Quotient Spaces modulo Algebraic Groups

In algebraic geometry one often encounters the problem of taking the quotient of a scheme by a group. Despite the frequent occurrence of such problems, there are few general results about the existence of such quotients. These questions come up again and again in the theory of moduli spaces. When we want to classify some type of algebraic objects, say varieties or vector bundles, the classification usually proceeds in two steps. First we classify the objects with some extra structure added. In classifying varieties we first parametrize embedded varieties in a fixed projective space P . In studying vector bundles we first describe them using a basis in the vector space V of global sections. Second, we have to get rid of the effect of the extra structure. In the case of varieties this usually means dividing out by the automorphism group of P . In the case of vector bundles we take the quotient by the automorphism group of the space of sections V . There have been several attempts to give a general theory of quotients in algebraic geometry. Mumford’s geometric invariant theory is designed to construct quotients by group actions. This approach works very successfully for the quotient problems arising in the theory of vector bundles. Actually, in this area geometric invariant theory solves two problems at once. It predicts the best equivalence relation by which a quotient might exist (a problem with semistable objects), and then it proceeds to construct the quotient. In the theory of the moduli of varieties, geometric invariant theory has been less successful. It does not seem to predict the correct class of varieties where a good quotient should exist. The existence of the quotient is rather difficult to establish, and it has been done only over C [Viehweg95]. Artin developed the theory of algebraic spaces in order to tackle more general quotient problems. It was noticed early on that the quotient of a scheme by a finite group is, in general, not a scheme. (The simplest example is smooth of dimension 3). Naively one can think of algebraic spaces as quotients of schemes by finite groups (this is completely correct for normal algebraic spaces). This approach is inconvenient in practice, and the working definition is different [Knutson71]. The theory of algebraic spaces is rather successful in handling general quotient problems. For instance, the quotient of an algebraic space by a flat equivalence relation is again an algebraic space [Artin74]. This implies that if g : S → S is faithfully flat then descent data for algebraic spaces over S are always effective.