A note on the Brill-Noether loci of small codimension in moduli space of stable bundles

TL;DR

This paper investigates the geometric properties of Brill-Noether loci within the moduli space of stable rank 2 vector bundles on algebraic curves, revealing their rationality and connectivity features for various genera.

Contribution

It establishes the stable-rationality, unirationality, and rational chain connectedness of Brill-Noether loci for generic curves and line bundles across different genera.

Findings

W^{1}_{X}(2,L) is stably-rational for genus 3

W^{1}_{X}(2,L) is unirational for genus 4

W^{1}_{X}(2,L) is rationally chain connected for genus ≥ 5

Abstract

Let $X$ be a smooth projective curve of genus $g$ over the field $\mathbb{C}$. Let $M_{X}(2,L)$ denote the moduli space of stable rank $2$ vector bundles on $X$ with fixed determinant $L$ of degree $2g-1$. Consider the Brill-Noether subvariety $W^{1}_{X}(2,L)$ of $M_{X}(2,L)$ which parametrises stable vector bundles having at least two linearly independent global sections. In this article, for generic $X$ and $L$, we show that $W^{1}_{X}(2,L)$ is stably-rational when $g=3$, unirational when $g=4$, and rationally chain connected by Hecke curves, when $g\geq 5$. We also show triviality of low dimensional rational Chow groups of an associated Brill-Noether hypersurface.