Higher syzygies on abelian surfaces

Abstract

Based on the theory of an infinitesimal Newton-Okounkov body, we extend the results of Lazarsfeld-Pareschi-Popa on abelian surfaces. Moreover, we show that the higher syzygies of $(X,L)$ are completely determined by its Seshadri constant when $L^{2}$ is large. As an application, we improve the existing lower bound of $(L^{2})$ for higher syzygies of a polarized abelian surface $(X,L)$.