Genus two curves on abelian surfaces

Abstract

This paper deals with singularities of genus 2 curves on a general (d1, d2)- 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 d2 is not divisible by 4. In the cases where d2 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 d1 and d2, we prove the existence of at least one nodal genus 2 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.