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 \( {\displaystyle U\subset Y} \) is open if and only if \( \pi ^{-1}(U) \) is open.
(iii) For any open subset \( {\displaystyle U\subset Y} \) , \( {\displaystyle \pi ^{\#}:k[U]\to k[\pi ^{-1}(U)]^{G}} \) 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 \( {\displaystyle {\mathcal {O}}_{Y}\simeq \pi _{*}({\mathcal {O}}_{X}^{G})} \). In particular, if X is irreducible, then so is Y and \( {\displaystyle k(Y)=k(X)^{G}} \) : 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 {\displaystyle G/H} 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 \( {\displaystyle \mathbb {A} ^{n+1}\setminus 0\to \mathbb {P} ^{n}} \) is a geometric quotient.
If L is a linearized line bundle on an algebraic G-variety X, then, writing \( {\displaystyle X_{(0)}^{s}} \) for the set of stable points with respect to L, the quotient
\( {\displaystyle X_{(0)}^{s}\to X_{(0)}^{s}/G} \)
is a geometric quotient.
References
Brion 2009, Definition 1.18
M. Brion, "Introduction to actions of algebraic groups" [1]
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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