Seshadri constants and hyperelliptic curves on abelian varieties

Given a principally polarized abelian variety, we give a sufficient condition for the van Geemen-van der Geer locus ${\mathrm{\Gamma }}_{00}$ 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. This is an extension to higher dimensions of the fact that smooth hyperelliptic curves on abelian surfaces have genus at most five. 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. In the process, we discuss further questions and conjectures.