Finite equivalence relations on algebraic varieties

Abstract

Let Γ⊂ V× k V be a finite, pseudo-equivalence relation on a nonsingular, projective variety V defined over an algebraically closed field k of characteristic zero. Suppose that every irreducible component of Γ surjects on V. It is then shown that Γ is isomorphic to ( V× W V) red for a normal, projective variety W and a finite morphism ƒ: V→ W. Furthermore, Γ is nonsingular if and only if Γ = V× W V and ƒ is étale.