Excess dimensions of Brill--Noether schemes of rank two stable bundles

TL;DR

This paper extends classical Brill--Noether results to rank two stable bundles, linking the smoothness of loci with expected dimensions across different degrees, using advanced algebraic geometry techniques.

Contribution

It generalizes known Brill--Noether results from line bundles to rank two stable bundles, establishing new equivalences in smoothness conditions.

Findings

Proves smoothness equivalence between $B^k_{2, d}$ and $B^{k}_{2, d+1}$ under expected dimension.

Extends results of Fulton--Harris--Lazarsfeld and Aprod-Sernesi to higher rank vector bundles.

Provides new insights into the geometry of Brill--Noether loci for stable bundles.

Abstract

We use results of M. Aprodu and E. Sernesi to extend a result by Fulton--Harris--Lazarsfeld in Brill--Noether theory of line bundles %and, as well, a result by Aprod-Sernesi in theory of Secant Loci, to Brill--Noether loci of stable bundles inside the moduli space of rank two stable vector bundles on a smooth projective algebraic curve. As a consequence; if $B^k_{2, d}$ is of expected dimension, then we prove that its smoothness is equivalent with the smoothness of $B^{k}_{2, d+1}$.