# variety

###### Definition 1

Let $X$ be a scheme over a field $k$. Then $X$ is said to be an *abstract variety* over $k$ if it is integral, separated, and of finite type over $k$. Usually we simply say $X$ is a *variety*. If $X$ is proper over $k$, it is said to be *complete ^{}*. If the dimension

^{}of $X$ is one, then $X$ is said to be a

*curve*.

Some authors also require $k$ to be algebraically closed^{}, and some authors require curves to be nonsingular.

Calling $X$ a variety would appear to conflict with the preexisting notion of an affine (http://planetmath.org/AffineVariety) or projective variety. However, it can be shown that if $k$ is algebraically closed, then there is an equivalence of categories between affine abstract varieties over $k$ and affine varieties^{} over $k$, and another between projective abstract varieties over $k$ and projective varieties over $k$.

This equivalence of categories identifies an abstract variety with the set of its $k$-points; this can be thought of as simply ignoring all the generic points. In the other direction, it identifies an affine variety with the prime spectrum of its coordinate ring: the variety in ${\mathbb{A}}^{n}$ defined by the ideal

$$ |

is identified with

$$ |

A projective variety is identified as the gluing together of the affine varieties obtained by taking the complements of hyperplanes^{}. To see this, suppose we have a projective variety in ${\mathbb{P}}^{n}$ given by the homogeneous ideal $$. If we delete the hyperplane ${X}_{i}=0$, then we obtain an affine variety: let ${T}_{j}={X}_{j}/{X}_{i}$; then the affine variety is the set of common zeros of

$$ |

In this way, we can get $n+1$ overlapping affine varieties that cover our original projective variety. Using the theory of schemes, we can glue these affine varieties together to get a scheme; the result will be projective.

For more on this, see Hartshorne’s book *Algebraic Geometry ^{}*; see the bibliography for algebraic geometry for more resources.

Title | variety |

Canonical name | Variety |

Date of creation | 2013-03-22 14:16:43 |

Last modified on | 2013-03-22 14:16:43 |

Owner | mps (409) |

Last modified by | mps (409) |

Numerical id | 9 |

Author | mps (409) |

Entry type | Definition |

Classification | msc 14-00 |

Synonym | abstract variety |

Related topic | Scheme |

Related topic | AffineVariety |

Related topic | ProjectiveVariety |

Defines | complete |

Defines | curve |