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Alternating groups and rational functions on surfaces

Published online by Cambridge University Press:  13 March 2006

Sonia Brivio  and
Gian Pietro Pirola
Sonia Brivio
Affiliation:
Dipartimento di Matematica, Universitá di Pavia, via Ferrata 1, 27100 Pavia, [email protected]
Gian Pietro Pirola
Affiliation:
Dipartimento di Matematica, Universitá di Pavia, via Ferrata 1, 27100 Pavia, [email protected]
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Abstract

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Let X be a smooth complex projective surface and let C(X) denote the field of rational functions on X. In this paper, we prove that for any m > M(X), there exists a rational dominant map $f \colon X \to Y$, which is generically finite of degree m, into a complex rational ruled surface Y, whose monodromy is the alternating group Am. This gives a finite algebraic extension C(X): C(x1, x2) of degree m, whose normal closure has Galois group Am.

Keywords

surfacesmonodromy grouprational functionsalternating group14H3012F05
Type
Research Article
Information
Compositio Mathematica , Volume 142 , Issue 2 , March 2006 , pp. 409 - 421
Copyright
Foundation Compositio Mathematica 2006