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Ulrich line bundles on Enriques surfaces with a polarization of degree four

Marian Aprodu, Yeongrak Kim

Abstract

In this paper, we prove the existence of an Enriques surface with a polarization of degree four with an Ulrich bundle of rank one. As a consequence, we prove that general polarized Enriques surfaces of degree four, with the same numerical polarization class, carry Ulrich line bundles.

Ulrich line bundles on Enriques surfaces with a polarization of degree four

Abstract

In this paper, we prove the existence of an Enriques surface with a polarization of degree four with an Ulrich bundle of rank one. As a consequence, we prove that general polarized Enriques surfaces of degree four, with the same numerical polarization class, carry Ulrich line bundles.

Paper Structure

This paper contains 4 sections, 7 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Ulrich bundles
  4. Construction of Ulrich bundles using $K3$ covers

Key Result

Proposition 3

Let $\varphi : X \to {\mathbb P}^n$ be a finite morphism and denote $H_X=\varphi^* {\mathcal{O}}_{{\mathbb P}^n}(1)$. If $X$ carries an Ulrich bundle $E$ with respect to $H_X$, then $X$ carries an Ulrich bundle with respect to $d H_X$ for any integer $d>0$.

Theorems & Definitions (18)

  • Definition 1: compare to ESW03, Proposition 2.1
  • Remark 2
  • Proposition 3
  • Example 4
  • Definition 5: ESW03
  • Proposition 6
  • Example 7
  • Example 8
  • Theorem 10
  • Theorem 11
  • ...and 8 more