figure 3: Example of a curve that is not a rosary but has an attached $2$-pointed rosary.
  • Figure 4:
  • Figure 5:
  • ...and 2 more figures

Theorems & Definitions (149)

  • Theorem A: \ref{['cor:stable-gms']}
  • Theorem B: \ref{['theo:classify-opens-hyp']}
  • Definition 1.1
  • Theorem 1.2: LunaetaleSliceStacks
  • Definition 1.3: ExistenceOfModuli
  • Theorem 1.4: ExistenceOfModuli
  • Definition 1.5: Basin of attraction
  • Definition 1.6
  • Lemma 1.7
  • proof
  • ...and 139 more