# 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 VeryAmple 2013-03-22 13:52:44 2013-03-22 13:52:44 Mathprof (13753) Mathprof (13753) 8 Mathprof (13753) Definition msc 14A99