Special determinants in higher-rank Brill-Noether theory

TL;DR

This paper develops a new framework for analyzing the dimensions of higher-rank Brill-Noether loci with special determinants, introducing generalized alternating Grassmannians and proving sharp bounds in several cases.

Contribution

It extends previous work by creating a general approach for arbitrary rank and establishing new dimension bounds using generalized alternating Grassmannians.

Findings

Proved modified expected dimension bounds for higher-rank Brill-Noether loci.

Introduced generalized alternating Grassmannians as a key tool.

Showed sharpness of bounds in specific rank 2 cases.

Abstract

Continuing our previous study of modified expected dimensions for rank-2 Brill-Noether loci with prescribed special determinants, we introduce a general framework which applies a priori for arbitrary rank, and use it to prove modified expected dimension bounds in several new cases, applying both to rank 2 and to higher rank. The main tool is the introduction of generalized alternating Grassmannians, which are the loci inside Grassmannians corresponding to subspaces which are simultaneously isotropic for a family of multilinear alternating forms on the ambient vector space. In the case of rank 2 with 2-dimensional spaces of sections, we adapt arguments due to Teixidor i Bigas to show that our new modified expected dimensions are in fact sharp.