Rational orbits on three-symmetric products of abelian varieties

Authors:
A. Alzati and G. P. Pirola

Journal:
Trans. Amer. Math. Soc. **337** (1993), 965-980

MSC:
Primary 14K05; Secondary 14C15, 14H40

DOI:
https://doi.org/10.1090/S0002-9947-1993-1106186-9

MathSciNet review:
1106186

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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.

- Mireille Martin-Deschamps and Renée Lewin-Ménégaux,
*Surfaces de type général dominées par une variété fixe*, Bull. Soc. Math. France**110**(1982), no. 2, 127–146 (French, with English summary). MR**667747** - Phillip Griffiths (ed.),
*Topics in transcendental algebraic geometry*, Annals of Mathematics Studies, vol. 106, Princeton University Press, Princeton, NJ, 1984. MR**756842** - Robin Hartshorne,
*Equivalence relations on algebraic cycles and subvarieties of small codimension*, Algebraic geometry (Proc. Sympos. Pure Math., Vol. 29, Humboldt State Univ., Arcata, Calif., 1974) Amer. Math. Soc., Providence, R.I., 1973, pp. 129–164. MR**0369359** - D. Mumford,
*Rational equivalence of $0$-cycles on surfaces*, J. Math. Kyoto Univ.**9**(1968), 195–204. MR**249428**, DOI https://doi.org/10.1215/kjm/1250523940 - Gian Pietro Pirola,
*Curves on generic Kummer varieties*, Duke Math. J.**59**(1989), no. 3, 701–708. MR**1046744**, DOI https://doi.org/10.1215/S0012-7094-89-05931-0
A. A. Roitman,

*On*$\Gamma$-

*equivalence of zero-dimensional cycles*, Math. USSR-Sb.

**15**(1971), 555-567. ---,

*Rational equivalence of zero-cycles*, Math. USSR-Sb.

**18**(1972), 571-588.

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Keywords:
Abelian varieties,
rational equivalence,
zero-cycles

Article copyright:
© Copyright 1993
American Mathematical Society