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 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 . In particular, if X is irreducible, then so is Y and : rational functions on Y may be viewed as invariant rational functions on X (i.e., rational-invariants of X).