Geometric quotient explained

\pi: X \to Y such that^[1]

(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

l{O} _{Y} \simeq \pi _{*(l{O}} ^{G)} ^{X} . 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

A ^{n+1} \setminus 0 \to 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 quotientX ^{s} _{( 0 )} \to X ^{s} _{( 0 )} /G  

is a geometric quotient.

Notes and References

1. Web site: Brion . M. . Introduction to actions of algebraic groups . Definition 1.18.