The Fano surface of the Veronese double cone

This article studies the Fano surface $\mathscr F$ of lines on the Veronese double cone $X$ branched in its intersection with a cubic in $P^6$; it is the last variety in the series of Fano 3-folds of index two. The irregularity of the surface $\mathscr F$ is computed, its Abel–Jacobi mapping $\Phi$ into the intermediate Jacobian of the body $X$ is constructed, the Gauss mapping for $\Phi(\mathscr F)$ is studied, and a theorem on uniquely recovering $X$ from $\Phi(\mathscr F)$ is proved.
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