Linear sections of Grassmannians and resonance of vector bundles

Abstract

This work revolves around the question of whether a given resonance variety is associated with a vector bundle. We show the existence of a family of natural morphisms on a stratification of the resonance variety to a suitable family of a Quot scheme and provide some applications in the curve case. The existence of this family of morphisms represents an obstruction to affirmatively answering the main question. In addition, we study the resonance of restricted universal rank-two quotient bundles over transversal linear sections of the Grassmann varieties $\operatorname{Gr}_2(\mathbb{C}^n)$, with a special attention to low-dimensional Grassmannians. These bundles are among the most natural to consider in this context. The analysis for $\operatorname{Gr}_2(\mathbb{C}^6)$ shows that any resonance variety in $\mathbb{P}^5$ consisting of fourteen disjoint lines is the resonance of some bundle which appeared in the work of Mukai.