# 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) For each
*y*in*Y*, the fiber is an orbit of*G*. - (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).

## 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.

## Examples

- The canonical map is a geometric quotient.
- If
*L*is a linearlized line bundle on an algebraic*G*-variety*X*, then, writing for the set of stable points with respect to*L*, the quotient

- is a geometric quotient.

## References

- ↑ Brion 2009, Definition 1.18