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Reduction type of genus-3 curves in a special stratum of their moduli space

Irene Bouw, Nirvana Coppola, Pınar Kılıçer, Sabrina Kunzweiler, Elisa Lorenzo García, Anna Somoza

Abstract

We study a 3-dimensional stratum $\mathcal{M}_{3,V}$ of the moduli space $\mathcal{M}_3$ of curves of genus $3$ parameterizing curves $Y$ that admit a certain action of $V= C_2\times C_2$. We determine the possible types of the stable reduction of these curves to characteristic different from $2$. We define invariants for $\mathcal{M}_{3,V}$ and characterize the occurrence of each of the reduction types in terms of them. We also calculate the $j$-invariant (resp. the Igusa invariants) of the irreducible components of positive genus of the stable reduction of $Y$ in terms of the invariants.

Reduction type of genus-3 curves in a special stratum of their moduli space

Abstract

We study a 3-dimensional stratum of the moduli space of curves of genus parameterizing curves that admit a certain action of . We determine the possible types of the stable reduction of these curves to characteristic different from . We define invariants for and characterize the occurrence of each of the reduction types in terms of them. We also calculate the -invariant (resp. the Igusa invariants) of the irreducible components of positive genus of the stable reduction of in terms of the invariants.

Paper Structure

This paper contains 16 sections, 31 theorems, 67 equations, 16 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Notation
  3. Stable reduction and admissible covers
  4. The set-up
  5. Stable reduction of covers
  6. Combinatorial description of the stable reduction
  7. Computing the stable reduction
  8. The smooth plane quartic case
  9. Invariants
  10. Main results
  11. Proofs of main results
  12. Main result with non-degenerate conic
  13. Main result with degenerate conic
  14. Hyperelliptic case
  15. Invariants
  16. ...and 1 more sections

Key Result

lemma 1

Let $Y/\overline{K}$ be a smooth projective curve of genus $3$, and $f:Y\to X\simeq \mathbb{P}^1_{\overline{K}}$ a Galois cover with Galois group isomorphic to $V$ such that all intermediate covers of degree $2$ have genus $1$.

Figures (16)

  • Figure 1: $\overline{X}$ of Type III.5.
  • Figure 2: $\overline{Y}$ corresponding to Type III.5.
  • Figure 3: Special fiber of the stable model.
  • Figure 4: Configuration of the branch points on the component $\overline{X}_0$.
  • Figure 5: Possible configurations of $\overline{X}_{\xi}$ and $\overline{X}_0$.
  • ...and 11 more figures

Theorems & Definitions (70)

  • lemma 1
  • proof
  • lemma 2
  • proof
  • remark 1
  • definition 1
  • proposition 1
  • definition 2
  • proposition 2
  • proof
  • ...and 60 more