# very ample

An invertible sheaf $\U0001d50f$ on a scheme $X$ over a field $k$ is called *very ample* if (1) at each point $x\in X$, there is a global section $s\in \U0001d50f(X)$ not vanishing at $x$, and (2) for each pair of points $x,y\in X$, there is a global section $s\in \U0001d50f(X)$ such that $s$ vanishes at exactly one of $x$ and $y$.

Equivalently, $\U0001d50f$ is very ample if there is an embedding^{} $f:X\to {\mathbb{P}}^{n}$ such that ${f}^{*}\mathcal{O}(1)=\U0001d50f$, that is, $\U0001d50f$ is the pullback^{} of the tautological bundle on ${\mathbb{P}}^{n}$.

If $k$ is algebraically closed^{}, Riemann-Roch (http://planetmath.org/RiemannRochTheorem) shows that on a curve $X$, any invertible sheaf of degree greater than or equal to twice the genus of $X$ is very ample.

Title | very ample |
---|---|

Canonical name | VeryAmple |

Date of creation | 2013-03-22 13:52:44 |

Last modified on | 2013-03-22 13:52:44 |

Owner | Mathprof (13753) |

Last modified by | Mathprof (13753) |

Numerical id | 8 |

Author | Mathprof (13753) |

Entry type | Definition |

Classification | msc 14A99 |