Rational Orbits on Three-Symmetric Products of Abelian Varieties

Abstract

Let $A$ be an $n$-dimensional Abelian variety, $n \geq 2$; let ${\text {CH}_0}(A)$ be the group of zero-cycles of $A$, modulo rational equivalence; 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 associated rational equivalence class, we get a map $\gamma :{S^k}(A) \to {\text {CH}_0}(A)$, whose fibres are called $\gamma$-orbits. For any $n \geq 2$, in this paper we determine the maximal dimension of the $\gamma$-orbits when $k = 2$ or $3$ (it is, respectively, $1$ and $2$), and the maximal dimension of families of $\gamma$-orbits; moreover, for generic $A$, we get some refinements and in particular we show that if $\dim (A) \geq 4$, ${S^3}(A)$ does not contain any $\gamma$-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: ${A_t} = {E_t} \times B$ (${E_t}$ is an elliptic curve with varying moduli) and we have constructed suitable projections between ${S^k}({A_t})$ and ${S^k}(B)$ which preserve the dimensions of the families 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 topic.