Rational orbits on threesymmetric products of abelian varieties
Authors:
A. Alzati and G. P. Pirola
Journal:
Trans. Amer. Math. Soc. 337 (1993), 965980
MSC:
Primary 14K05; Secondary 14C15, 14H40
MathSciNet review:
1106186
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Abstract: Let be an dimensional Abelian variety, ; let be the group of zerocycles of , modulo rational equivalence; by regarding an effective, degree , zerocycle, as a point on (the symmetric product of ), and by considering the associated rational equivalence class, we get a map , whose fibres are called orbits. For any , in this paper we determine the maximal dimension of the orbits when or (it is, respectively, and ), and the maximal dimension of families of orbits; moreover, for generic , we get some refinements and in particular we show that if , does not contain any orbit; note that it implies that a generic Abelian fourfold 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: ( is an elliptic curve with varying moduli) and we have constructed suitable projections between and which preserve the dimensions of the families of orbits; then we have done induction on . For the proof is based upon the papers of Mumford and Roitman on this topic.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199311061869
PII:
S 00029947(1993)11061869
Keywords:
Abelian varieties,
rational equivalence,
zerocycles
Article copyright:
© Copyright 1993
American Mathematical Society
