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On the Prym map of degree 4 cyclic covers of hyperelliptic curves

Anatoli Shatsila

Abstract

In this paper, we study the Prym map associated to degree 4 étale cyclic covers of genus $g$ hyperelliptic curves restricted to the irreducible component $\mathcal{RH}_g[4]^{hyp}$ of the moduli space of such covers where an intermediate cover is hyperelliptic. We show that for $g \geq 3$ the Prym map is injective on $\mathcal{RH}_g[4]^{hyp}$. In the case $g=2$ (where $\mathcal{RH}_2[4]^{hyp} = \mathcal{RH}_2[4]$) we prove that non-empty fibers of the Prym map, apart from two exceptional fibers, are isomorphic to the projective line without 8 points. Moreover, we obtain a new description of the space $\mathcal{RH}_g[4]^{hyp}$ in terms of tuples of complex numbers and find equations of hyperelliptic curves arising from such covers.

On the Prym map of degree 4 cyclic covers of hyperelliptic curves

Abstract

In this paper, we study the Prym map associated to degree 4 étale cyclic covers of genus hyperelliptic curves restricted to the irreducible component of the moduli space of such covers where an intermediate cover is hyperelliptic. We show that for the Prym map is injective on . In the case (where ) we prove that non-empty fibers of the Prym map, apart from two exceptional fibers, are isomorphic to the projective line without 8 points. Moreover, we obtain a new description of the space in terms of tuples of complex numbers and find equations of hyperelliptic curves arising from such covers.

Paper Structure

This paper contains 5 sections, 16 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Degree 4 cyclic covers of hyperelliptic curves
  3. Geometry of the Prym variety
  4. Fibers of the Prym map in genus 2
  5. The Prym map on $\mathcal{RH}_g[4]^{hyp}$

Key Result

Theorem 1.1

i) The Prym map $\mathcal{P}_g[4]: \mathcal{RH}_g[4]^{hyp} \to\mathcal{A}_{3g-3}^{(1,\ldots,1,4,\ldots, 4)}$ is injective for $g \geq 3$. ii) Each non-empty fiber of $\mathcal{P}_2[4]$, except for two exceptional fibers, is isomorphic to the projective line without 8 points.

Theorems & Definitions (29)

  • Theorem 1.1: Theorem \ref{['discr']}, Theorem \ref{['injective']}
  • Lemma 2.1
  • Lemma 2.2: BO17
  • Corollary 2.3
  • Proposition 2.4
  • proof
  • Lemma 2.6
  • proof
  • Remark 2.7
  • Theorem 2.8
  • ...and 19 more