Resumen de Plücker forms and the theta map

Sonia Brivio, Alessandro Verra

• Let ${\rm SU}_X(r,0)$ 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 $\theta_r: {\rm SU}_X(r,0) \rightarrow \Bbb{P}^N$ is the rational map defined by the ample generator of ${\rm Pic}{\rm SU}_X(r,0)$. The main result of the paper is that $\theta_r$ is generically injective if $g \gg r$ and $X$ is general. This partially answers the following conjecture proposed by Beauville: $\theta_r$ is generically injective if $X$ is not hyperelliptic. The proof relies on the study of the injectivity of the determinant map $d_E: {\wedge}^r H^0(E) \rightarrow H^0(\det E)$, 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 Pl\"ucker form. The method applies to other moduli spaces of vector bundles on a projective variety $X$.