Genus two curves on abelian surfaces

Abstract

This paper deals with singularities of genus 2 curves on a general (d_1,d_2)-polarized abelian surface (S,L). In analogy with Chen's results concerning rational curves on K3 surfaces [Ch1,Ch2], it is natural to ask whether all such curves are nodal. We prove that this holds true if and only if d_2 is not divisible by 4. In the cases where d_2 is a multiple of 4, we exhibit genus 2 curves in |L| that have a triple, 4-tuple or 6-tuple point. We show that these are the only possible types of unnodal singularities of a genus 2 curve in |L|. Furthermore, with no assumption on d_1 and d_2, we prove the existence of at least a nodal curve in |L|. As a corollary, we obtain nonemptiness of all Severi varieties on general abelian surfaces and hence generalize [KLM, Thm 1.1] to nonprimitive polarizations.