A. Alzati et Gp. Pirola, RATIONAL ORBITS ON 3-SYMMETRICAL PRODUCTS OF ABELIAN-VARIETIES, Transactions of the American Mathematical Society, 337(2), 1993, pp. 965-980
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.