On the irreducible components of some Brill-Noether loci of rank-two, stable bundles over a general $ν$-gonal curve
Youngook Choi, Flaminio Flamini, Seonja Kim
TL;DR
This work advances higher-rank Brill-Noether theory on general ν-gonal curves by classifying irreducible components of Brill-Noether loci B_d^{k_i} ∩ U^s_C(d) for rank-two stable bundles in the degree window 2g−2 ≤ d ≤ 4g−4, focusing on speciality i=2 and i=3. The authors develop a geometric extension framework, leveraging presentations 0 → N → 𝔽 → L → 0 and the associated unisecant geometry on ruled surfaces to produce and distinguish regular and superabundant components, describe their general points, and compute dimensions and birational structures; they also provide Serre-dual perspectives and fixed-determinant consequences. The results yield a thorough classification in many ranges, prove regular components are uniruled and smooth, identify superabundant families with explicit parametrizations, and rule out extra components in many cases, thereby extending and correcting prior work (notably CFK2) and informing determinant-fixed Brill-Noether theory. The methods combine line-bundle Brill-Noether data on ν-gonal curves, degeneration techniques, and degeneracy loci analyses, culminating in a comprehensive picture of rank-two Brill-Noether loci on general ν-gonal curves with detailed descriptions of component types and their geometric presentations.
Abstract
In this paper we consider Brill-Noether loci of rank-two, stable vector bundles of given degree $d$, with speciality $2$ and $3$, on a general $ν$-gonal curve $C$ of genus $g$ with the aim of studying their irreducible components in the whole range of interest for $d$, namely $2g-2 \leq d \leq 4g-4$. For speciality $2$, either we prove that such a Brill-Noether locus is empty or we completely classify all of its irreducible components, giving also some extra information about their dimensions (exhibiting both regular and superabundant components), about their birational geometry as well as giving precise description of their general point. Similarly, in speciality $3$, either we prove that such a Brill-Noether locus is empty or we exhibit irreducible components of several types according to their regularity or superabundance but also according to the precise descriptions of their general points, proving e.g. that in some degrees $d$ there are more than one superabundant component together with the presence of also a regular component. We moreover give extra information about the birational geometry of the constructed components as well as about their local behavior. As a by-product of our general results, we deduce also some consequences on Brill-Noether loci in rank two with fixed general determinant line bundle instead of fixed degree.