# 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 |