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

TL;DR

This paper introduces jets-separation thresholds on abelian varieties, relates them to Seshadri constants, and uses Fourier-Mukai techniques to connect these thresholds with the surjectivity of higher Gauss-Wahl maps, providing new criteria.

Contribution

It defines jets-separation thresholds and links them to Seshadri constants and higher Gauss-Wahl maps on abelian varieties using Fourier-Mukai methods.

Findings

Jets-separation thresholds are related to the Seshadri constant of the ideal.

A criterion for the surjectivity of higher Gauss-Wahl maps is established.

Fourier-Mukai techniques are used to connect geometric and algebraic properties.

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 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 $\ell.$