Rational orbits on 3-symmetric products of Abelian varieties

Let A be an n-dimensional Abelian variety, n ≥ 2; let CH0(A) be the group of zero-cycles of A, modulo rational equivalence; by regarding an effective, degree k, zero-cycle, as a point on Sk (A) (the k-symmetric product of A), and by considering the associated rational equivalence class, we get a map γ: Sk(A) →CH0(A), whose fibres are called γ-orbits. For any n ≥ 2, in this paper we determine the maximal dimension of the γ-orbits when k = 2 or 3 (it is, respectively, 1 and 2), and the maximal dimension of families of γ-orbits; moreover, for generic A, we get some refinements and in particular we show that if dim(A) ≥ 4, S3(A) does not contain any γ-orbit; note that it implies that a generic Abelian four-fold does not contain any trigonal curve. We also show that our bounds are sharp by some examples. The used technique is the following: we have considered some special families of Abelian varieties: At = Et×B (Et is an elliptic curve with varying moduli) and we have constructed suitable projections between Sk(At) and Sk(B) which preserve the dimensions of the families of γ-orbits; then we have done induction on n. For n = 2 the proof is based upon the papers of Mumford and Roitman on this topic. © 1993 American Mathematical Society.