projective variety

Given a homogeneous polynomial $F$ of degree $d$ in $n+1$ variables $X_0,\mathrm{\dots },X_n$ and a point $[x_0:\mathrm{\cdots }:x_n]$, we cannot evaluate $F$ at that point, because it has multiple such representations, but since $F(\lambda x_0,\mathrm{\dots },\lambda x_n)={\lambda }^{d}F(x_0,\mathrm{\dots },x_n)$ 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 space ${\mathbb{P}}_k^n$ over $k$ which can be described as the common vanishing locus of finitely many homogeneous polynomials with coefficients 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	Variety
Related topic	ChowsTheorem
Defines	quasi-projective variety