projective variety

Given a homogeneous polynomialMathworldPlanetmath F of degree d in n+1 variablesMathworldPlanetmath X0,,Xn and a point [x0::xn], we cannot evaluate F at that point, because it has multiple such representations, but since F(λx0,,λxn)=λdF(x0,,xn) we can say whether any such representation (and hence all) vanish at that point.

A projective variety over an algebraically closed field k is a subset of some projective spaceMathworldPlanetmath kn over k which can be described as the common vanishing locus of finitely many homogeneous polynomials with coefficientsMathworldPlanetmath in k, and which is not the union of two such smaller loci. Also, a quasi-projective variety is an open subset of a projective variety.

Title projective variety
Canonical name ProjectiveVariety
Date of creation 2013-03-22 12:03:58
Last modified on 2013-03-22 12:03:58
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 9
Author mathcam (2727)
Entry type Definition
Classification msc 14-00
Related topic AffineVariety
Related topic Scheme
Related topic AlgebraicGeometry
Related topic VarietyPlanetmathPlanetmathPlanetmath
Related topic ChowsTheorem
Defines quasi-projective variety