Seshadri constants and hyperelliptic curves on abelian varieties

Abstract

Given a principally polarized abelian variety, we give a sufficient condition for the van Geemen-van der Geer locus ${\Gamma }_{00}$ to be positive dimensional. Along the way, we prove a sharp Castelnuovo-type inequality for hyperelliptic curves on abelian varieties, and characterize the cases in which we have equality. These results are consequences of a general relation between Seshadri constants and the surjectivity of high order Gauß-Wahl maps on curves inside abelian varieties, which we also prove. Applying such relation to Abel-Jacobi curves we obtain new results about the surjectivity of higher Gauß-Wahl maps on smooth curves, sharpening a previous result of Bertram, Ein and Lazarsfeld. In the process, we discuss further questions and conjectures.

arXiv comment18 pages. Small exposition changes. Extra application to GW maps on curves. Submitted version. Comments are welcome and encouraged