On the connectedness of some degeneracy loci and of Ulrich subvarieties

TL;DR

This paper investigates the connectedness properties of degeneracy loci and Ulrich subvarieties on smooth varieties, providing criteria based on Chern classes and the concept of V-bigness, with specific results for surfaces.

Contribution

It offers new characterizations of connectedness for degeneracy loci using Chern class vanishing and establishes connectedness conditions for Ulrich bundles, especially on surfaces.

Findings

Connectedness characterized by Chern class vanishing for certain degeneracy loci.

Connectedness of degeneracy loci proven under V-bigness condition.

Detailed results for Ulrich bundles on surfaces.

Abstract

We study connectedness of degeneracy loci $D_{r-k}(\varphi)$ of morphisms $\varphi : {\mathcal O}_X^{\oplus (r+1-k)} \to \mathcal E$, where $\mathcal E$ is a rank $r$ globally generated bundle on a smooth $n$-dimensional variety $X$ and $k \le 3$. For $k \le 2$ we give a characterization of connectedness in terms of vanishing of Chern classes. Moreover we prove that they are connected, for $k \le \min\{2, r-1,n-1\}$, if $\mathcal E$ is V-big. In the case of Ulrich bundles more precise results are given, both in general and in the case of surfaces.