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Clusters, toric ranks, and 2-ranks of hyperelliptic curves in the wild case

Leonardo Fiore, Jeffrey Yelton

Abstract

Given a Galois cover $Y \to X$ of smooth projective geometrically connected curves over a complete discrete valuation field $K$ with algebraically closed residue field, we define a semistable model of $Y$ over the ring of integers of a finite extension of $K$ which we call the \emph{relatively stable model} $\mathcal{Y}^{\mathrm{rst}}$ of $Y$, and we discuss its properties, eventually focusing on the case when $Y : y^2 = f(x)$ is a hyperelliptic curve viewed as a degree-$2$ cover of the projective line $X := \mathbb{P}_K^1$. Over residue characteristic different from $2$, it follows from known results that the toric rank (i.e.\ the number of loops in the graph of components) of the special fiber of $\mathcal{Y}^{\mathrm{rst}}$ can be computed directly from the knowledge of the even-cardinality clusters of roots of the defining polynomial $f$. We instead consider the "wild" case of residue characteristic $2$ and demonstrate an analog to this result, showing that each even-cardinality cluster of roots of $f$ gives rise to a loop in the graph of components of the special fiber of $\mathcal{Y}^{\mathrm{rst}}$ if and only if the depth of the cluster exceeds some threshold, and we provide a computational description of and bounds for that threshold. As a bonus, our framework also allows us to provide a formula for the $2$-rank of the special fiber of $\mathcal{Y}^{\mathrm{rst}}$.

Clusters, toric ranks, and 2-ranks of hyperelliptic curves in the wild case

Abstract

Given a Galois cover of smooth projective geometrically connected curves over a complete discrete valuation field with algebraically closed residue field, we define a semistable model of over the ring of integers of a finite extension of which we call the \emph{relatively stable model} of , and we discuss its properties, eventually focusing on the case when is a hyperelliptic curve viewed as a degree- cover of the projective line . Over residue characteristic different from , it follows from known results that the toric rank (i.e.\ the number of loops in the graph of components) of the special fiber of can be computed directly from the knowledge of the even-cardinality clusters of roots of the defining polynomial . We instead consider the "wild" case of residue characteristic and demonstrate an analog to this result, showing that each even-cardinality cluster of roots of gives rise to a loop in the graph of components of the special fiber of if and only if the depth of the cluster exceeds some threshold, and we provide a computational description of and bounds for that threshold. As a bonus, our framework also allows us to provide a formula for the -rank of the special fiber of .
Paper Structure (38 sections, 58 theorems, 35 equations)

This paper contains 38 sections, 58 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Our main problem
  3. A summary of our main results
  4. Comparison to other works
  5. Outline of the paper
  6. Notations and conventions
  7. Rings, fields, and valuations
  8. Lines, hyperelliptic curves, and models
  9. Polynomials, discs, and clusters
  10. Acknowledgements
  11. The relatively stable model of a Galois covering
  12. Preliminaries on models of curves
  13. Semistable models
  14. Contracting components of special fibers
  15. Invariants attached to a component of the special fiber
  16. ...and 23 more sections

Key Result

Theorem 1.1

Every smooth projective geometrically connected curve $C$ over $K$ achieves semistable reduction over a finite extension $K' \supseteq K$, i.e. $C$ admits a model $\mathcal{C}^{\mathrm{ss}}$ over $R'$, where $R'$ is the ring of integers in $K'$ whose special fiber is a reduced curve with at worst no

Theorems & Definitions (154)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • proof
  • Definition 1.4
  • Remark 1.5
  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • ...and 144 more