Ulrich bundles on intersections of two 4-dimensional quadrics

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.