projective variety ProjectiveVariety 2001-12-21 04:44:08 2004-06-04 14:58:53 Definition quasi-projective variety \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{graphicx} \usepackage{xypic} 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 \emph{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 \emph{quasi-projective variety} is an open subset of a projective variety.