A note on homogeneous ideals of polarized 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 $d\ge 9$. We prove 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.