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In algebraic geometry, a '''geometric quotient''' of an [[algebraic variety]] ''X'' with the action of an [[algebraic group]] ''G'' is a [[morphism of varieties]] <math>\pi: X \to Y</math> such that<ref>{{harvnb|Brion|2009|loc=Definition 1.18}}</ref>
:(i) For each ''y'' in ''Y'', the fiber <math>\pi^{-1}(y)</math> is an orbit of ''G''.
:(ii) The topology of ''Y'' is the [[quotient topology]]: a subset <math>U \subset Y</math> is open if and only if <math>\pi^{-1}(U)</math> is open.
:(iii) For any open subset <math>U \subset Y</math>, <math>\pi^{\#}: k[U] \to k[\pi^{-1}(U)]^G</math> 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 <math>\mathcal{O}_Y \simeq (\pi_*\mathcal{O}_X)^G</math>. In particular, if ''X'' is irreducible, then so is ''Y'' and <math>k(Y) = k(X)^G</math>: rational functions on ''Y'' may be viewed as invariant rational functions on ''X'' (i.e., [[rational-invariant]]s of ''X'').

For example, if ''H'' is a closed subgroup of ''G'', then <math>G/H</math> 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.

== Example ==
* The canonical map <math>k^n - 0 \to \mathbb{P}^n</math> is a geometric quotient.

== References ==
{{reflist}}
*M. Brion, "Introduction to actions of algebraic groups" [http://www-fourier.ujf-grenoble.fr/~mbrion/notes_luminy.pdf]

[[Category:Algebraic geometry]]