very ample

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

Equivalently, 𝔏 is very ample if there is an embeddingPlanetmathPlanetmath f:Xn such that f*𝒪(1)=𝔏, that is, 𝔏 is the pullbackPlanetmathPlanetmath of the tautological bundle on n.

If k is algebraically closedMathworldPlanetmath, Riemann-Roch ( 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