Pluecker forms and the theta map

Let SU X (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 θ r: SU X (r, 0) → P N is the rational map defined by the ample generator of Pic SU X (r, 0). The main result of the paper is that θ r is generically injective if g ≫ r and X is general. This partially answers the following conjecture proposed by Beauville: θ r is generically injective if X is not hyperelliptic. The proof relies on the study of the injectivity of the determinant map d E: ∧ 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