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 $\pi :X\to Y$ such that

(i) The map $\pi$ is surjective, and its fibers are exactly the G-orbits in X.

(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 $U\subset Y$, ${\pi }^{\mathrm{\#}}:k[U]\to k[{\pi }^{-1}(U){]}^G$ 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 ${{𝒪}}_Y\simeq {\pi }_{*}({{𝒪}}_X^G)$. 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.

Examples

• The canonical map ${\mathbb{𝔸}}^{n+1}\setminus 0\to {\mathbb{\mathbb{P}}}^n$ is a geometric quotient.
• If L is a linearized line bundle on an algebraic G-variety X, then, writing $X_{(0)}^s$ for the set of stable points with respect to L, the quotient

$X_{(0)}^s\to X_{(0)}^s/G$  

is a geometric quotient.