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On the theorem of de Franchis. (English) Zbl 0534.14016

Let X be a compact Riemann surface of genus \(g>1\). Let Hol(X) denote the set of surjective holomorphic mappings whose domain is X and whose range has genus \(>1\); note that the range is allowed to vary. The well known theorem of de Franchis asserts that Hol(X) is a finite set. In this paper an explicit bound on Hol(X) that depends only on g is given. The bound is probably far from being sharp; the worst term in the bound is of the order \(g^{2g^ 2}\). A generalization for the set of maps from a fixed higher dimensional X to Riemann surfaces of genus \(>2\) is also given.

MSC:

14E05 Rational and birational maps
14N10 Enumerative problems (combinatorial problems) in algebraic geometry
30F10 Compact Riemann surfaces and uniformization

References:

[1] T.M. Bandman , Surjective holomorphic mappings of projective manifolds , Siberian Math. J. , 22 ( 1981 ), pp. 204 - 210 . MR 610768 | Zbl 0491.32019 · Zbl 0491.32019 · doi:10.1007/BF00968417
[2] M. De Franchis , Un teorema sulle involuzioni irrazionali , Rend. Circ. Mat. Palermo , 36 ( 1913 ), pg. 368 . JFM 44.0657.02 · JFM 44.0657.02
[3] P. Griffiths - J. Harris , Principles of Algebraic Geometry , John Wiley , New York , 1978 . MR 507725 | Zbl 0408.14001 · Zbl 0408.14001
[4] D. Lieberman - D. Mumford , Matsusaka’s big theorem , Amer. Math. Soc. Proceedings of Symposia in Pure Mathematics , 29 ( 1975 ), pp. 513 - 530 . MR 379494 | Zbl 0321.14004 · Zbl 0321.14004
[5] P. Samuel , Lectures on old and new results on algebraic curves , Tata Institute of Fundamental Research , Bombay , 1966 . MR 222088 | Zbl 0165.24102 · Zbl 0165.24102
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