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

TL;DR

The paper studies the Brill-Noether locus $W^{1}_{X}(2,L)$ inside the moduli space $M_{X}(2,L)$ of stable rank-2 bundles with fixed determinant $\,L$ of degree $2g-1$ on a curve $X$ of genus $g$. It constructs the extension space $\\mathbb{P}_{L}$ with a universal extension and a Brill-Noether hypersurface $\\mathcal{H}$ whose geometry controls $W^{1}_{X}(2,L)$ via Bertram’s birational map to $M_{X}(2,L)$, and analyzes $\\mathcal{H}$ to deduce rationality properties of $W^{1}_{X}(2,L)$ for small $g$: $W^{1}_{X}(2,L)$ is stably rational for $g=3$, unirational for $g=4$, and rationally chain connected (via Hecke curves) for $g\ge 5$. For $g\ge 5$ the paper also shows $\\mathcal{H}$ is rationally chain connected and determines the minimal length of rational chains needed to connect points, with $\\mathcal{H}$ being chain connected by lines of length two. Finally, it establishes vanishing and structure results for the rational Chow groups $CH_{k}(\\mathcal{H})_{\\mathbb{Q}}$ in low degrees, leveraging Otwinowska’s theorem and the birational link to $W^{1}_{X}(2,L)$. These results provide explicit rationality and connectivity profiles for Brill-Noether loci in low-genus regimes and contribute to the understanding of Chow groups of associated Brill-Noether hypersurfaces.

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.

Theorems & Definitions (36)

• Theorem 1.1
• Theorem 1.2
• Lemma 3.1
• proof
• Theorem 3.2
• Lemma 3.3
• proof
• Proposition 3.4
• proof
• Corollary 3.5