Jets Separation Thresholds, Seshadri Constants, and Higher Gauss–Wahl Maps on Abelian Varieties

Source abstract

Abstract Given a closed subscheme $Z$ of a polarized abelian variety $(A,\ell )$ we define its vanishing threshold with respect to $\ell$ and relate it to the Seshadri constant of the ideal defining $Z.$ As a particular case, we introduce the notion of jets separation threshold, which naturally arises as the vanishing threshold of the $p$-infinitesimal neighborhood of a point. Afterwards, by means of Fourier–Mukai methods we relate the jets separation thresholds with the surjectivity of certain higher Gauss–Wahl maps. As a consequence we obtain a criterion for the surjectivity of those maps in terms of the Seshadri constant of the polarization $\ell .$