Ulrich bundles on non-special surfaces with $p_g=0$ and $q=1$

TL;DR

This paper demonstrates the existence of large families of non-isomorphic, indecomposable Ulrich bundles on certain non-special surfaces with specific invariants, and constructs stable rank 2 Ulrich bundles under genus conditions.

Contribution

It establishes the existence of arbitrarily large families of Ulrich bundles on surfaces with $p_g=0$, $q=1$, and provides conditions for stable rank 2 Ulrich bundles.

Findings

Supports large families of Ulrich bundles of arbitrary dimension.

Constructs stable rank 2 Ulrich bundles when the genus condition is met.

Shows existence of Ulrich bundles on non-special surfaces with specific invariants.

Abstract

Let $S$ be a surface with $p_g(S)=0$, $q(S)=1$ and endowed with a very ample line bundle $\mathcal O_S(h)$ such that $h^1\big(S,\mathcal O_S(h)\big)=0$. We show that such an $S$ supports families of dimension $p$ of pairwise non-isomorphic, indecomposable, Ulrich bundles for arbitrary large $p$. Moreover, we show that $S$ supports stable Ulrich bundles of rank $2$ if the genus of the general element in $\vert h\vert$ is at least $2$.