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A Hurwitz divisor on the Moduli of Prym curves

Andrei Bud

TL;DR

For even genus g=2i≥4\documentclass[12pt]{minimal] \usepackage{amsmath} \use package{wasysym} £1,000,000 ¬2,500,Ã’¬3,000 \setlength{\oddsidemargin}{-69pt}

Abstract

For genus $g=2i\geq4$ and the length $g-1$ partition $μ= (4,2,\ldots,2,-2,\ldots,-2)$ of 0, we compute the first coefficients of the class of $\overline{D}(μ)$ in $\mathrm{Pic}_\mathbb{Q}(\overline{\mathcal{R}}_g)$, where $D(μ)$ is the divisor consisting of pairs $[C,η]\in \mathcal{R}_g$ with $η\cong \mathcal{O}_C(2x_1+x_2+\cdots + x_{i-1}-x_i-\cdots-x_{2i-1})$ for some points $x_1,\ldots, x_{2i-1}$ on $C$. We further provide several enumerative results that will be used for this computation.

A Hurwitz divisor on the Moduli of Prym curves

TL;DR

For even genus g=2i≥4\documentclass[12pt]{minimal] \usepackage{amsmath} \use package{wasysym} £1,000,000 ¬2,500,Ã’¬3,000 \setlength{\oddsidemargin}{-69pt}

Abstract

For genus and the length partition of 0, we compute the first coefficients of the class of in , where is the divisor consisting of pairs with for some points on . We further provide several enumerative results that will be used for this computation.

Paper Structure

This paper contains 16 sections, 13 theorems, 157 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Admissible covers and enumerative geometry
  3. Various enumerative results
  4. A Hurwitz space over the elliptic curves I
  5. A Hurwitz space over the elliptic curves II
  6. Combinatorial identities
  7. Test curves
  8. Test curve $A$
  9. Test curve $B$
  10. Test curves $C_i$
  11. Intersection numbers
  12. Intersection with test curves of type $A$
  13. Intersection with test curves of type $C_{g-1}$
  14. Intersection with test curves of type $B$
  15. Conclusions (Proof of Theorem \ref{['maintheorem']}):
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $g =2i\geq4$ and $\mu$ the length $g-1$ partition $(4,2,\ldots,2,-2,\ldots,-2)$ of $0$. Then for the class in $\mathrm{Pic}_\mathbb{Q}(\overline{\mathcal{R}}_g)$ of the divisor we have the equalities:

Figures (1)

  • Figure 1: A curve $X$ corresponding to an admissible cover in $\overline{H}_{g,\mu}$ over $[C_{/x\sim y}]$

Theorems & Definitions (24)

  • Theorem 1.1
  • Lemma 2.1
  • Remark 2.2
  • Theorem 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • ...and 14 more