Ulrich bundles on intersections of two 4-dimensional quadrics

TL;DR

This paper proves that smooth intersections of two 4-dimensional quadrics in projective 5-space always admit Ulrich bundles of any rank at least 2, using geometric and derived category methods.

Contribution

It introduces two novel methods to establish the existence of Ulrich bundles on these intersections and describes their moduli space.

Findings

Existence of Ulrich bundles of all ranks ≥ 2 on the intersection.

Construction of Ulrich bundles via ACM curves and Serre correspondence.

Analysis of the moduli space of stable Ulrich bundles.

Abstract

In this paper, we investigate the existence of Ulrich bundles on a smooth complete intersection of two $4$-dimensional quadrics in $\mathbb P^5$ by two completely different methods. First, we find good ACM curves and use Serre correspondence in order to construct Ulrich bundles, which is analogous to the construction on a cubic threefold by Casanellas-Hartshorne-Geiss-Schreyer. Next, we use Bondal-Orlov's semiorthogonal decomposition of the derived category of coherent sheaves to analyze Ulrich bundles. Using these methods, we prove that any smooth intersection of two 4-dimensional quadrics in $\mathbb P^5$ carries an Ulrich bundle of rank $r$ for every $r \ge 2$. Moreover, we provide a description of the moduli space of stable Ulrich bundles.