| The rank two Brill-Noether Theory deals with the linear series of rank 2 on curve C, more precisely with the cycles BN_C(d,k) in the moduli space of semistable rank 2 vector bundles on C of degree d, defined by the condition of admitting at least k sections. Unlike the classical Brill-Noether theory,the dimension of BN_C(d,k) on a general curve, is not governed by the Brill Noether number. Related to the non-emptiness problem of BN_C(d,k), the Mercat's conjecture gives an uniform bound for the number of independent sections on a rank 2 vector bundle. In this talk I will explain the link between rank 2 Brill-Noether theory and the Koszul geometry of the curve C. I will show that, for any odd number g > 10, there exists a smooth curve C of genus g and maximal Clifford index, as well as a stable rank 2 vector bundle on C which contradicts Mercat´s conjecture. In particular, the locus of curves which fail Mercat´s conjecture is not a Brill-Noether locus in the sense of line bundles. This a joint work with G. Farkas. |