Finite Equivalence Relations on Algebraic Varieties and Hidden Symmetries

Abstract

This paper can be considered as a continuation of Miyanishi's paper which contains a theorem on existence of a quotient of an affine normal or a projective smooth variety by a finite equivalence relation such that every component of the relation projects onto the variety (we call such an equivalence relation a wide finite equivalence relation). Later papers of Kollar and Keel-Mori shed new light on the subject and can serve as a base for further studies. The results of the present paper are based on the fact that every wide finite equivalence relation on a normal variety V is determined by an action of a finite group on the normalization of V in some Galois extension of k(V). Hence, such an equivalence relation hides some symmetry of a (ramified) cover of V. One may find some analogy of the situation with the concept of a hidden symmetry considered in physics. An important part of the paper is examples described in Section 6 which show that the main result of the paper (Theorem 2.3) is valid neither in the seminormal case, nor under the additional assumptions that there exists a finite morphism whose fibers contain equivalence classes of a given finite relation. In the nonnormal case, identification of some points described by a finite wide equivalence relation may force identification of some other nonequivalent points. This seems to show that the class of normal varieties and wide equivalence relation is a proper frame for considering the general problems of quotients by finite equivalence relations.