Hyperelliptic curves on $(1,d)$ abelian surfaces for $d\le 15$

Abstract

Bryan, Oberdieck, Pandharipande and Yin predicted the number of hyperelliptic curves on general abelian surfaces under a transversality condition. Some of the numbers have already been confirmed. In this paper we construct singular hyperelliptic curves on $(1,d)$ polarised abelian surfaces for $d\le 15$ confirming a few predicted numbers, namely of (geometric) genus $g=d$ and of geometric genus $k+3$ on $(1,2k)$ and $(1,2k+1)$ polarised surfaces. Moreover, we have found that all genus 7 hyperelliptic curves on general $(1,8)$ polarised surfaces are double coverings of curves locally embedded into $(1,4)$ polarised surfaces. More interestingly, there is a unique genus 9 hyperelliptic curve (up to translation) locally embedded into a general $(1,12)$ polarised surface and it is a Klein covering of a curve locally embedded into a $(1,3)$ polarised surface.

arXiv comment19 pages