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 $$U \subset Y$$, $$\pi^{\#}: k[U] \to k[\pi^{-1}(U)]^G$$ 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.

Examples

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