Genus two curves on Abelian surfaces

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