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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 $\pi :X\to Y$ such that

(i) For each y in Y, the fiber ${\pi }^{-1}(y)$ is an orbit of G.

(ii) The topology of Y is the quotient topology: a subset $U\subset Y$ is open if and only if ${\pi }^{-1}(U)$ is open.

(iii) For any open subset $U\subset Y$, ${\pi }^{\#}:k[U]\to k[{\pi }^{-1}(U){]}^G$ 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 ${\mathcal{𝒪}}_Y\simeq ({\pi }_{*}{\mathcal{𝒪}}_X{)}^G$. In particular, if X is irreducible, then so is Y and $k(Y)=k(X{)}^G$: 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 $G/H$ 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 $k^n-0\to {\mathbb{\mathbb{P}}}^n$ is a geometric quotient.

References

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• M. Brion, "Introduction to actions of algebraic groups" [1]