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 | References | Similar Articles | Additional Information

Abstract: Let be an -dimensional Abelian variety, ; let be the group of zero-cycles of , modulo rational equivalence; by regarding an effective, degree , zero-cycle, 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 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: ( 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.

**[D-M]**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****[G]**Phillip Griffiths (ed.),*Topics in transcendental algebraic geometry*, Annals of Mathematics Studies, vol. 106, Princeton University Press, Princeton, NJ, 1984. MR**756842****[H]**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****[M]**D. Mumford,*Rational equivalence of 0-cycles on surfaces*, J. Math. Kyoto Univ.**9**(1968), 195–204. MR**0249428**, https://doi.org/10.1215/kjm/1250523940**[P]**Gian Pietro Pirola,*Curves on generic Kummer varieties*, Duke Math. J.**59**(1989), no. 3, 701–708. MR**1046744**, https://doi.org/10.1215/S0012-7094-89-05931-0**[R]**A. A. Roitman,*On*-*equivalence of zero-dimensional cycles*, Math. USSR-Sb.**15**(1971), 555-567.**[R]**-,*Rational equivalence of zero-cycles*, Math. USSR-Sb.**18**(1972), 571-588.

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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