# isomorphism of varieties

###### Definition 1.

Let ${V}_{\mathrm{1}}$ and ${V}_{\mathrm{2}}$ be algebraic varieties. We say that ${V}_{\mathrm{1}}$ and ${V}_{\mathrm{2}}$ are isomorphic, and write ${V}_{\mathrm{1}}\mathrm{\cong}{V}_{\mathrm{2}}$, if there are regular maps

$$\varphi :{V}_{1}\to {V}_{2},\psi :{V}_{2}\to {V}_{1}$$ |

such that the compositions^{} $\psi \mathrm{\circ}\varphi $ and $\varphi \mathrm{\circ}\psi $ are the identity maps on ${V}_{\mathrm{1}}$ and ${V}_{\mathrm{2}}$ respectively.

###### Definition 2.

Let ${V}_{\mathrm{1}}$ and ${V}_{\mathrm{2}}$ be varieties^{} defined over a field $K$. We say that ${V}_{\mathrm{1}}\mathrm{/}K$ and ${V}_{\mathrm{2}}\mathrm{/}K$ are isomorphic over $K$ if ${V}_{\mathrm{1}}$ and ${V}_{\mathrm{2}}$ are isomorphic as in Definition 1 and the regular maps $\varphi $ and $\psi $ can be defined over $K$.

Title | isomorphism of varieties |
---|---|

Canonical name | IsomorphismOfVarieties |

Date of creation | 2013-03-22 15:06:22 |

Last modified on | 2013-03-22 15:06:22 |

Owner | alozano (2414) |

Last modified by | alozano (2414) |

Numerical id | 4 |

Author | alozano (2414) |

Entry type | Definition |

Classification | msc 14A10 |

Related topic | JInvariantClassifiesEllipticCurvesUpToIsomorphism |