
GIT quotient

In algebraic geometry, an affine GIT quotient, or affine geometric invariant theory quotient, of an affine scheme formula_1 with an action by a group scheme "G" is the affine scheme formula_2, the prime spectrum of the ring of invariants of "A", and is denoted by formula_3. A GIT quotient is a categorical quotient: any invariant morphism uniquely factors through it.
Taking Proj (of a graded ring) instead of formula_4, one obtains a projective GIT quotient (which is a quotient of the set of semistable points.)
A GIT quotient is a categorical quotient of the locus of semistable points; i.e., "the" quotient of the semistable locus. Since the categorical quotient is unique, if there is a geometric quotient, then the two notions coincide: for example, one has 
for an algebraic group "G" over a field "k" and closed subgroup "H".
If "X" is a complex smooth projective variety and if "G" is a reductive complex Lie group, then the GIT quotient of "X" by "G" is homeomorphic to the symplectic quotient of "X" by a maximal compact subgroup of "G" (Kempf–Ness theorem).
Construction of a GIT quotient.
Let "G" be a reductive group acting on a quasi-projective scheme "X" over a field and "L" a linearized ample line bundle on "X". Let 
be the section ring. By definition, the semistable locus formula_7 is the complement of the zero set formula_8 in "X"; in other words, it is the union of all open subsets formula_9 for global sections "s" of formula_10, "n" large. By ampleness, each formula_11 is affine; say formula_12 and so we can form the affine GIT quotient
Note that formula_14 is of finite type by Hilbert's theorem on the ring of invariants. By universal property of categorical quotients, these affine quotients glue and result in
which is the GIT quotient of "X" with respect to "L". Note that if "X" is projective; i.e., it is the Proj of "R", then the quotient formula_16 is given simply as the Proj of the ring of invariants formula_17.
The most interesting case is when the stable locus formula_18 is nonempty; formula_18 is the open set of semistable points that have finite stabilizers and orbits that are closed in formula_7. In such a case, the GIT quotient restricts to
which has the property: every fiber is an orbit. That is to say, formula_22 is a genuine quotient (i.e., geometric quotient) and one writes formula_23. Because of this, when formula_18 is nonempty, the GIT quotient formula_25 is often referred to as a "compactification" of a geometric quotient of an open subset of "X".
A difficult and seemingly open question is: which geometric quotient arises in the above GIT fashion? The question is of a great interest since the GIT approach produces an "explicit" quotient, as opposed to an abstract quotient, which is hard to compute. One known partial answer to this question is the following: let formula_26 be a locally factorial algebraic variety (for example, a smooth variety) with an action of formula_27. Suppose there are an open subset formula_28 as well as a geometric quotient formula_29 such that (1) formula_25 is an affine morphism and (2) formula_31 is quasi-projective. Then formula_32 for some linearlized line bundle "L" on "X". (An analogous question is to determine which subring is the ring of invariants in some manner.)
Examples.
Finite group action by formula_33.
A simple example of a GIT quotient is given by the formula_33-action on formula_35 sending
Notice that the monomials formula_37 generate the ring formula_38. Hence we can write the ring of invariants as
Scheme theoretically, we get the morphism
which is a singular subvariety of formula_41 with isolated singularity at formula_42. This can be checked using the differentials, which are
hence the only point where the differential and the polynomial formula_44 both vanish is at the origin. The quotient obtained is a conical surface with an ordinary double point at the origin.
Torus action on plane.
Consider the torus action of formula_45 on formula_46 by formula_47. Note this action has a few orbits: the origin formula_48, the punctured axes, formula_49, and the affine conics given by formula_50 for some formula_51. Then, the GIT quotient formula_52 has structure sheaf formula_53 which is the subring of polynomials formula_54, hence it is isomorphic to formula_55. This gives the GIT quotientformula_56Notice the inverse image of the point formula_57 is given by the orbits formula_58, showing the GIT quotient isn't necessarily an orbit space. If it were, there would be three origins, a non-separated space.