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very ample (Definition)

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

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

If $ k$ is algebraically closed, 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.



"very ample" is owned by Mathprof. [ full author list (2) | owner history (1) ]
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Cross-references: genus, degree, curve, algebraically closed, tautological, pullback, embedding, vanishes, global section, point, field, scheme, invertible sheaf
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This is version 5 of very ample, born on 2003-08-19, modified 2007-04-28.
Object id is 4621, canonical name is VeryAmple.
Accessed 1840 times total.

Classification:
AMS MSC: 14A99 (Algebraic geometry :: Foundations :: Miscellaneous)
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