Vector bundles on bielliptic surfaces: Ulrich bundles and degree of irrationality

This paper deals with two problems about vector bundles on bielliptic surfaces. The first is to give a classification of Ulrich bundles on such surfaces $S$, which depends on the topological type of $S$. In doing so, we study the weak Brill-Noether property for moduli spaces of sheaves with isotropic Mukai vector. Adapting an idea of Moretti, we also interpret the problem of determining the degree of irrationality of bielliptic surfaces in terms of the existence of certain stable vector bundles of rank 2, completing the work of Yoshihara.