very ample

An invertible sheaf $\mathfrak{L}$ 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 $s\in\mathfrak{L}(X)$ not vanishing at $x$ , and (2) for each pair of points $x,y\in X$ , there is a global section $s\in\mathfrak{L}(X)$ such that $s$ vanishes at exactly one of $x$ and $y$ .

Equivalently, $\mathfrak{L}$ is very ample if there is an embedding $f:X\to\mathbb{P}^{n}$ such that $f^{*}\mathcal{O}(1)=\mathfrak{L}$ , that is, $\mathfrak{L}$ 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