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On Ulrich bundles on projective bundles

Andreas Hochenegger

TL;DR

This work develops a cohomological framework to study Ulrich bundles on projective bundles P(E) → X with polarisation D = π^*A + H, linking Ulrich properties on the base to those on the total space. It provides a precise vanishings criterion in terms of Sym^k E and the base’s cohomology, yielding concrete existence results for curves and surfaces: over curves, Ulrichs on P(E) correspond to base bundles with H^⋅(C, F)=0; over surfaces, they require vanishings of H^⋅(S, F) and H^⋅(S, F(−D')), with D' depending on E and A, enabling rank-two Ulrich bundles on P(E) for bases like P^2 and Hirzebruch surfaces. The approach recovers and extends known results (Aprodu-etal, Fania-etal) and applies to blowups of projective space, while also clarifying limitations and providing a framework for further Ulrich constructions on projective bundles. An erratum corrects a previous claim about pullback constructions, highlighting the nuanced interaction between base and bundle geometry, and guiding future investigations for more general polarisations.

Abstract

In this article, the existence of Ulrich bundles on projective bundles $\mathbb P(E) \to X$ is discussed. In the case, that the base variety $X$ is a curve or surface, a close relationship between Ulrich bundles on $X$ and those on $\mathbb P(E)$ is established for specific polarisations. This yields the existence of Ulrich bundles on a wide range of projective bundles over curves and surfaces.

On Ulrich bundles on projective bundles

TL;DR

This work develops a cohomological framework to study Ulrich bundles on projective bundles P(E) → X with polarisation D = π^*A + H, linking Ulrich properties on the base to those on the total space. It provides a precise vanishings criterion in terms of Sym^k E and the base’s cohomology, yielding concrete existence results for curves and surfaces: over curves, Ulrichs on P(E) correspond to base bundles with H^⋅(C, F)=0; over surfaces, they require vanishings of H^⋅(S, F) and H^⋅(S, F(−D')), with D' depending on E and A, enabling rank-two Ulrich bundles on P(E) for bases like P^2 and Hirzebruch surfaces. The approach recovers and extends known results (Aprodu-etal, Fania-etal) and applies to blowups of projective space, while also clarifying limitations and providing a framework for further Ulrich constructions on projective bundles. An erratum corrects a previous claim about pullback constructions, highlighting the nuanced interaction between base and bundle geometry, and guiding future investigations for more general polarisations.

Abstract

In this article, the existence of Ulrich bundles on projective bundles is discussed. In the case, that the base variety is a curve or surface, a close relationship between Ulrich bundles on and those on is established for specific polarisations. This yields the existence of Ulrich bundles on a wide range of projective bundles over curves and surfaces.

Paper Structure

This paper contains 8 sections, 15 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Projective bundles
  4. Projective bundles over arbitrary varieties
  5. Projective bundles over curves
  6. Projective bundles over surfaces
  7. Projective bundles over ${\mathbb P}^2$
  8. Projective bundles over Hirzebruch surfaces

Key Result

Theorem A

Let $\pi \colon {\mathbb P}({\mathcal{E}}) \to C$ be a projective bundle over a smooth projective curve, and let $D = \pi^* A + H$ be very ample. Then a locally free sheaf $\pi^* {\mathcal{F}} (D)$ is Ulrich if and only if ${\mathrm H}^\bullet(C,{\mathcal{F}})=0$. In particular, there are Ulrich lin

Theorems & Definitions (41)

  • Theorem A
  • Theorem B
  • Definition 1
  • Remark 1
  • Lemma 1
  • proof
  • Lemma 2: Hartshorne
  • Proposition 1: Orlov
  • Remark 2
  • Example 1
  • ...and 31 more