From MR546468 AMS Math Sci Net: To any smooth cubic threefold Y in P4 k one associates in case char(k) 6= 2 the group A2(Y ) of the 1-cycles algebraically equivalent to zero modulo those linearly equivalent to zero. The group A2(Y ) has a natural structure of an abelian variety with a principal polarization divisor . It is well known that Y is not rational if (A2(Y ), ) is not a product of Jacobians of curves. In the present paper the author constructs a family X/S of cubic threefolds with the following properties: (i) S is a smooth, but not necessarily complete curve; (ii) there is a closed point s0 2 S such that X0 := Xs0 has exactly one ordinary double point; (iii) Xt is smooth for all t 2 T := S r {s0}. To this family there is associated a family of (generalized) Prym varieties, A/S, such that (iv) At = A2(Xt) for all t 2 T and (v) A0 is an extension of an abelian variety B by a multiplicative groupGm. Moreover, a relative Cartier divisorD is constructed on A/S with the properties (vi)Dt induces the same polarization on At as the theta divisor t; (vii) D0 = 20 in NS(B) = NS(A0), where 0 is a divisor which induces a principal polarization on the Prym variety B (NS denote the N´eron-Severi group0. Then for the whole construction the following result is shown: If allXt were rational, then, passing to the limit, (A0,D0) should be a polarized generalized Jacobian of a curve C0 with ordinary double points. Moreover, (B,0) is the polarized Jacobian of the normalization of C0. The proof uses a generalization of a theorem ofW. Hoyt [Ann. of Math. (2) 77 (1963), 415–423; MR0150145 (27 #148)] which is also given in the present paper. It says roughly that a limit of polarized products of Jacobians is, in a certain sense, a polarized product of generalized Jacobians. Furthermore, it is shown that (A0,D0) and (B,0) cannot have the properties deduced above, hence most fibres Xt are not rational. The author describes his proof of the irrationality of most cubic threefolds as “cheap” because it makes no use of the deep analysis of theta divisors on intermediate Jacobians or of the singularities of the theta divisor on Prym varieties.
A cheap proof of the irrationality of most cubic threefolds.Boll.Un. Mat. Ital. B (5) 16 (1979), no. 2, 451–465.
COLLINO, Alberto
1979-01-01
Abstract
From MR546468 AMS Math Sci Net: To any smooth cubic threefold Y in P4 k one associates in case char(k) 6= 2 the group A2(Y ) of the 1-cycles algebraically equivalent to zero modulo those linearly equivalent to zero. The group A2(Y ) has a natural structure of an abelian variety with a principal polarization divisor . It is well known that Y is not rational if (A2(Y ), ) is not a product of Jacobians of curves. In the present paper the author constructs a family X/S of cubic threefolds with the following properties: (i) S is a smooth, but not necessarily complete curve; (ii) there is a closed point s0 2 S such that X0 := Xs0 has exactly one ordinary double point; (iii) Xt is smooth for all t 2 T := S r {s0}. To this family there is associated a family of (generalized) Prym varieties, A/S, such that (iv) At = A2(Xt) for all t 2 T and (v) A0 is an extension of an abelian variety B by a multiplicative groupGm. Moreover, a relative Cartier divisorD is constructed on A/S with the properties (vi)Dt induces the same polarization on At as the theta divisor t; (vii) D0 = 20 in NS(B) = NS(A0), where 0 is a divisor which induces a principal polarization on the Prym variety B (NS denote the N´eron-Severi group0. Then for the whole construction the following result is shown: If allXt were rational, then, passing to the limit, (A0,D0) should be a polarized generalized Jacobian of a curve C0 with ordinary double points. Moreover, (B,0) is the polarized Jacobian of the normalization of C0. The proof uses a generalization of a theorem ofW. Hoyt [Ann. of Math. (2) 77 (1963), 415–423; MR0150145 (27 #148)] which is also given in the present paper. It says roughly that a limit of polarized products of Jacobians is, in a certain sense, a polarized product of generalized Jacobians. Furthermore, it is shown that (A0,D0) and (B,0) cannot have the properties deduced above, hence most fibres Xt are not rational. The author describes his proof of the irrationality of most cubic threefolds as “cheap” because it makes no use of the deep analysis of theta divisors on intermediate Jacobians or of the singularities of the theta divisor on Prym varieties.
I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
