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Fujita’s approximation theorem in positive characteristics. (English) Zbl 1136.14004
Let $$X$$ be a projective variety of dimension $$n$$ over an algebraically closed field $$k$$. For a line bundle $$L$$ on $$X$$ the volume of $$L$$, denoted by vol$$_X(L)$$, measures, when $$L$$ is big, the asymptotic growth of the linear series $$| mL|$$ for $$m\gg 0$$. It is defined as lim sup$$_{m \to \infty}{h^0(X, L^{\otimes m}) \over m^n/n!}$$ and extends uniquely to a continuous function vol$$_X:N^1(X)_{\mathbb{R}} \to \mathbb{R}$$, where $$\mathbb{R}$$ is the field of real numbers and $$N^1(X)_{\mathbb{R}}$$ is the group of numerical equivalence classes of $$\mathbb{R}$$-divisors. The paper under review extends to positive characteristic the Fujita’s approximation theorem, known to be true in characteristic zero. The idea is to substitute in the proof of the approximation theorem the Hironaka’s desingularization theorem by de Jong’s alteration theorem, which works in positive characteristic. The theorem establishes the following: for a rational big class $$\zeta \in N^1(X)_{\mathbb{Q}}$$ and any real number $$\epsilon >0$$ there exist a projective birational morphism $$\pi:X' \to X$$ and a decomposition $$\pi^*\zeta=\alpha+e \in N^1(X')_{\mathbb{Q}}$$ such that $$\alpha$$ is ample, $$e$$ is effective and vol$$_{X'}(\zeta)-$$vol$$_X(\alpha)<\epsilon$$. As an application (see Thm. 3.2) a sort of uniform convergence of the volume is shown: Let $$H$$ be an ample $$\mathbb{Q}$$-divisor on $$X$$ and denote by $$d=$$deg$$_H(L)$$. For any $$\epsilon>0$$ there exists $$d_0>0$$ such that if $$d>d_0$$ then $${h^0(X,L) \over d^n}<\epsilon +{\text{ vol}_X(L/d) \over n!}$$.

##### MSC:
 14C20 Divisors, linear systems, invertible sheaves 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
##### Keywords:
big line bundles; volume; ample line bundles; global sections
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