Plucker forms and the theta map

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 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(detE), 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 Pluecker form. The method applies to other moduli spaces of vector bundles on a projective variety X