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 $\pi: X \to Y$ such that
:(i) For each ''y'' in ''Y'', the fiber $\pi^(y)$ is an orbit of ''G''.
:(ii) The topology of ''Y'' is the quotient topology: a subset $U \subset Y$ is open if and only if $\pi^(U)$ is open.
:(iii) For any open subset $U \subset Y$, \pi^: k \to k pi^(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 $\mathcal_Y \simeq \pi_*(\mathcal_X^G)$. 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).

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 $\mathbb^ \setminus 0 \to \mathbb^n$ is a geometric quotient.
* If ''L'' is a linearized line bundle on an algebraic ''G''-variety ''X'', then, writing $X^s_$ for the set of stable points with respect to ''L'', the quotient
::$X^s_ \to X^s_/G$  
:is a geometric quotient.

References

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Algebraic geometry