Remarks on Ruled surfaces and rank two bundles with canonical determinant and 4 sections

TL;DR

This paper investigates the geometry of rank two vector bundles with canonical determinant and four sections on a general algebraic curve, revealing an irreducible component with specific dimension and describing the structure of general elements.

Contribution

It identifies an irreducible component of the Brill-Noether locus with four sections and describes the exact sequence structure of its general elements on a general curve.

Findings

Existence of an irreducible component of dimension 3g-13 in B^4(2,K_C)

General elements fit into a specific exact sequence involving a degree three divisor

The coboundary map has a cokernel of dimension three for general elements

Abstract

Let $C$ be a smooth irreducible complex projective curve of genus $g$ and let $B^k(2,K_C)$ be the Brill-Noether loci parametrizing classes of (semi)-stable vector bundles $E$ of rank two with canonical determinant over $C$ with $h^0(C,E)\geq k$. We show that $B^4(2,K_C)$ it has an irreducible component $\mathcal B$ of dimension $3g-13$ on a general curve $C$ of genus $g\geq 8$. Moreover, we show that for the general element $[E]$ of $\mathcal B$, $E$ fits in a exact sequence $0\to\mathcal O_C(D)\to E\to K_C(-D)\to 0$, with $D$ a general effective divisor of degree three, and the corresponding coboundary map $\partial: H^0(C,K_C(-D))\to H^1(C,\mathcal O_C(D))$ has cokernel of dimension three.