Existence of Ulrich Bundle on general Surfaces

Abstract

Let $X$ be a smooth projective algebraic surface of Picard rank one with very ample canonical bundle $K_X$. We further assume that $q-1\le \chi ({\mathcal{O}}_X$. In this article, we will study the existence of the Ulrich bundle and its stability property of it with respect to $K_X$.

Revised version. Lemma 4.1 and Proposition 4.3 has been rewritten