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Abstract. Let SU (r, 1) be the moduli space of stable vector bundles, on a smooth curve C of genus g ≥ 2, with rank r ≥ 3 and determinant OC (p), p ∈ C; let L be the generalized theta divisor on SU (r, 1). In this paper we prove that the map φL, defined by L, is a morphism and has degree 1. §0. Introduction In this paper, we will assume r ≥ 3 and we will consider SU (r, 1), where L = O C (p) and p is a given point of C, our first result is the following:Theorem 0.0.1. For any curve C of genus g ≥ 2: deg(φ L ) = 1, the linear system |L| on SU (r, 1) is base points free, i.e. the map φ L is a morphism.As a second result we prove the following:The paper is organized as follows. The first section is devoted to proving theorem (0.0.1). In section 2, we study rank r-bundles with r + 1

Let E be a stable vector bundle of rank r and slope 2g − 1 on a smooth irreducible complex projective curve C of genus g ≥ 3. In this paper we show a relation between theta divisor E and the geometry of the tautological model P E of E. In particular, we prove that for r > g − 1, if C is a Petri curve and E is general in its moduli space then E defines an irreducible component of the variety parametrizing (g − 2)-linear spaces which are g-secant to the tautological model P E. Conversely, for a stable, (g − 2)-very ample vector bundle E, the existence of an irreducible non special component of dimension g − 1 of the above variety implies that E admits theta divisor.

Let C be an algebraic smooth complex curve of genus g > 1. The object of this paper is the study of the birational structure of certain moduli spaces of vector bundles and of coherent systems on C and the comparison of different type of notions of stability arising in moduli theory. Notably we show that in certain cases these moduli spaces are birationally equivalent to fibrations over simple projective varieties, whose fibers are GIT quotients (P r−1 ) rg //PGL(r), where r is the rank of the considered vector bundles. This allows us to compare different definitions of (semi-)stability (slope stability, α-stability, GIT stability) for vector bundles, coherent systems and point sets, and derive relations between them. In certain cases of vector bundles of low rank when C has small genus, our construction produces families of classical modular varieties contained in the Coble hypersurfaces. a double covering of P 8 branched along a hypersurface of degree six C 6 called the Coble-Dolgachev sextic [25]. Our result is the following. Theorem 1.3. The Coble-Dolgachev sextic C 6 is birational to a fibration over P 4 whose fibers are Igusa quartics. More precisely, C 6 contains a four-dimensional family of Igusa quartics parametrized by an open subset of P 4 .We recall that an Igusa quartic is a modular quartic hypersurface in P 4 that is related to some classical GIT quotients (see e.g. [15]) and moduli spaces. Its dual variety is a cubic three-fold called the Segre cubic, that is isomorphic to the GIT quotient (P 1 ) 6 //PGL(2).If r = 2 and g = 3, then SU C (2) is embedded by θ in P 7 as a remarkable quartic hypersurface C 4 called the Coble quartic [27]. Our methods also allow us to give a quick proof of the following fact.

Let SUX (r, 0) be the moduli space of semistable vector bundles of rank r and trivial determinant over a smooth, irreducible, complex projective curve X. The theta map θr : SUX (r, 0) → P N is the rational map defined by the ample generator of Pic SUX (r, 0). The main result of the paper is that θr is generically injective if g >> r and X is general. This partially answers the following conjecture proposed by Beauville: θr is generically injective if X is not hyperelliptic. The proof relies on the study of the injectivity of the determinant map dE : ∧ r H 0 (E) → H 0 (det E), for a vector bundle E on X, and on the reconstruction of the Grassmannian G(r, rm) from a natural multilinear form associated to it, defined in the paper as the Plücker form. The method applies to other moduli spaces of vector bundles on a projective variety X.