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[1]

(i) For each y in Y, the fiber is an orbit of G.
(ii) The topology of Y is the quotient topology: a subset is open if and only if is open.
(iii) For any open subset , is an isomorphism. (Here, k is the base field.)

The notion appears in geometric invariant theory. (i), (ii) say that Y is an orbit space of X in topology. (iii) 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 (i.e., rational-invariants of 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 (without them being the same).

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.


is a geometric quotient.


  1. Brion 2009, Definition 1.18
This article is issued from Wikipedia - version of the 4/18/2015. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.