RATIONAL ORBITS ON 3-SYMMETRICAL PRODUCTS OF ABELIAN-VARIETIES

Let A be an n-dimensional Abelian variety, n greater-than-or-equal-to 2 ; let CH0(A) be the group of zero-cycles of A , modulo rational equi valence; by regarding an effective, degree k , zero-cycle, as a point on S(k)(A) (the k-symmetric product of A) , and by considering the ass ociated rational equivalence class, we get a map gamma: S(k)(A) --> CH 0(A) , whose fibres are called gamma-orbits. For any n greater-than-or -equal-to 2, in this paper we determine the maximal dimension of the g amma-orbits when k = 2 or 3 (it is, respectively, 1 and 2) , and the m aximal dimension of families of gamma-orbits; moreover, for generic A, we get some refinements and in particular we show that if dim(A) grea ter-than-or-equal-to 4, S3(A) does not contain any gamma-orbit; note t hat it implies that a generic Abelian four-fold does not contain any t rigonal 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: A(t) = E(t) x B (E(t) is an elliptic c urve with varying moduli) and we have constructed suitable projections between S(k)(A(t)) and S(k)(B) which preserve the dimensions of the f amilies of gamma-orbits; then we have done induction on n . For n = 2 the proof is based upon the papers of Mumford and Roitman on this topi c.