On the Chow group and the intermediate Jacobian of a conic bundle
The author studies conic bundles X over a nonsingular surface S. Let \(A^ 2(X)\) denote the group of codimension 2 cycles which are algebraically equivalent to zero, and let J(X) be the intermediate Jacobian. The main result is that \(A^ 2(X)\) is naturally isogenous to \(A^ 2(S)\oplus A^ 1(S)\oplus P_ X\), where \(P_ X\) is the (generalized) Prym variety associated to X [this extends a result of \textit{A. Beauville}, Ann. Sci. Éc. Norm. Supér., IV. Sér. 10, 309-391 (1977; Zbl 0368.14018) where it is assumed that \(S={\mathbb{P}}^ 2]\). It is also shown that J(X) is isogenous to \(Alb(S)\oplus Pic^ 0(S)\oplus P_ X\).
conic bundles over a nonsingular surface
scientific article; zbMATH DE number 3946314
scientific article