algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

, a geometric quotient of an

algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Mo ...

''X'' with the action of an

algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Man ...

''G'' is a

morphism of varieties In algebraic geometry, a morphism between algebraic varieties is a function between the varieties that is given locally by polynomials. It is also called a regular map. A morphism from an algebraic variety to the affine line is also called a regular ...

\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 In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient t ...

: 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 In mathematics, geometric invariant theory (or GIT) is a method for constructing quotients by Group action (mathematics), group actions in algebraic geometry, used to construct moduli spaces. It was developed by David Mumford in 1965, using ideas ...

. (i), (ii) say that ''Y'' is an

orbit space In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a p ...

of ''X'' in

topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...

. (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 In algebraic geometry, an affine GIT quotient, or affine geometric invariant theory quotient, of an affine scheme X = \operatorname A with an action by a group scheme ''G'' is the affine scheme \operatorname(A^G), the prime spectrum of the ring of i ...

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 In algebraic geometry, given a category ''C'', a categorical quotient of an object ''X'' with action of a group ''G'' is a morphism \pi: X \to Y that :(i) is invariant; i.e., \pi \circ \sigma = \pi \circ p_2 where \sigma: G \times X \to X is the ...

. This is proved in Mumford's geometric invariant theory. A geometric quotient is precisely a

good quotient This is a glossary of algebraic geometry. See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory. For the number-theoretic applications, see glossary of arithmetic and Diophantine geometry. ...

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 In mathematics, given an action \sigma: G \times_S X \to X of a group scheme ''G'' on a scheme ''X'' over a base scheme ''S'', an equivariant sheaf ''F'' on ''X'' is a sheaf of \mathcal_X-modules together with the isomorphism of \mathcal_-modules ...

on an algebraic ''G''-variety ''X'', then, writing

X^s_

for the set of

stable point In mathematics, geometric invariant theory (or GIT) is a method for constructing quotients by Group action (mathematics), group actions in algebraic geometry, used to construct moduli spaces. It was developed by David Mumford in 1965, using ideas ...

s with respect to ''L'', the quotient ::

X^s_ \to X^s_/G

:is a geometric quotient.

References

{{reflist Algebraic geometry