Seshadri constants and hyperelliptic curves on abelian varieties

Given a smooth curve $C$ on a polarized abelian variety $(A,\theta),$ we show that that quotient between the Seshadri constant of $(A,\theta)$ and the degree of $C$ is bounded above by an intrinsic invariant of $C.$ This invariant is defined in terms of the (non) surjectivity of certain high order Gau\ss-Wahl maps on the curve. As a consequence, 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. At the end, we compare this result to the conjectural picture about polarized abelian varieties with small Seshadri constant.