Albanese variety

Abstract: The method of Picard and Poincare in studying an algebraic surface V consists in applying the theory of curves to the curves of an auxililary linear pencil {Cj on V. The same method was employed later by Lefschetz in his theory of algebraic surfaces (cf. [19], Chapters 6-7). If we examine closely their method, we can see that they considered implicitly a certain variety cJ which the author proposes to call the Teron variety of V associated with {CJ. In fact Neron was the first who considered g explicitly in his algebraic proof of the theorem of the base [12]. The variety #J is the graph of the correspondence u ->Ju between u and the Jacobian variety JX of Cu. Here, we restrict our attention to such linear pencils whose members are all irreducible and whose general members are nonsingular. If V does not carry any multiple curve, we can always find such a linear pencil. Also, if V is nonsingular, we can assume that singular members of the pencil are curves with ordinary double points. Now, the main part of the paper is devoted to determining the " degenerate fibres " of the fibre system {u X J,} on # at those finite number of values of u where Cu become singular. We note that in Neron's case such degenerate fibres are not considered explicitly. The same thing can be said about Chow's investigations on Abelian varieties over function fields [4]. However, in some problems in algebraic geometry it becomes necessary to consider those degenerate fibres and also the behavior of fibres along the degenerate fibres. We shall show that the degenerate fibres are certain completions of the generalized Jacobian varieties of the singular curves in the sense of Rosenlicht [15]. The singular locus of #J is contained in the union of singular loci of degenerate fibres. Also we can define in a natural way a birational map p from V into g., and we can show that p gives isomorphisms of the Albanese varieties and of the spaces of linear differential forms of the first kind of V and l. This result has already been applied to show that the dimension of the Albanese variety of an arbitrary

Abstract: In this paper we continue our study of rational equivalence of zero-cycles on algebraic varieties. In particular, we study the case where the dimension of the group of classes of zero-cycles of degree zero modulo rational equivalence is finite, and prove that it coincides with the Albanese variety in this case. Bibliography: 3 items.

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