TWO-DIMENSIONAL VERSAL G-COVERS AND CREMONA EMBEDDINGS OF FINITE GROUPS

Abstract

Let G be a finite group. A Galois cover ω : X → Y with Galois group G is called a versal G-cover if any G-cover is induced from ω : X → Y . In this paper, we prove that, if the essential dimension of G is equal to 2, then there exists a versal G-cover ω : X → Y such that (i) X is a smooth rational surface, (ii) G is a finite automorphism group of X, (iii) the action of G on X is minimal, and (iv) Y = X/G. We also give some examples of finite groups which have non-conjugate embeddings into the Cremona group Cr2(C).