Geometry of certain Brill-Noether locus on a very general sextic surface and Ulrich bundles

Let $X \subset \mathbb P^3$ be a very general sextic surface over complex numbers. In this paper we study certain Brill-Noether problems for moduli of rank $2$ stable bundles on $X$ and its relation with rank $2$ weakly Ulrich and Ulrich bundles. In particular, we show the non-emptiness of certain Brill-Noether loci and using the geometry of the moduli and the notion of the Petri map on higher dimensional varieties, we prove the existence of components of expected dimension. We also give sufficient conditions for the existence of rank $2$ weakly Ulrich bundles $\mathcal E$ on $X$ with $c_1(\mathcal E) =5H$ and $c_2 \geq 91$ and partially address the question of whether these conditions really hold. We then study the possible implication of the existence of an weakly Ulrich bundle in terms of non-emptiness of Brill-Noether loci. Finally, using the existence of rank $2$ Ulrich bundles on $X$ we obtain some more non-empty Brill-Noether loci and investigate the possibility of existence of higher rank simple Ulrich bundles on $X$.