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{{cite web |last=Brion |first=M. |title=Introduction to actions of algebraic groups |url=http://www-fourier.ujf-grenoble.fr/~mbrion/notes_luminy.pdf |at=Definition 1.18}}

:(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.

References

Category:Algebraic geometry