Editing Projective space (section)

== Algebraic geometry ==
Originally, [[algebraic geometry]] was the study of common zeros of sets of [[multivariate polynomial]]s. These common zeros, called [[algebraic varieties]] belong to an [[affine space]]. It appeared soon, that in the case of real coefficients, one must consider all the complex zeros for having accurate results. For example, the [[fundamental theorem of algebra]] asserts that a univariate [[square-free polynomial]] of degree {{mvar|n}} has exactly {{mvar|n}} complex roots. In the multivariate case, the consideration of complex zeros is also needed, but not sufficient: one must also consider ''zeros at infinity''. For example, [[Bézout's theorem]] asserts that the intersection of two plane [[algebraic curve]]s of respective degrees {{mvar|d}} and {{mvar|e}} consists of exactly {{mvar|de}} points if one consider complex points in the projective plane, and if one counts the points with their multiplicity.{{efn|The correct definition of the multiplicity if not easy and dates only from the middle of 20th century}} Another example is the [[genus–degree formula]] that allows computing the genus of a plane [[algebraic curve]] from its [[singular point of a curve|singularities]] in the ''complex projective plane''.

So a [[projective variety]] is the set of points in a projective space, whose [[homogeneous coordinates]] are common zeros of a set of [[homogeneous polynomial]]s.{{efn|Homogeneous required in order that a zero remains a zero when the homogeneous coordinates are multiplied by a nonzero scalar.}}

Any affine variety can be ''completed'', in a unique way, into a projective variety by adding its [[points at infinity]], which consists of [[homogenization of a polynomial|homogenizing]] the defining polynomials, and removing the components that are contained in the hyperplane at infinity, by [[saturation (commutative algebra)|saturating]] with respect to the homogenizing variable.

An important property of projective spaces and projective varieties is that the image of a projective variety under a [[morphism of algebraic varieties]] is closed for [[Zariski topology]] (that is, it is an [[algebraic set]]). This is a generalization to every ground field of the compactness of the real and complex projective space.

A projective space is itself a projective variety, being the set of zeros of the zero polynomial.

=== Scheme theory ===
Scheme theory, introduced by [[Alexander Grothendieck]] during the second half of 20th century, allows defining a generalization of algebraic varieties, called [[scheme (mathematics)|schemes]], by gluing together smaller pieces called [[affine schemes]], similarly as [[manifold]]s can be built by gluing together open sets of {{math|'''R'''<sup>''n''</sup>}}.  The [[Proj construction]] is the construction of the scheme of a projective space, and, more generally of any projective variety, by gluing together affine schemes. In the case of projective spaces, one can take for these affine schemes the affine schemes associated to the charts (affine spaces) of the above description of a projective space as a manifold.
{{See also|Algebraic geometry of projective spaces}}