Special Ulrich bundles on regular surfaces with non-negative Kodaira dimension

TL;DR

This paper proves the existence of special Ulrich bundles of rank 2 on certain regular surfaces with specific line bundle conditions, extending results inspired by recent work on K3 surfaces and exploring various surface classes.

Contribution

It introduces new conditions under which regular surfaces support special Ulrich bundles, expanding the understanding of their existence beyond K3 surfaces.

Findings

Existence of special Ulrich bundles on regular surfaces under technical conditions.

Applications to embeddings of regular surfaces and elliptic surfaces.

Analysis of the families and low-degree cases of Ulrich bundles.

Abstract

Let $S$ be a regular surface endowed with a very ample line bundle $\mathcal O_S(h_S)$. Taking inspiration from a very recent result by D. Faenzi on $K3$ surfaces, we prove that if $\mathcal O_S(h_S)$ satisfies a short list of technical conditions, then such a polarized surface supports special Ulrich bundles of rank $2$. As applications, we deal with general embeddings of regular surfaces, pluricanonically embedded regular surfaces and some properly elliptic surfaces of low degree in $\mathbb P^N$. Finally, we also discuss about the size of the families of Ulrich bundles on $S$ and we inspect the existence of special Ulrich bundles on surfaces of low degree.