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\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.