EQUIVARIANT JACOBIAN CONJECTURE IN DIMENSION TWO

Abstract

Let G be a small finite subgroup of GL(2,ℂ) and let $\overset{\sim }{\varphi }:{\mathbbm{A}}^2\to {\mathbbm{A}}^2$ be a G- equivariant étale endomorphism of the affine plane. We show that $\overset{\sim }{\varphi }$ is an automorphism if the order of G is even. The proof depends on an analysis of a quasi-étale endomorphism φ induced by $\overset{\sim }{\varphi }$ on the singular quotient surface ${\mathbbm{A}}^2/G$ whose smooth part X° has the standard ${\mathbbm{A}}_{\ast}^1$-fibration p°: X° → ℙ¹. If φ preserves the standard ${\mathbbm{A}}_{\ast}^1$-fibbration p° then both φ and $\overset{\sim }{\varphi }$ are automorphisms. We look for the condition with which φ preserves the standard ${\mathbbm{A}}_{\ast}^1$-fibration and prove as a consequence that $\overset{\sim }{\varphi }$ is an automorphism if G has even order.