GEOGRAPHY OF BRILL-NOETHER LOCI FOR SMALL SLOPES

Let X be a nonsingular projective curve of genus g greater than or equ al to 2 over an algebraically closed field of characteristic zero. Let M(n, d) denote the moduli space of stable bundles of rank n and degre e d on X and let W-n,d(k-1) denote the Brill-Noether loci in M(n, d). We prove that, if 0 less than or equal to d less than or equal to n an d W-n,d(k-1) is nonempty, then it is irreducible of the expected dimen sion and smooth outside W-n,d(k). We prove further that, in this range , W-n,d(k-1) is nonempty if and only if d > 0, n less than or equal to d + (n - k)g, and (n, d, k) not equal (n, n, n). We also prove irredu cibility and nonemptiness for the semistable Brill-Noether loci.