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 π : X → Y {displaystyle pi :X o Y} such that 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 o Y} such that 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). 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.