Geodesic
Singularities of the theta divisor of the intermediate Jacobian of a~double cover of~$P^3$ of index two
Izvestiya. Mathematics , Tome 21 (1983) no. 2, pp. 355-373.

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In this paper a theorem is proved on the singularities of the Poincaré theta divisor $\Theta$ of the intermediate Jacobian of a body $X$, a double cover of $P^3$ of index two: the codimension of $\Theta$ in $J_3(X)$ is two. Hence a) $X$ is not rational, b) $(J_3(X),\Theta)$ is not a Prym variety, and, as a consequence, c) $X$ has no structure of a bundle of conics. Bibliography: 13 titles. 
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A. S. Tikhomirov. Singularities of the theta divisor of the intermediate Jacobian of a~double cover of~$P^3$ of index two. Izvestiya. Mathematics , Tome 21 (1983) no. 2, pp. 355-373. http://geodesic.mathdoc.fr/item/IM2_1983_21_2_a7/

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