# Non-emptiness of Brill-Noether Loci over very general quintic hypersurface

## TL;DR

This paper investigates the non-emptiness of Brill-Noether loci on very general quintic hypersurfaces in projective 3-space, introducing a Petri map analogy to produce components of expected dimension.

## Contribution

It establishes the non-emptiness of specific Brill-Noether loci on quintic hypersurfaces and develops a Petri map framework analogous to the curve case.

## Key findings

- Confirmed non-emptiness of certain Brill-Noether loci
- Constructed components of expected dimension using Petri map
- Extended Brill-Noether theory to surfaces in projective space

## Abstract

In this article we study Brill-Noether loci of moduli space of stable bundles over smooth surfaces. We define Petri map as an analogy with the case of curves. We show the non-emptiness of certain Brill-Noether loci over very general quintic hypersurface in $\mathbb{P}^3$, and use the Petri map to produce components of expected dimension.