A note on homogeneous ideals of polarized abelian surfaces

TL;DR

This paper proves that the homogeneous ideal of a polarized abelian surface is generated by quadrics and cubics for most cases, confirming a conjecture for certain polarizations using projective normality.

Contribution

It establishes the generation of the ideal by quadrics and cubics for (1,d)-polarized abelian surfaces with d ≥ 9, and extends to general embeddings with three exceptions.

Findings

Homogeneous ideal generated by quadrics and cubics for most cases

Confirmed Gross and Popescu's conjecture for d ≥ 9

Identified three exceptions in the general case

Abstract

Gross and Popescu conjectured that the homogeneous ideal of an embedded $(1,d)$-polarized abelian surface is generated by quadrics and cubics for $d\geq 9$. 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.