# Higher rank BN-theory for curves of genus 6

## TL;DR

This paper explores higher rank Brill-Noether theory for genus 6 curves, revealing new phenomena and providing upper bounds for Brill-Noether loci, including examples with negative Brill-Noether numbers.

## Contribution

It introduces new upper bounds for non-emptiness of Brill-Noether loci in genus 6 and constructs examples approaching these bounds, highlighting phenomena absent in lower genus.

## Key findings

- New upper bounds for Brill-Noether loci in genus 6
- Existence of curves with negative Brill-Noether numbers
- Examples approaching theoretical bounds

## Abstract

Higher rank Brill-Noether theory for genus 6 is especially interesting as, even in the general case, some unexpected phenomena arise which are absent in lower genus. Moreover, it is the first case for which there exist curves of Clifford dimension greater than 1 (smooth plane quintics). In all cases, we obtain new upper bounds for non-emptiness of Brill-Noether loci and construct many examples which approach these upper bounds more closely than those that are well known. Some of our examples of non-empty Brill-Noether loci have negative Brill-Noether numbers.