Finite equivalence relations on algebraic varieties (Q1263620)

Let V be a nonsingular projective variety over an algebraically closed field k of characteristic zero. An equivalence relation on V is defined by a closed subscheme \(\Gamma\) of \(V\times_ kV\) if for any \(k- scheme\quad T,\) \(\Gamma\) (T) defines functorially in T an equivalence relation on V(T) as follows: \(P\sim Q\) if and only if (P,Q)\(\in \Gamma (T)\). Let \(p_ i\) be the projection to the i-th factor \((i=1,2)\). \(\Gamma\) is said to be finite and reduced if \(p_ 1:\quad \Gamma \to V\) is finite and \(\Gamma_{red}=\Gamma\). A pseudo-equivalence relation on V is defined as a reduced closed subscheme \(\Gamma\) of \(V\times_ kV\) such that \(\Gamma\) (T) defines an equivalence relation on V(T) for all reduced k-schemes T. In this paper the author shows that for a pseudo-equivalence relation \(\Gamma\) such that every irreducible componant of \(\Gamma\) surjects on V: (1) \(\Gamma\) is isomorphic to \((V\times_ WV)_{red}\) for some normal projective variety W with a finite morphism \(f:\quad V\to W\) and (2) \(\Gamma\) is nonsingular if and only if \(\Gamma \cong V\times_ WV\) and f is étale.