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We study the second fundamental form of the Siegel metric in $\mathcal A_5$restricted to the locus of intermediate Jacobians of cubic threefolds. We provethat the image of this second fundamental form, which is known to benon-trivial, is contained in the kernel of a suitable multiplication map. Someingredients are: the conic bundle structure of cubic threefolds, Prym theory,Gaussian maps and Jacobian ideals.

The second fundamental form of the moduli space of cubic threefolds in $\mathcal A_5$