A note on homogeneous ideals of abelian surfaces

Gross and Popescu conjectured that the homogeneous ideal of an embedded (1,d)-polarized abelian surface is generated by quadrics and cubics for d⩾9. We prove a generalization of this using the projective normality of the embedding. It follows that the homogeneous ideal of an abelian surface embedded by a complete linear system is generated by quadrics and cubics, with three exceptions.