Parametrization of Sing Theta for a Fano 3-fold of genus 7 by moduli of vector bundles

摘要

According to Mukai, any prime Fano threefold X of genus 7 is a linear section of the spinor tenfold in the projectivized half-spinor space of Spin(10). The orthogonal linear section of the spinor tenfold is a canonical genus-7 curve Gamma, and the intermediate Jacobian J(X) is isomorphic to the Jacobian of Gamma. It is proven that, for a generic X, the Abel-Jacobi map of the family of elliptic sextics on X factors through the moduli space of rank-2 vector bundles with c(1) = -K-X and deg c(2) = 6 and that the latter is birational to the singular locus of the theta divisor of J(X).