Stable Ulrich bundles

The existence of stable ACM vector bundles of high rank on algebraic varieties is a challenging problem. In this paper, we study stable Ulrich bundles (that is, stable ACM bundles whose corresponding module has the maximum number of generators) on nonsingular cubic surfaces $X \subset \mathbb{P}^3.$ We give necessary and sufficient conditions on the first Chern class $D$ for the existence of stable Ulrich bundles on $X$ of rank $r$ and $c_1=D$. When such bundles exist, we prove that that the corresponding moduli space of stable bundles is smooth and irreducible of dimension $D^2-2r^2+1$ and consists entirely of stable Ulrich bundles (see Theorem 1.1). As a consequence, we are also able to prove the existence of stable Ulrich bundles of any rank on nonsingular cubic threefolds in $\mathbb{P}^4$.