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 such that

Contents

• Relation to other quotients
• Examples
• References

(i) For each y in Y, the fiber π − 1 ( y ) is an orbit of G. (ii) The topology of Y is the quotient topology: a subset U ⊂ Y is open if and only if π − 1 ( U ) is open. (iii) For any open subset U ⊂ Y , π # : k [ U ] → k [ π − 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 O Y ≃ ( π ∗ O X ) G . In particular, if X is irreducible, then so is Y and 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 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).