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

Nelson Alvarado

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 $\ell.$

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

Abstract

Given a closed subscheme of a polarized abelian variety we define its vanishing threshold with respect to and relate it to the Seshadri constant of the ideal defining As a particular case, we introduce the notion of jets-separation thresholds, which naturally arise as the vanishing threshold of the -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
Paper Structure (11 sections, 23 theorems, 120 equations)

This paper contains 11 sections, 23 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $\mathbb{Q}$-twisted objects
  4. Fourier-Mukai transform
  5. Cohomological rank functions
  6. Vanishing thresholds
  7. Seshadri constants
  8. Gauss-Wahl maps
  9. Effective surjectivity of Gauss-Wahl maps
  10. Further questions
  11. Acknowledgements

Key Result

Theorem 1.1

Let $A$ be a $g$-dimensional abelian variety and $Z\subset A$ a closed subscheme with ideal $I_{Z}.$ For $p\in\mathbb{Z}_{\geq 0}$ write $Z^{(p)}$ for the closed subscheme of $A$ defined by the ideal $I_{Z}^{p+1}.$ Let $L$ be an ample line bundle on $A$ with class $\ell\in\mathrm{NS}(A),$ then:

Theorems & Definitions (53)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Example 2.5
  • ...and 43 more