  We describe applications of Koszul cohomology to the Brill-Noether theory of
rank 2 vector bundles. Among other things, we show that in every genus g>10,
there exist curves invalidating Mercat's Conjecture for rank 2 bundles. On the
other hand, we prove that Mercat's Conjecture holds for general curves of
bounded genus, and its failure locus is a Koszul divisor in the moduli space of
curves. We also formulate a conjecture concerning the minimality of Betti
diagrams of suitably general curves, and point out its consequences to rank 2
Brill-Noether theory.
