We study coherent systems of type (n, d, n + 1) on a Petri curve X of genus g ≥ 2. We describe the geometry of the moduli space of such coherent systems for large values of the parameter α. We determine the top critical value of α and show that the corresponding "flip" has positive codimension. We investigate also the nonemptiness of the moduli space for smaller values of α, proving in many cases that the condition for non-emptiness is the same as for large α. We give some detailed results for g ≤ 5 and applications to higher rank Brill-Noether theory and the stability of kernels of evaluation maps, thus proving Butler's conjecture in some cases in which it was not previously known.

Abstract. Let X be a non-singular algebraic curve of genus g. We prove that theThese results hold for an arbitrary curve of genus ≥ 2, and allow us to construct a region in the associated "Brill-Noether (µ, λ)-map" of points for which the Brill-Noether loci are non-empty.Even for the generic case, the region so constructed extends beyond that defined by the so-called "Teixidor parallelograms." For hyperelliptic curves, the same methods give more extensive and precise results.

Let C be a smooth irreducible projective curve of genus g and L a line bundle of degree d generated by a linear subspace V of H 0 (L) of dimension n + 1. We prove a conjecture of D. C. Butler on the semistability of the kernel of the evaluation map V ⊗ O C → L and obtain new results on the stability of this kernel. The natural context for this problem is the theory of coherent systems on curves and our techniques involve wall crossing formulae in this theory.

IntroductionLet E be a vector bundle of rank 2 on a smooth projective curve C of genus g ≥ 2 over an algebraically closed field K of arbitrary characteristic.The invariantwhere the maximum is taken over all line subbundles L of E, is just the minimum of the self intersection numbers of all sections of the ruled surface P(E) → C. Note that E is stable (respectively semistable) if and only if s 1 (E) ≥ 1 (respectively ≥ 0). According to a Theorem of C. Segre s 1 (E) ≤ g. Moreover, the function s 1 is lower semicontinuous. Thus s 1 gives a stratification of the moduli space M(2, d) of stable vector bundles of rank 2 and degree d on C, into locally closed subsets M(2, d, s) according to the value s of s 1 (E). 1 Let M 1 (E) denote the set of line subbundles of E of maximal degree. The set M 1 (E) can be considered as an algebraic scheme in a natural way. Maruyama proved in [10] (see also [8] for different proofs) the following statement: It is shown inIt is the aim of the present paper, to generalize statements (A) and (B) to vector bundles of arbitrary rank r ≥ 2.Let now E be a vector bundle of rank r ≥ 2 over C. For any integer with 1 ≤ k ≤ r − 1 we definewhere the maximum is taken over all subbundles F of rank k of E. There is also an interpretation of the invariant s k (E) in terms of self intersection numbers in the ruled variety P(E) (see [9]). According to a theorem of Hirschowitz (see [4])Moreover, the function s k is lower semicontinuous. Thus s k gives a stratification of the moduli space M(r, d) of stable vector bundles of rank r and degree on d on C into locally closed subsets M(r, d, k, s) according to the value of s and k. There is a component M 0 (r, d, k, s) of M(r, d, k, s) distinguish by the fact that a general E ∈ M 0 (r, d, k, s) admits a stable subbundle F such that E/F is also stable.In this paper we prove (see Theorem 4.2): non-empty, and its componentThe bound g ≥ r+1 2works for all k, 1 ≤ k ≤ r − 1 simultaneously. For some special k the result is better. For example, for k = 1 or r − 1 statement (C) is valid for all g ≥ 2 (see Remark 3.3). 2Let M k (E) denote the set of maximal subbundles of rank k of E and M k (E) denote the closure of the set of stable subbundles F of rank k of E of maximal degree such that E/F is also stable in M k (E). Also M k (E) can be considered as an algebraic scheme in a natural way and M k (E) is a union of components of M k (E).Theorem 4.4 below says:again under the hypothesis g ≥ Acknowledgement Part of the work was done during a visit of both authors to CIMAT, Gto. Mexico. They wish to acknowledge the hospitality of CIMAT. We would like to thank P.E. Newstead for his comments on the first version of this paper and Remark 4.5. The first author also want to thank P. E. Newstead and B. Russo for previous discussions on the subject and the University of Liverpool where those discussion were done.

Let X be a smooth algebraic curve of genus g ≥ 2. A stable vector bundle over X of degree d, rank n with at least k sections is called a Brill-Noether bundle of type (n, d, k). By tensoring coherent systems, we prove that most of the known Brill-Noether bundles define coherent systems of type (n, d, k) that are α-stables for all allowable α.