Universal extension spaces and modular maps: unveiling irreducible components of Brill-Noether loci of stable bundles on a general $\nu$-gonal curve

We investigate the Brill-Noether theory of rank-two, degree-$d$ stable vector bundles of speciality $3$ on a general $\nu$-gonal curve of genus $g$, $3\le \nu <\lfloor \frac{g+3}{2}\rfloor$. Our approach leverages universal extension spaces, modular maps, and recent advancements in rank-one Brill-Noether theory over Hurwitz spaces. We establish existence criteria for the corresponding Brill-Noether loci and provide a comprehensive description of their irreducible components. We moreover prove that these components exhibit diverse geometric behaviors, categorized by their regularity, superabundance, and the properties of their general points. Notably, for specific degrees $d$, we prove the coexistence of multiple superabundant components alongside a regular one. Using specialization techniques, we uncover a stratification into locally closed subschemes within these components and provide insights into their birational geometry and local structure. Furthermore, our results yield also consequences for Brill-Noether loci of stable, rank-two bundles with a fixed general determinant.