principle that this *second order* BN analysis is connected in a precise forms (formulated in Section 5) to the *rank two* BN theory of the curve and the various predictions on the two sides of this correspondence are remarkably compatible!

As a word of caution however, proving $(MRC)_{g, d}^r$ when $\rho(g, r, d)\geq 1$ (let alone Conjecture \[minres\]), seems considerably more difficult that proving the original Harris Conjecture. When $\rho(g, r, d)=0$ the two statements are equivalent, see [@F3] Theorem 1.5.

We discuss the structure of the paper. In Section 2 we review the Maximal Rank Conjecture and some of its consequences. Section 3 contains the most important results of the paper. Using $K3$ surfaces, we disprove Mercat’s Conjecture (\[mercat2\]) (Theorems \[K3surf1\] and \[K3surf2\]) and set-up a link between rank $2$ vector bundles and MRC. We also prove Mercat’s Conjecture (\[mercat1\]) for general curves of bounded genus. In Section 4 we complete the proof of Theorem \[existence\] concerning non-emptiness of Brill-Noether loci, while Section 5 is devoted entirely to Koszul cohomology and its applications to rank two Brill-Noether theory. We end the introduction by thanking Herbert Lange and Peter Newstead for pertinent comments made on an earlier version of this 