The intermediate Jacobian of a cubic threefold with one ordinary double point; an algebraic-geometric approach. II.Nederl. Akad.Wetensch. Proc. Ser. A 81=Indag.Math. 40 (1978), no. 1, 56–71.

from MR0498605 (58 #16694b) :The authors study a cubic threefold X in projective four-space P4 over an algebraically closed field of characteristic 6= 2. They assume X to be smooth except for a single ordinary double point x0. Their aim is to find and prove, by algebraic and geometric methods, results analogous to those proved by C. H. Clemens and P. A. Griffiths especially those relating the group of 1-cycles algebraically equivalent to zero on X (the intermediate Jacobian in transcendental terms) to a certain Prym variety defined by D. Mumford [ Much of this treatment involves modifications of earlier work by the second author which extended to nonzero characteristic the work of Clemens and Griffiths in the smooth case. Let S be the Fano surface of lines on X. Fix a line l0 on X not passing through x0 and such that no plane in P4 containing l0 intersects X in a double line. Now the planes containing l0 are parametrized by a plane N. Let be the subset of N whose corresponding planes intersect X in l0 and two other lines. This is a quintic curve, smooth except for an ordinary double point whose corresponding plane is the one through x0. Let 0 be the subset of S of points whose lines meet l0 properly, and let q:0 ! be the natural morphism. Then 0 is a complete irreducible curve with two ordinary double points, and q is an ´etale covering. Let W0 be the cone of lines in X through x0. For a fixed hyperplane P3 of P4, let C = W0 \ P3, so thatW0 is the cone over C with vertex x0. The curve C is a sextic, isomorphic to the curve D0 of S corresponding to the lines on X through x0. Define the map : Sym2(C)!S by sending two points of C to the point of S corresponding to the residual intersection of the plane generated by the points and x0 with X. Then is an isomorphism off D0 and −1(D0) = D1 [D2, a union of two disjoint curves. Let Z denote the blow-up of X with center x0. Then Z is also the blow-up of P3 along C, and A2(Z), the group of 1-cycles on Z algebraically equivalent to zero, is isomorphic to the (rational points of the) Jacobian J(C) of C. Then they show that J(C) is isomorphic to the Prym variety of 0 n/n where n denotes normalization. Combining these two results gives the isomorphism A2(Z) = Prym(0 n/n), a result analogous to the main result of the second author In the final section, the authors define Prym(0/) in the generalized Jacobian of0 in a manner analogous to the definition of the Prym variety of a double covering of smooth curves. On the other hand, Sym2(C)−(D1 [D2) has a generalized Albanese variety G [J. P. Serre, S´eminaire C. Chevalley, 3i`eme ann´ee: 1958/59. Vari´et´es de Picard, Exp. No. 11, ´ ]. The authors call G the Albanese variety of S. They then show that G is isomorphic to Prym(0/), a result analogous to the main result of the second author