In algebraic geometry, an affine GIT quotient, or affine geometric invariant theory quotient, of an affine scheme
, the prime spectrum of the ring of invariants of A, and is denoted by
A GIT quotient is a categorical quotient: any invariant morphism uniquely factors through it.
Taking Proj (of a graded ring) instead of
, one obtains a projective GIT quotient (which is a quotient of the set of semistable points.)
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.[clarification needed] 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).
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.
By definition, the semistable locus
in X; in other words, it is the union of all open subsets
and so we can form the affine GIT quotient Note that
is of finite type by Hilbert's theorem on the ring of invariants.
is given simply as the Proj of the ring of invariants
The most interesting case is when the stable locus[1]
is the open set of semistable points that have finite stabilizers and orbits that are closed in
In such a case, the GIT quotient restricts to which has the property: every fiber is an orbit.
is nonempty, the GIT quotient
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:[2] let
for some linearlized line bundle L on X.
(An analogous question is to determine which subring is the ring of invariants in some manner.)
A simple example of a GIT quotient is given by the
sending Notice that the monomials
Hence we can write the ring of invariants as Scheme theoretically, we get the morphism which is a singular subvariety of
The quotient obtained is a conical surface with an ordinary double point at the origin.
Note this action has a few orbits: the origin
Notice the inverse image of the point
, showing the GIT quotient isn't necessarily an orbit space.
If it were, there would be three origins, a non-separated space.