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Since BLS (C) is smooth, π is an isomorphism. 11) H 3 (S , Z) ⊕ H 1 (C, Z). 12) The factor H 3 (S , Z) ⊂ H 3 (P, Z) maps to zero under the map p2∗ : H 3 (P, Z) → H 3 (X, Z). Proof H 3 (S ) sits in H 3 (P) via p∗1 = j∗ q∗1 . Since p2∗ j∗ = k∗ q2∗ , it suffices to show that q2∗ q∗1 H 3 (S ) = 0. But this group sits in H 3 (V, Z) = (0) by Lefschetz theory. 13) The map p2∗ : H 3 (P, Z) → H 3 (X, Z) is surjective. Before giving the proof of this, let me show how it implies the theorem. 12) we see the correspondence r2∗ r1∗ : H 1 (C, Z) → H 3 (X, Z) is surjective, so the corresponding map on jacobians J(C) → J 2 (X) has connected fibres.

Roitman, Γ-equivalence of zero-dimensional cycles (in Russian), Mat. Sb. ), 86 (128) (1971), 557–570. ] [6] A. A. Roitman, Rational equivalence of zero-dimensional cycles (in Russian), Mat. Sb. ), 89 (131) (1972), 569–585, 671. [Translation: Math. ] [7] A. Mattuck, Ruled surfaces and the Albanese mapping, Bull. Amer. Math. , 75 (1969), 776–779. [8] A. Mattuck, On the symmetric product of a rational surface, Proc. Amer. Math. , 21 (1969), 683–688. For a more positive point of view, see [9] S. Bloch, K2 of Artinian Q-algebras with application to algebraic cycles, Comm.

Weil, Courbes Alg´ebriques et Vari´et´es Ab´eliennes, Hermann, Paris (1971). [10] D. Quillen, Higher algebraic K-theory. I, pp. , no. 341, Springer, Berlin (1973). 1 Zero-cycles on surfaces In this first section I want to consider the question of zero-cycles on an algebraic surface from a purely geometric point of view. I will consider a number of explicit examples, and give a heuristic description of a result of Mumford 0 implies A0 (X) is “very large”. In particular A0 (X) is not an [4] that Pg abelian variety in this case.

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