Suplico a vuesa merced, señor don Quijote, que mire bien y especule con cien ojos lo que hay allá dentro: quizá habrá cosas que las ponga yo en el libro de mis Transformaciones (El ingenioso hidalgo don Quijote de la Mancha, Book 2, Chapter XXII) I beg you, don Quixote sir: look carefully, inspect with a hundred eyes what you see down there. Who knows, maybe you will find something that I can put in my book on Transformations.

For smooth projective curves of genus g ≥ 4, the Clifford index is an important invariant which provides a bound for the dimension of the space of sections of a line bundle. This is the first step in distinguishing curves of the same genus. In this paper we generalise this to introduce Clifford indices for semistable vector bundles on curves. We study these invariants, giving some basic properties and carrying out some computations for small ranks and for general and some special curves. For curves whose classical Clifford index is two, we compute all values of our new Clifford indices.

We compute the E-polynomials of the moduli spaces of representations of the fundamental group of a complex surface into SL(2, C), for the case of small genus g, and allowing the holonomy around a fixed point to be any matrix of SL(2, C), that is Id , − Id , diagonalisable, or of either of the two Jordan types.For this, we introduce a new geometric technique, based on stratifying the space of representations, and on the analysis of the behaviour of the E-polynomial under fibrations.

IntroductionIn this paper we shall construct moduli spaces of pairs consisting of a torsion-free sheaf E over an algebraic curve X (possibly singular, but always of pure dimension) and a vector subspace Λ of its space of sections H 0 (E). Such pairs are the common generalisation of the classical notion of a linear system, for which E is (the sheaf of sections of) a line bundle, and of Bradlow pairs [3], for which one may take Λ to be 1-dimensional -although strictly speaking Λ should be replaced by a single non-zero section φ. These more general pairs have also been studied by Bertram [2], Raghavendra and Vishwanath [13] (under the name of "pairs") and by Le Potier [8] (under the name of "coherent systems"), who works over varieties (or schemes) of arbitrary dimension. In this paper, we shall use the term "Brill-Noether pair", to emphasise the role that we hope they will play in higher rank Brill-Noether theory.There are three numerical invariants which make up the 'type' of a Brill-Noether pair: the rank r and degree d of the sheaf E, which can be defined even on a singular curve provided it is polarised (see Definition 2.1), and the dimension l of Λ. In the case r = 1 (and X smooth) there is a well-known parameter space for all linear systems -usually denoted G r d (e.g.[1] Chap IV), where r + 1 is the dimension of the space of sections! In higher rank, one must introduce a notion of semistability, which depends on a single parameter α (see Definition 2.3.2). The same definition is used in [13], while in [3] an equivalent definition is used involving τ = d+αl r . The result we shall prove is the following. Theorem 1. Let X be a polarised algebraic curve and α any positive rational number. For each (r, d, l) there exists a projective scheme G α (r, d, l) which is a coarse moduli space for families of α-semistable Brill-Noether pairs of type (r, d, l). The (closed) points of G α (r, d, l) are in one-one correspondence with S-equivalence classes of pairs.

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