A quadratic forms approach to curves in linear systems on abelian surfaces

Let A be a complex abelian surface with polarizing line bundle L of type (1, 4). There is an isomorphism of moduli spaces due to Yoshioka that relates 0-dimensional length 4 subschemes on A to certain sheaves supported on curves in the linear system | ˆL| on the dual abelian surface ˆA. When a subscheme is symmetric with respect to the [−1] involution on A, so is the associated curve. We use quadratic forms to exploit this symmetry and classify a distinguished finite set of curves according to the 2-torsion points they pass through, and the singularities they acquire at these points. As a consequence of this analysis, we prove that if L admits a symmetric theta structure, then the dual polarizing line bundle ˆL is odd.