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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
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[1] Lawrence Ein and Robert Lazarsfeld, Seshadri constants on smooth surfaces, Astérisque 218 (1993), 177 – 186. Journées de Géométrie Algébrique d’Orsay (Orsay, 1992). · Zbl 0812.14027
[2] Lawrence Ein, Robert Lazarsfeld, and Karen E. Smith, Uniform bounds and symbolic powers on smooth varieties, Invent. Math. 144 (2001), no. 2, 241 – 252. · Zbl 1076.13501 · doi:10.1007/s002220100121 · doi.org
[3] Lawrence Ein, Oliver Küchle, and Robert Lazarsfeld, Local positivity of ample line bundles, J. Differential Geom. 42 (1995), no. 2, 193 – 219. · Zbl 0866.14004
[4] William Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984. · Zbl 0541.14005
[5] Gerd Faltings and Gisbert Wüstholz, Diophantine approximations on projective spaces, Invent. Math. 116 (1994), no. 1-3, 109 – 138. · Zbl 0805.14011 · doi:10.1007/BF01231559 · doi.org
[6] Michael Nakamaye, Seshadri constants and the geometry of surfaces, J. Reine Angew. Math. 564 (2003), 205 – 214. · Zbl 1052.14038 · doi:10.1515/crll.2003.091 · doi.org
[7] Yum-Tong Siu, Effective very ampleness, Invent. Math. 124 (1996), no. 1-3, 563 – 571. · Zbl 0853.32034 · doi:10.1007/s002220050063 · doi.org
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