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 ${\displaystyle \pi :X\to Y}$ such that[1]

(i) For each y in Y, the fiber ${\displaystyle \pi ^{-1}(y)}$ 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 ${\displaystyle \pi ^{-1}(U)}$ 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 ${\displaystyle G/H}$ 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).