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[1]
- (i) The map is surjective, and its fibers are exactly the G-orbits in X.
- (ii) The topology of Y is the quotient topology: a subset is open if and only if is open.
- (iii) For any open subset , 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 . 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).
For example, if H is a closed subgroup of G, then 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).