I find this concept in Kollar and Mori's book {\em Birational Geometry of Algebraic Varieties}, but cant search the precise definition in the book or google. Can you tell me the definition? Thanks for any comments or references.
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$\begingroup$ Maybe you should say where precisely in the book you found this concept... $\endgroup$– Francesco PolizziCommented Apr 3, 2013 at 14:53
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1$\begingroup$ A wild guess is that for any ample line bundle $L$, there exists some $N$, such that for all $n\geq N$, $L^{\otimes n}$ is sufficiently ample. This "definition" only makes sense if the term is used in certain contexts like "If $L$ is sufficiently ample, then .." and not "If ..., then $L$ is sufficiently ample." $\endgroup$– Will SawinCommented Apr 3, 2013 at 15:21
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There are many theorems of the form
Theorem Frame If something holds and $\mathscr L$ is an ample line bundle, then there exists an $n_0\in \mathbb N$ such that for all $n\geq n_0$, something else holds with $\mathscr L^{\otimes n}$ in it.
You should think of things like
- a sheaf being generated by global sections
- Serre vanishing
- transversality statements
- avoiding certain points or properties
- etc.
Sufficiently ample means that whatever they are claiming holds for $\mathscr L^{\otimes n}$ for all $n\geq n_0$ with some $n_0\in \mathbb N$ for any ample line bundle $\mathscr L$.
answered Apr 3, 2013 at 15:41