A Reider-type theorem for higher syzygies on abelian surfaces

Building on the theory of infinitesimal Newton--Okounkov bodies and previous work of Lazarsfeld--Pareschi--Popa, we present a Reider-type theorem for higher syzygies of ample line bundles on abelian surfaces. As an application of our methods we confirm a conjecture of Gross and Popescu on abelian surfaces with a very ample primitive polarization of type $(1,d)$, whenever $d\geq 23$.