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 
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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Voir la notice de l'article provenant de la source Math-Net.Ru

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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[8] Clemens C. H., Double solids, preprint, 1979 | MR

[9] Clemens C. H., Griffiths P. A., “The intermediate Jacobian of the cubic threefold”, Ann. of Math. (2), 95:2 (1972), 281–356 | DOI | MR | Zbl

[10] Griffiths P. A., Harris J., Principles of algebraic geometry, N. Y., 1977 | MR

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[13] Welters G. E., Abel–Jacobi isogenics for certain types of Fano threefolds, thesis, Amsterdam, 1981 | MR | Zbl