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: X → 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): G(x1, x2) of degree m, whose normal closure has Galois group Am. © Foundation Compositio Mathematica 2006.
Brivio, S., Pirola, G. (2006). Alternating groups and rational functions on surfaces. COMPOSITIO MATHEMATICA, 142(2), 409-421 [10.1112/S0010437X05001715].
Alternating groups and rational functions on surfaces
Brivio, S;
2006
Abstract
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: X → 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): G(x1, x2) of degree m, whose normal closure has Galois group Am. © Foundation Compositio Mathematica 2006.
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