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Seshadri constants at very general points. (English) Zbl 1084.14008
Let $$X$$ be a smooth projective variety, let $$A$$ be an ample line bundle on $$X$$, and let $$x$$ be a point of $$X$$. Then $\epsilon (x, A):= \inf_{C\ni x} \frac{c_1(A)\cap C}{{\text{mult}}_x(C)},$ where the infimum runs over all integral curves $$C\subset X$$ passing through $$x$$, is called the Seshadri constant of $$A$$ at $$x$$. In the case, where $$X$$ is a surface, L. Ein and R. Lazarsfeld [in: Journées de géométrie algébrique d’Orsay, France 1992, Astérisque 218, 177–186 (1993; Zbl 0812.14027)], proved for a very general point $$\eta\in X$$ that $$\varepsilon(\eta, A)\geq 1$$. Then in [L. Ein, O. Küchle and R. Lazarsfeld, J. Differ. Geom. 42, 193–219 (1995; Zbl 0866.14004)], in arbitrary dimension, it was proved that $$\varepsilon(\eta, A)\geq 1/\dim X$$. In the present paper, the author proves that if $$X$$ is a smooth threefold, then $$\varepsilon(\eta, A)\geq 1/2$$, and if $$\dim X=d\geq 4$$, then $$\epsilon(\eta, A)>(3d+1)/3d^2$$.

##### MSC:
 14C20 Divisors, linear systems, invertible sheaves
##### Keywords:
line bundle; projective variety; blow-up
Full Text:
##### References:
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