On the connectedness of some degeneracy loci and of Ulrich subvarieties

TL;DR

The paper analyzes when degeneracy loci $D_{r-k}(\varphi)$ associated to injective maps into globally generated bundles are connected, focusing on small codimensions ($k\le3$) and the Ulrich bundle setting. It relates connectedness and nonemptiness to Chern class data, base loci, and positivity notions, and provides detailed results for Ulrich subvarieties, including classifications on surfaces and criteria on threefolds. A key contribution is a general framework showing that all suitable degeneracy loci in a given situation share the same number of connected components in families, with precise theorems (e.g., Theorem connesse) linking component counts to $c_k(\mathcal{E})$ and $c_{k+1}(\mathcal{E})$. The work combines base-loci theory, Eagon–Northcott resolutions, and Serre-type constructions to produce concrete connectedness statements and sharp examples across dimensions, offering both general principles and dimension-specific classifications for Ulrich subvarieties.

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.

Theorems & Definitions (82)

• Theorem 1
• Theorem 2
• Corollary 1
• Theorem 3
• Lemma 3.1
• proof
• Definition 3.2
• Definition 3.3
• Remark 3.4
• Lemma 3.5