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Ulrich bundles on ruled surfaces

Marian Aprodu, Laura Costa, Rosa Maria Miro-Roig

Abstract

In this short note, we study the existence problem for Ulrich bundles on ruled surfaces, focusing our attention on the smallest possible rank. We show that existence of Ulrich line bundles occurs if and only if the coefficient $α$ of the minimal section in the numerical class of the polarization equals one. For other polarizations, we prove the existence of rank two Ulrich bundles.

Ulrich bundles on ruled surfaces

Abstract

In this short note, we study the existence problem for Ulrich bundles on ruled surfaces, focusing our attention on the smallest possible rank. We show that existence of Ulrich line bundles occurs if and only if the coefficient of the minimal section in the numerical class of the polarization equals one. For other polarizations, we prove the existence of rank two Ulrich bundles.

Paper Structure

This paper contains 3 sections, 6 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Ulrich line bundles
  3. Rank-two Ulrich bundles

Key Result

Theorem 2.1

Let $X$ be a geometrically ruled surface over a smooth curve $C$ of genus $g$ and with $e>0$. Let $H_{\alpha}=\alpha C_0+\beta F$ be any very ample divisor on $X$. Then, there are Ulrich line bundles with respect to $H_{\alpha}$ if and only if $\alpha=1$. In this case, there exists exactly two types and with $\mathcal{L}_1,\mathcal{L}_2\in\mathrm{Pic}^0(C)$ general.

Theorems & Definitions (10)

  • Theorem 2.1
  • proof
  • Proposition 3.1
  • Lemma 3.2
  • Proposition 3.3
  • Theorem 3.4
  • Remark 3.5
  • Remark 3.6
  • Remark 3.7
  • Corollary 3.8