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 [math]\displaystyle{ \pi: X \to Y }[/math] such that

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

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

(iii) For any open subset [math]\displaystyle{ U \subset Y }[/math], [math]\displaystyle{ \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]\displaystyle{ \mathcal{O}_Y \simeq \pi_*(\mathcal{O}_X^G) }[/math]. In particular, if X is irreducible, then so is Y and [math]\displaystyle{ k(Y) = k(X)^G }[/math]: 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 [math]\displaystyle{ 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.

Examples

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

[math]\displaystyle{ X^s_{(0)} \to X^s_{(0)}/G }[/math]  

is a geometric quotient.