In this chapter we study general questions about geometric quotients by the finite set theoretic equivalence relations that were used in Chapter 5. These results are somewhat technical and probably the best introduction is to study the examples (9.7) and (9.61). While they do not help with the proofs, they show that the questions are quite subtle and that the theorems are likely to be optimal.

Section 9.1 studies the existence question for geometric quotients. Usually they do not exist, but we identify some normality and seminormality properties of stratified spaces that ensure the existence of quotients (9.21).

The finiteness of equivalence relations is studied in Section 9.4.

The seminormality of subvarieties of geometric quotients is investigated in Section 9.2.

In Section 9.3 we give necessary and sufficient conditions for a line bundle to descend to a geometric quotient. This makes it possible to decide when such a quotient is projective.

Assumptions The quotient theorems in this chapter hold for quasi-excellent algebraic spaces in characteristic 0. See (9.5) for remarks about schemes.

We do not study the positive characteristic case for two reasons: there are much stronger quotient theorems (9.6) and the assumptions (HN) and (HSN) are not known to hold. Our method crucially relies on the descent of normality of subschemes by finite, generically étale morphisms (10.26) which fails in positive characteristic (10.28).