New examples of twisted Brill-Noether loci II

TL;DR

The paper broadens the construction of twisted Brill-Noether loci on curves of genus $g\ge2$ by leveraging dual-span and kernel-bundle techniques to generate new, often negative, BN-number examples. It completes Butler's Conjecture for coherent systems of type $(n,d,v)$ with $v=n+1$ on general curves, proving birationality, smoothness, and irreducibility of the relevant BN strata and establishing nonemptiness results via morphisms between BN loci. The authors introduce a unifying framework of BN maps, including $B^k(\mathcal{U}_1,\mathcal{U}_2)$, and exploit BN regions such as the BPN and BMNO zones to produce new BN-map points, sometimes outside classical regions. Concrete constructions based on dual spans and kernel bundles yield families of bundles with $h^0(E_1\otimes E_2)\ge k$, including cases with $n_i\ge2$, and extend to $n=1$ to obtain a rich supply of twisted BN loci. Overall, the work advances the understanding of nonemptiness, stability, and geometry of BN loci in higher rank, offering systematic methods to generate novel examples and new BN-map points.

Abstract

Our purpose in this paper is to construct new examples of twisted Brill Noether loci on curves of genus g greater than 2 with negative expected dimension. We begin by completing the proof of Butler's conjecture for coherent systems of certain type establishing the birationality, smoothness, and irreducibility of the corresponding loci. We also produce new points on the BN map.

Theorems & Definitions (59)

• Theorem 1.1
• Theorem 1.2: Theorem \ref{['t01']}
• Corollary 1.3: Corollary \ref{['tl']}
• Theorem 1.4
• Conjecture 1
• Remark 2.1
• Theorem 2.2
• Theorem 2.3
• proof
• Remark 2.4