Ulrich bundles on a general blow-up of the plane

Abstract

We prove that on X_n, the plane blown-up at n very 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 mleqslant 2sqrt{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 ngeqslant 2 and any r geqslant 1 and we compute the dimensions of their moduli spaces. These computations imply that X_n is Ulrich wild.