# The Strong Maximal Rank Conjecture and higher rank Brill--Noether theory

## TL;DR

This paper verifies the non-emptiness of special maximal-rank loci in algebraic geometry, providing new evidence for the Strong Maximal Rank Conjecture and higher-rank Brill--Noether theory.

## Contribution

It computes cohomology classes of special loci and confirms their non-vanishing, advancing understanding of the Strong Maximal Rank Conjecture and higher-rank Brill--Noether theory.

## Key findings

- Non-zero cohomology classes of special loci are established.
- Non-emptiness of these loci in many cases is verified.
- Supports existence conjectures in higher-rank Brill--Noether theory.

## Abstract

In this paper, we compute the cohomology class of certain "special maximal-rank loci" originally defined by Aprodu and Farkas. By showing that such classes are nonzero, we are able to verify the non-emptiness portion of the Strong Maximal Rank Conjecture in a wide range of cases. As an application, we obtain new evidence for the existence portion of a well-known conjecture due to Bertram, Feinberg and independently Mukai in higher-rank Brill--Noether theory.