Ulrich sheaves and higher-rank Brill-Noether theory

We show that the existence of an Ulrich sheaf on a projective variety X⊆PN is equivalent to the solution of a (possibly) higher-rank Brill-Noether problem for a curve on X that is rarely general in moduli. In addition, we exhibit a large family of curves for which this Brill-Noether problem admits a solution, and we show that existence of an Ulrich sheaf for a finite morphism of smooth projective varieties of any dimension implies sharp numerical constraints involving the degree of the map and the ramification divisor.