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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
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.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1993-1106186-9
Keywords: Abelian varieties, rational equivalence, zero-cycles
Article copyright: © Copyright 1993 American Mathematical Society