Geometric quotient
In algebraic geometry, a geometric quotient of an algebraic variety X with the action of an algebraic group G is a morphism of varieties such that
The notion appears in geometric invariant theory., say that Y is an orbit space of X in topology. may also be phrased as an isomorphism of sheaves. In particular, if X is irreducible, then so is Y and : rational functions on Y may be viewed as invariant rational functions on X.
For example, if H is a closed subgroup of G, then is a geometric quotient. A GIT quotient may or may not be a geometric quotient: but both are categorical quotients, which is unique; in other words, one cannot have both types of quotients.Relation to other quotients
A geometric quotient is a categorical quotient. This is proved in Mumford's geometric invariant theory.
A geometric quotient is precisely a good quotient whose fibers are orbits of the group.Examples
