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

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 $m\le 2\sqrt{n}$ 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 $n\ge 2$ and any $r\ge 1$ and we compute the dimensions of their moduli spaces. These computations imply that $X_n$ is {Ulrich wild}.