Abstract. In this paper, we compute the genus of the variety of linear series of rank r and degree d on a general curve of genus g, with ramification at least α and β at two given points, when that variety is 1-dimensional. Our proof uses degenerations and limit linear series along with an analysis of random staircase paths in Young tableaux, and produces an explicit scheme-theoretic description of the limit linear series of fixed rank and degree on a generic chain of elliptic curves when that scheme is itself a curve.

Abstract. Denote by B k 2,K the locus of vector bundles of rank two that have canonical determinant and at least k sections. We show that for a generic curve of genus g, B k 2,K is non-empty and has a component of the expected dimension if g is sufficiently large. IntroductionDenote by U (r, d) the moduli space of stable vector bundles of rank r and degree d on a fixed curve C of genus g. The Brill-Noether loci B k r,d are defined as the subsets of U (r, d) consisting of vector bundles with k sections (note that we use k rather than k − 1 in the notation). These can be given a locally determinantal scheme structure. As such the expected dimension of B k r,d for a generic curve is given by the Brill-Noether number ρ = r 2 (g − 1) + 1 − k(k − d + r(g − 1)) and the expected singular locus is B k+1 d,r . The case r = 1 is classical and all the expected results hold for the generic curve. For higher rank though, several cases are known where ρ is positive and the locus is empty (cf [BGN]) or ρ is negative and the locus is non-empty (use [Me]) or the singular locus is larger than expected (see [T1]). We shall add to the list of abnormalities by showing that B k 2,2g−2 is sometimes of dimension larger than expected for the generic curve and therefore reducible (see also [BF]).Denote now by U (r, L) the moduli space of stable vector bundles of rank r and determinant L. Define similarly B k r,L as the subset of U (r, L) consisting of vector bundles with k sections . For k > r (and d ≤ 2g − 2), one expects that the geometry of B k r,L will very much depend on the position of L itself as a point of various B j 1,d . The case L = O is not very interesting as a stable vector bundle of degree zero cannot have sections. It is then natural to look at the special case when L = K is the canonical bundle. When r = 2, this locus has a natural structure as a symmetric degeneracy locus (see [Mu]. Using this description, the expected dimension of B k 2,K is given by ρ k K = 3g − 3 − k+1 2 . By its definition, this is the minimum dimension of any component of these loci.One can again ask the question ([Mk1]) of whether the locus is non-empty for ρ k K > 0 and of dimension ρ k K and empty for ρ k K < 0. It was conjectured in [BF] that this would be the case. Some evidence for this conjecture using curves of small genus was provided in [R]. The main purpose of this paper is to partly prove these statements.We want to mention that the study of the loci B k 2,K , apart from its intrinsic interest, seem to contribute a lot to our knowledge of the geometry of the curves. Mukai for example showed in [Mk2] that if C is a curve of genus eleven, then W 7 2,K 1

Abstract. This paper gives an overview of the main results of Brill-Noether Theory for vector bundles on algebraic curves.

Denote by Jf the locus in the moduli space of curves of genus g of those curves which have a theta-characteristic of (projective) dimension at least r. We give an upper bound for the dimension of Jtr and we determine this dimension completely for r ^ 4. For r < 4, we prove also that a generic point in every component of J(r has a single theta-characteristic of this dimension. 0. Introduction. Let Jt g be the moduli space of smooth, complete curves of genus g over the complex field C. We investigate the subloci Jlrg of J(' These are defined as the loci of curves having a theta-characteristic (i.e. a line bundle L such that L ® L = Kc) of (projective) dimension at least r and of the same parity as r. By Clifford's Theorem, it is clear that J(r is empty if r > \(g-1). On the other hand, one can easily see that hyperelliptic curves have theta-characteristics of all dimensions r with 0 < r < \~(g-1). In [H], Harris proves (0.1) Theorem (cf. [H, Theorem (1.10)]). Let X-> S be a family of curves and La line bundle on X such that the restriction L(s) to every fiber X(s) satisfies L2(s)-Kx(s)-Then the subset of S {s G S\k°(X(s),L(s))> r+ landh°(X(s),L(s)

We prove the Bertram-Feinberg-Mukai conjecture for a generic curve C of genus g and a semistable vector bundle E of rank two and determinant K on C, namely we prove the injectivity of the Petri-canonical map S 2 (H 0 (E)) → H 0 (S 2 (E)).