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Prym varieties of triple cyclic covers. (English) Zbl 0628.14034

Prym varieties associated to unramified triple cyclic covers are studied. We construct the Prym map from the moduli space of curves of genus \(g\) with such a cover to a certain moduli space of abelian varieties of dimension 2g-2 with a non-principal polarization and with an automorphism of order three.
The main theorem is that for \(g=4\) this Prym map is generically finite of degree \(16;\) moreover, the general fiber carries a \((16_ 6)\) Kummer structure.

MSC:

14K10 Algebraic moduli of abelian varieties, classification
14H40 Jacobians, Prym varieties
14H30 Coverings of curves, fundamental group
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References:

[1] Barth, W.: Abelian surfaces with (1,2)-polarization. Preprint, Univ. Erlangen (1986) · Zbl 0639.14023
[2] Beauville, A.: Prym varieties and the Schottky problem. Invent. Math.41, 149-196 (1977) · Zbl 0354.14013 · doi:10.1007/BF01418373
[3] Beauville, A.: Sous-variétés spéciales des variétés de Prym. Compos. Math.45, 357-383 (1982) · Zbl 0504.14022
[4] Deligne, P., Mumford, D.: The irreducibility of the space of curves of given genus. Publ. Math. IHES36, 75-110 (1969) · Zbl 0181.48803
[5] Donagi, R.: The tetragonal construction. Bull. Am. Math. Soc.4, 181-185 (1981) · Zbl 0491.14016 · doi:10.1090/S0273-0979-1981-14875-8
[6] Donagi, R., Smith, R.C.: The structure of the Prym map. Acta Math.146, 25-102 (1981) · Zbl 0538.14019 · doi:10.1007/BF02392458
[7] Griffith, P.A., Harris, J.: Principles of Algebraic Geometry. New York: Wiley 1978 · Zbl 0408.14001
[8] Masiewicki, L.: Universal properties of Prym varieties with an application to algebraic curves of genus five. Trans. Am. Math. Soc.222, 221-240 (1976) · Zbl 0333.14012 · doi:10.1090/S0002-9947-1976-0422289-6
[9] Matsusaka, T.: On a theorem of Torelli. Am. J. Math.80, 784-800 (1958) · Zbl 0100.35602 · doi:10.2307/2372783
[10] Mumford, D.: Prym varieties I. In: Contributions to analysis. New York: Academic Press 1974 · Zbl 0299.14018
[11] Shimura, G.: On analytic families of polarized abelian varieties and automorphic functions. Ann. Math.78, 149-192 (1963) · Zbl 0142.05402 · doi:10.2307/1970507
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