Ulrich bundles on a general blow--up of the plane

TL;DR

This paper constructs and analyzes Ulrich bundles on the blow-up of the plane at general points, showing their abundance and stability, and establishing the wildness of the surface in terms of Ulrich bundles.

Contribution

It demonstrates the existence of infinitely many Ulrich line bundles and stable vector bundles on the blow-up of the plane, and computes their moduli spaces, revealing the surface's Ulrich wildness.

Findings

Existence of infinitely many Ulrich line bundles on $X_n$.

Construction of slope-stable rank-$r$ Ulrich vector bundles for all $r \\geq 1$.

Proof that $X_n$ is Ulrich wild.

Abstract

We prove that on $X_n$, the plane blown--up at $n$ general points, there are Ulrich line bundles with respect to a line bundle corresponding to curves of degree $m$ passing simply through the $n$ blown--up points, with $m\leq 2\sqrt{n}$ and such that the line bundle in question is very ample on $X_n$. We prove that the number of these Ulrich line bundles tends to infinity with $n$. We also prove the existence of slope--stable rank--$r$ Ulrich vector bundles on $X_n$, for $n\geq 2$ and any $r \geq 1$ and we compute the dimensions of their moduli spaces. These computations imply that $X_n$ is {Ulrich wild}.