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On the Prym map for cyclic covers of genus two curves

Daniele Agostini

Abstract

The Prym map assigns to each covering of curves a polarized abelian variety. In the case of unramified cyclic covers of curves of genus two, we show that the Prym map is ramified precisely on the locus of bielliptic covers. The key observation is that we can naturally associate to such a cover an abelian surface with a cyclic polarization, and then the codifferential of the Prym map can be interpreted in terms of multiplication of sections on the abelian surface. Furthermore, we give a different proof of a result by Ramanan that a genus two cyclic cover of degree sufficiently high is never hyperelliptic.

On the Prym map for cyclic covers of genus two curves

Abstract

The Prym map assigns to each covering of curves a polarized abelian variety. In the case of unramified cyclic covers of curves of genus two, we show that the Prym map is ramified precisely on the locus of bielliptic covers. The key observation is that we can naturally associate to such a cover an abelian surface with a cyclic polarization, and then the codifferential of the Prym map can be interpreted in terms of multiplication of sections on the abelian surface. Furthermore, we give a different proof of a result by Ramanan that a genus two cyclic cover of degree sufficiently high is never hyperelliptic.

Paper Structure

This paper contains 6 sections, 6 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Background
  3. General facts on cyclic covers
  4. Cyclic covers of curves and the Prym map
  5. The non bielliptic case
  6. The bielliptic case

Key Result

Lemma \oldthetheorem

Suppose that $Y$ is projective and connected and let $f\colon X \to Y$ be an étale cyclic cover given by a $d$-torsion line bundle $\eta$. Then, the kernel of the pullback map $f^*\colon \operatorname{Pic}(Y) \to \operatorname{Pic}(X)$ is precisely the subgroup generated by $\eta$.

Theorems & Definitions (17)

  • Remark \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Remark \oldthetheorem
  • Proposition \oldthetheorem: Lange-Ortega
  • proof
  • Remark \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • ...and 7 more