projective variety

Given a homogeneous polynomial $F$ of degree $d$ in $n+1$ variables $X_{0},\ldots,X_{n}$ and a point $[x_{0}:\cdots:x_{n}]$, we cannot evaluate $F$ at that point, because it has multiple such representations, but since $F(\lambda x_{0},\ldots,\lambda x_{n})=\lambda^{d}F(x_{0},\ldots,x_{n})$ 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 $\mathbb{P}^{n}_{k}$ 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