Quotients by finite equivalence relations

Summary

We study the existence of geometric quotients by finite set-theoretic equivalence relations. We show that such geometric quotients always exist in positive characteristic but not in characteristic 0. The appendix gives some examples of unexpected behavior for scheme-theoretic equivalence relations.

Let be a finite morphism of schemes. Given Y, one can easily describe X by the coherent sheaf of algebras. Here our main interest is the converse. Given X, what kind of data do we need to construct Y? For this question, the surjectivity of f is indispensable.

The fiber product defines an equivalence relation on X, and one might hope to reconstruct Y as the quotient of X by this equivalence relation. Our main interest is in the cases when is not flat. A typical example we have in mind is when Y is not normal and X is its normalization. In these cases, the fiber product X x _Y X can be rather complicated. Even if Y and X are pure-dimensional and CM, X x _y X can have irreducible components of different dimension and its connected components need not be pure-dimensional. None of these difficulties appear if is flat [Raynaud 1967; SGA3 1970] or if Y is normal (Lemma 21).

Finite equivalence relations appear in moduli problems in two ways. First, it is frequently easier to construct or to understand the normalization of a moduli space M. Then one needs to construct M as a quotient of by a finite equivalence relation. This method was used in [Kollár 1997] and finite equivalence relations led to some unsolved problems in [Viehweg 1995, Section 9.5]; see also [Kollár 2011].

Second, in order to compactify moduli spaces of varieties, one usually needs nonnormal objects. The methods of the minimal model program seem to apply naturally to their normalizations. It is quite subtle to descend information from the normalization to the nonnormal variety, see [Kollár 2012, Chapter 5].

In Sections 1, 2, 3 and 6 of this article we give many examples, review (and correct) known results and pose some questions. New results concerning finite equivalence relations are in Sections 4 and 5 and in the Appendix.