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^[1]

(i) For each y in Y, the fiber

\pi ^{{-1}}(y)

is an orbit of G.

(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 $X_{(0)}^{s}$ for the set of stable points with respect to L, the quotient

X_{(0)}^{s}\to X_{(0)}^{s}/G

is a geometric quotient.

References

↑ Brion 2009, Definition 1.18

M. Brion, "Introduction to actions of algebraic groups"

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[1] Brion 2009, Definition 1.18