An approach to curves in abelian surfaces using Fourier--Mukai and quadratic forms

Abstract

It was proven by Yoshioka that given a complex abelian surface $A$ of Picard rank $1$ whose primitive polarization is non-principal of type $(1,d)$, there is an isomorphism $\Psi :{\mathrm{Hilb}}_A^d\times \hat{A}\to M_{\hat{A}}(0,\hat{l},-1)$ where ${\mathrm{Hilb}}_A^d$ is the Hilbert scheme of lenth-$d$ subschemes of $A$ and $M_{\hat{A}}(0,\hat{l},-1)$ is a moduli space of Gieseker-stable sheaves on the dual abelian surface. Specifically, $M_{\hat{A}}(0,\hat{l},-1)$ parametrizes rank $1$ torsion-free sheaves with Euler characteristic $-1$ that are supported on a curve whose Néron--Severi class is dual to that of the polarization on $A$. The Fourier--Mukai transform is a crucial component of $\Psi$. In this paper, we use the isomorphism $\Psi$ to deduce information about curves on abelian surfaces. In the case that $Z\in {\mathrm{Hilb}}_A^d$ is symmetric, i.e., fixed by the inverse map $\iota$ on $A$, we use quadratic forms to compute information about the sheaf $\Psi (Z)$ and its supporting curve. It was recently shown by Knutsen and Lelli-Chiesa that any singularity on a curve of geometric genus $2$ contained in a general $(d_1,d_2)$-polarized abelian surface must have multiplicity at most $6$, among other constraints. In contrast, we demonstrate there are curves with singularities of arbitrarily high multiplicity contained in general $(1,d)$-polarized abelian surfaces for sufficiently large $d$. Furthermore, we identify the isolated fixed points of $\iota$ in acting on the variety of Kummer type $K_{\hat{A}}(0,\hat{l},-1)$ when $d=4$. Along the way, we prove some structural results on symmetric line bundles, showing that if $d$ is even, the dual of an odd line bundle is odd.

30 pages