Geometry of certain Brill-Noether locus on a very general sextic surface and Ulrich bundles

TL;DR

This paper investigates the geometry of Brill-Noether loci on a very general sextic surface, establishing non-emptiness results and conditions for Ulrich bundles, thereby advancing understanding of vector bundles on algebraic surfaces.

Contribution

It provides new non-emptiness results for Brill-Noether loci and establishes conditions for the existence of rank 2 weakly Ulrich bundles on a sextic surface.

Findings

Non-emptiness of certain Brill-Noether loci confirmed.

Existence conditions for rank 2 weakly Ulrich bundles with specific Chern classes.

Existence of rank 2 Ulrich bundles used to find additional Brill-Noether loci.

Abstract

Let $X \subset \mathbb P^3$ be a very general sextic surface over complex numbers. In this paper we study certain Brill-Noether problems for moduli of rank $2$ stable bundles on $X$ and its relation with rank $2$ weakly Ulrich and Ulrich bundles. In particular, we show the non-emptiness of certain Brill-Noether loci and using the geometry of the moduli and the notion of the Petri map on higher dimensional varieties, we prove the existence of components of expected dimension. We also give sufficient conditions for the existence of rank $2$ weakly Ulrich bundles $\mathcal E$ on $X$ with $c_1(\mathcal E) =5H$ and $c_2 \geq 91$ and partially address the question of whether these conditions really hold. We then study the possible implication of the existence of an weakly Ulrich bundle in terms of non-emptiness of Brill-Noether loci. Finally, using the existence of rank $2$ Ulrich bundles on $X$ we obtain some more non-empty Brill-Noether loci and investigate the possibility of existence of higher rank simple Ulrich bundles on $X$.