We had a slogan that the goal of GIT is to construct quotients in algebraic geometry; one would like to consider the quotient topological space (which is the natural way to endow the set of orbits with a topology) and give it the structure of an algebraic variety. This is not always possible. On the other hand, at least in the affine setting, one is tempted to consider the spectrum of the algebra of invariants as a quotient, but this sometimes happen to record little information about the topological quotient. What we are going to do is to get rid of bad orbits and obtain a reasonable quotient space of the remaining ones. Let's now go through some definitions of reasonable quotients and the properties we would expect from them.
Let be a category of algebro-geometric objects (we have in mind 1. (quasi-)affine algebraic varieties, 2. (quasi-)projective varieties, 3. schemes over some base, say ). Let be a group (object in ) acting on . A morphism in is called a categorical quotient if, for every -invariant arrow (i.e. equivariant with respect to the trivial action of on ) in , there exists a map that factorizes through .
Uniqueness follows from the universal property.
Exercise: let be the action map. Write down the diagrams that this map is required to satisfy (i.e. identity, compatibility).
A map is called a good quotient if
- is -invariant (i.e. constant on orbits) and surjective;
- is affine and, for every affine open , identifies as the subring of -invariant functions inside ;
- if is closed and -invariant in , then is closed (i.e. submersive; it is sometimes required to be universally submersive);
- if and are closed, -invariants subsets of such that they do not intersect, then also .
A map is called a geometric quotient if it is a good quotient and is an orbit space.
Basically, a good quotient may fail to be a geometric quotient by grouping together a bunch of orbits. The basic results of GIT for reductive groups are the following:
- in the affine setting, one obtains a good quotient by taking ;
- in the projective case, one can throw away some orbits and obtain a good quotient (that is also a projective variety) of the other ones (semistable orbits); the quotient map restricts to a geometric quotient on the set of stable orbits (the quotient is only quasi-projective).
Furthermore, it is possible to give a numerical characterization of (semi)-stable points.