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Positivity of Ulrich bundles in the ample and free case

Valerio Buttinelli

Abstract

We study the positivity of an Ulrich vector bundle defined with respect to a globally generated ample line bundle. First we prove a generalization of a Lopez theorem on the first Chern class and the bigness of an Ulrich bundle. Then, under some additional assumptions on the polarization, we give a description of its augmented base locus, which consequently leads to a characterization of the V-bigness and of the ampleness of an Ulrich bundle in this setting.

Positivity of Ulrich bundles in the ample and free case

Abstract

We study the positivity of an Ulrich vector bundle defined with respect to a globally generated ample line bundle. First we prove a generalization of a Lopez theorem on the first Chern class and the bigness of an Ulrich bundle. Then, under some additional assumptions on the polarization, we give a description of its augmented base locus, which consequently leads to a characterization of the V-bigness and of the ampleness of an Ulrich bundle in this setting.
Paper Structure (6 sections, 24 theorems, 93 equations)

This paper contains 6 sections, 24 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Notations
  3. Preliminary definitions and generalities on Ulrich vector bundles
  4. General facts about Seshadri constants
  5. Proof of Theorem \ref{['thm:main']} and some remarks
  6. Augmented base locus of an Ulrich vector bundle

Key Result

Theorem 1

Let $X$ be a smooth projective variety of dimension $n\geq1$ and let $B$ a globally generated ample line bundle on $X$ with $B^n=d.$ Let $\mathcal{E}$ be a vector bundle of rank $r$ which is $0$-regular with respect to $B.$ Then holds for every $x\in X$ and for every subvariety $Z\subset X$ of dimension $k\geq1$ passing through $x$ provided that the following conditions are satisfied: In particu

Theorems & Definitions (89)

  • Definition 1.1
  • Theorem 1
  • Definition 1.2
  • Theorem 2
  • Corollary 3
  • Theorem 4
  • Definition 3.1
  • Remark 3.2
  • Definition 3.3
  • Remark 3.4
  • ...and 79 more