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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 \(\epsilon(\eta, A)>(3d+1)/3d^2\).

14C20 Divisors, linear systems, invertible sheaves
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