In algebraic geometry, a geometric quotient of an algebraic variety X with the action of an algebraic group G is a morphism of varieties π : X → Y {\displaystyle \pi :X\to Y} such that
(i) The map π {\displaystyle \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 ⊂ Y {\displaystyle U\subset Y} is open if and only if π − 1 ( U ) {\displaystyle \pi ^{-1}(U)} is open. (iii) For any open subset U ⊂ Y {\displaystyle U\subset Y} , π # : k [ U ] → k [ π − 1 ( U ) ] G {\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 O Y ≃ π ∗ ( O X G ) {\displaystyle {\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 {\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} 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 ∖ 0 → P n {\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 X ( 0 ) s {\displaystyle X_{(0)}^{s}} for the set of stable points with respect to L, the quotient
X ( 0 ) s → X ( 0 ) s / G {\displaystyle X_{(0)}^{s}\to X_{(0)}^{s}/G} is a geometric quotient.
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