Jets-separation thresholds, Seshadri constants and higher Gauss-Wahl maps on abelian varieties

Given a closed subscheme $Z$ of a polarized abelian variety $(A,\mathrm{\ell })$ we define its vanishing threshold with respect to $\mathrm{\ell }$ and relate it to the Seshadri constant of the ideal defining $Z\mathrm{.}$ As a particular case, we introduce the notion of jets-separation thresholds, which naturally arise 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 $\mathrm{\ell }\mathrm{.}$