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 formula_1 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 formula_7. In particular, if "X" is irreducible, then so is "Y" and formula_8: 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 formula_9 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.