Complete classification of irreducible components of the Brill-Noether locus of rank-$2$ vector bundles of degree $d$ and speciality $2$ on a general $\nu$-gonal curve

Abstract

This paper replaces the previous longer version and focuses on the specialty $2$ case. More precisely, in this paper we address the Brill-Noether theory for rank-two, degree $d$ stable bundles of speciality $2$ on a general $\nu$-gonal curve $C$ of genus $g$, $3\le \nu <\lfloor \frac{g+3}{2}\rfloor$, leveraging universal extension spaces, modular maps and recent developments in rank-one Brill-Noether theory over Hurwitz spaces on $C$. We completely classify the irreducible components of such Brill-Noether loci in the whole range of interest for $d$, namely $2g-2\le d\le 4g-4$. Using specialization techniques, we further uncover a stratification into locally closed subsets within some of these components, and we also provide additional insight into the birational geometry and the local structure of every such a component. Our methods yield descriptions of the irreducible components of any such Brill-Noether locus and, as a by-product of our more general results, also derive interesting consequences for Brill-Noether loci of stable, rank-two bundles with a fixed general determinant, rather than fixed degree.

52 pages. Submitted preprint. Authors acknowledge support from funds "AAAGH" (P.I. G. Pareschi) during visit of first and third author at the Dept. Mathematics-Univ. Rome "Tor Vergata" when collaboration began; from MIUR Excellence Proj. MatMod@TOV 2023-27, and from Organizers of "Workshop in Classical Algebraic Geometry", Daejeon-South Korea December 2023 (D. S. Hwang, J.-M. Hwang, Y. Lee)