Alternating groups and rational functions on surfaces

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 $f \colon X \to Y$, which is generically finite of degree m, into a complex rational ruled surface Y, whose monodromy is the alternating group A_m. This gives a finite algebraic extension C(X): C(x₁, x₂) of degree m, whose normal closure has Galois group A_m.