Ranks of maps of vector bundles

TL;DR

This paper extends techniques from line bundles to vector bundles to analyze their properties, proving key results about the Petri map's injectivity and deformation theory related maps.

Contribution

It introduces a generalized method for vector bundles and applies it to prove injectivity and surjectivity results in algebraic geometry.

Findings

Proved the injectivity of the Petri map for vector bundles.

Established the surjectivity of a deformation-related map.

Extended techniques from line bundles to vector bundles.

Abstract

We generalize to vector bundles the techniques introduced for line bundles in prior work of the author with Liu, Osserman and Zhang. We then use this method to prove the injectivity of the Petri map for vector bundles and the surjectivity of a map related to deformation theory of Poincar\'e sheaves.