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Effective base point freeness. (English) Zbl 0818.14002
If $$X$$ is a smooth projective variety of dimension $$n$$ and $$L$$ is a nef divisor on $$X$$ such that $$aL - K_ X$$ is nef and big for some $$a \geq 0$$, then the base point free theorem says that the linear system $$| bD |$$ is base point free for $$b \gg 0$$. The author makes this statement effective by showing that the condition $$b \geq 2 (n + 2)!(a + n)$$ is sufficient to guarantee base point freeness. The importance of this result is not the actual value of the bound but the fact that it depends only on $$n = \dim X$$ and $$a$$; a conjectured bound is $$b \geq a + n + 1$$. The theorem is formulated in the more general case where $$X$$ has log terminal singularities.
The author applies his results to get explicit bounds for the number of irreducible families of $$n$$-dimensional smooth Fano varieties and polarized varieties.

##### MSC:
 14C20 Divisors, linear systems, invertible sheaves 14J10 Families, moduli, classification: algebraic theory 14N10 Enumerative problems (combinatorial problems) in algebraic geometry
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