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On the correspondence between Ulrich bundles and curves on surfaces

Sofia Bordoni

TL;DR

The paper establishes a curve-based correspondence for Ulrich bundles on smooth surfaces by showing that an Ulrich bundle of rank r on S is equivalent to a Lazarsfeld-Mukai bundle K_{C,W,L}^* associated to a smooth curve C ⊂ S and an r-dimensional base-point-free linear series W ⊂ H^0(L), under explicit cohomological and numerical conditions. This framework yields concrete genus bounds for the curve C, restricts the connectedness of C, and specializes to surfaces in P^3 to give explicit degree formulas and a Noether-Lefschetz perspective on Ulrich line bundles. It further analyzes the Noether-Lefschetz locus, showing that the Ulrich condition defines a component of NL(d) and describes when such curves arise from ruled surface models. A refined genus bound is obtained for quartic K3 surfaces, proving g ≥ 3 r^2 for minimal Ulrich bundles, and the results illuminate the geometry of Ulrich bundles in low dimensions and their interplay with ACM curves and determinant representations.

Abstract

This work provides a curve-based approach to Ulrich bundles on surfaces, establishing a correspondence that characterizes their existence, with a focus on applications to surfaces in $\mathbb{P}^3$.

On the correspondence between Ulrich bundles and curves on surfaces

TL;DR

The paper establishes a curve-based correspondence for Ulrich bundles on smooth surfaces by showing that an Ulrich bundle of rank r on S is equivalent to a Lazarsfeld-Mukai bundle K_{C,W,L}^* associated to a smooth curve C ⊂ S and an r-dimensional base-point-free linear series W ⊂ H^0(L), under explicit cohomological and numerical conditions. This framework yields concrete genus bounds for the curve C, restricts the connectedness of C, and specializes to surfaces in P^3 to give explicit degree formulas and a Noether-Lefschetz perspective on Ulrich line bundles. It further analyzes the Noether-Lefschetz locus, showing that the Ulrich condition defines a component of NL(d) and describes when such curves arise from ruled surface models. A refined genus bound is obtained for quartic K3 surfaces, proving g ≥ 3 r^2 for minimal Ulrich bundles, and the results illuminate the geometry of Ulrich bundles in low dimensions and their interplay with ACM curves and determinant representations.

Abstract

This work provides a curve-based approach to Ulrich bundles on surfaces, establishing a correspondence that characterizes their existence, with a focus on applications to surfaces in .
Paper Structure (8 sections, 11 theorems, 66 equations)

This paper contains 8 sections, 11 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Preliminaries and Notation
  4. Proof of Theorem \ref{['main']}
  5. Curves of Theorem \ref{['main']}: Connectivity and Genus
  6. Applications to surfaces in $\mathbb{P}^3$
  7. Ulrich bundles and Noether-Lefschetz Loci
  8. Refining the lower genus bound for the curve $C$ in Theorem \ref{['main']} on quartic surfaces in $\mathbb{P}^3$

Key Result

Theorem 1.1

Let $S \subset \mathbb{P}^N$ be a smooth projective surface of degree $d\geq 2$, embedded by the linear system $|H|$, where $H\in |\mathcal{O}_S(1)|$. Then, there exists an Ulrich bundle $\mathcal{E}$ of rank $r$ on $S$ if and only if there exists a smooth (possibly disconnected) curve $C \subset S$

Theorems & Definitions (50)

  • Theorem 1.1
  • Corollary 1.2
  • Definition 2.1: Ulrich bundle
  • Definition 2.2: Ulrich complexity
  • Remark 2.3
  • Remark 2.4: Stability of Ulrich bundles
  • Definition 2.5: Lazarsfeld-Mukai bundles
  • Remark 2.6
  • Lemma 2.7
  • proof
  • ...and 40 more