Ulrich bundles on a general blow-up of the plane

Abstract

AbstractWe prove that on $$X_n$$ 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 $$m\leqslant 2\sqrt{n}$$ m ⩽ 2 n and such that the line bundle in question is very ample on $$X_n$$ 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 $$X_n$$ X n , for $$n\geqslant 2$$ n ⩾ 2 and any $$r \geqslant 1$$ r ⩾ 1 and we compute the dimensions of their moduli spaces. These computations imply that $$X_n$$ X n is Ulrich wild.