×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] Benveniste, X.: Sur les varietes de dimension 3.... Math. Ann.266, 479-497 (1984) · Zbl 0531.14005 · doi:10.1007/BF01458541
[2] Benveniste, X.: Sur les applications pluricanoniques.... Am. J. Math.108, 433-449 (1986) · Zbl 0601.14035 · doi:10.2307/2374679
[3] Catanese, F.: Chow varieties, Hilbert schemes and moduli spaces of surfaces of general type. J. Algebra Geom.1, 561-596 (1992) · Zbl 0807.14006
[4] Clemens, H., Koll?r, J., Mori, S.: Higher dimensional complex geometry. (Asterisque, vol. 166) Paris: Soc. Math. Fr. 1988 · Zbl 0689.14016
[5] Demailly, J.P.: A numerical criterion for very ample line bundles. J. Differ. Geom.37, 323-374 (1993) · Zbl 0783.32013
[6] Ein, L., Lazarsfeld, R.: Global generation of pluricanonical and adjoint linear systems on smooth projective threefolds. J. Am. Math. Soc. (to appear) · Zbl 0803.14004
[7] Esnault, H., Viehweg, E.: Rev?tements cycliques II., In: Arocca, J.-M. et al. (eds.) G?om?trie Alg?brique et Applications II. La R?bida, pp. 81-94. Paris: Herman 1987
[8] Fujita, T.: On polarized manifolds whose adjoint bundles are not semipositive. In: Oda, T. (ed.) Algebraic Geometry, Sendai. (Adv. Stud. Pure Math., vol. 10, pp. 167-178) Tokyo: Kinokuniya and Amsterdam: North-Holland 1987 · Zbl 0659.14002
[9] Kawamata, Y., Matsuda, K., Matsuki, K.: Introduction to the minimal model problem. In: Oda, T. (ed.) Algebraic Geometry, Sendai. (Adv. Stud. Pure Math., vol 10, pp. 283-360) Tokyo: Kinokuniya and Amsterdam: North-Holland 1987 · Zbl 0672.14006
[10] Kawamata, Y.: On the finiteness of generators of the pluri-canonical ring for a three-fold of general type. Am. J. Math.106, 1503-1512 (1984) · Zbl 0587.14027 · doi:10.2307/2374403
[11] Koll?r, J.: Log surfaces of general type; some conjectures. In: L’Aquilla Conference Proceedings. Contemp. Math. (to appear) · Zbl 0860.14014
[12] Koll?r, J.: Shafarevich maps and plurigenera of algebraic varieties. Invent. Math. (to appear) · Zbl 0819.14006
[13] Koll?r, J.: Higher direct images of dualizing sheaves. I. Ann. Math.123, 11-42 (1986); II. Ann. Math.124, 171-202 (1986) · Zbl 0598.14015 · doi:10.2307/1971351
[14] Koll?r, J. et al.: Flips and abundance for algebraic threefolds. Ast?risque (to appear)
[15] Koll?r, J., Miyaoka, Y., Mori, S.: Rational connectedness and boundedness of Fano manifolds. J. Differ. Geom.36, 765-779 (1992) · Zbl 0759.14032
[16] Matsuki, K.: On pluricanonical maps for threefolds of general type. J. Math. Soc. Japan38, 339-359 (1986) · Zbl 0606.14033 · doi:10.2969/jmsj/03820339
[17] Matsusaka, T.: Polarised varieties with a given Hilbert polynomial. Am. J. Math.94, 1027-1077 (1972) · Zbl 0256.14004 · doi:10.2307/2373563
[18] Matsusaka, T.: On polarized normal varieties. I. Nagoya Math. J.104, 175-211 (1986) · Zbl 0595.14033
[19] Milnor, J.: On the Betti numbers of real varieties. Proc. Am. Math. Soc.15, 275-280 (1964) · Zbl 0123.38302 · doi:10.1090/S0002-9939-1964-0161339-9
[20] Oguiso, K.: On polarised Calabi- Yau 3-folds. J. Fac. Sci. Univ. Tokyo38, 395-429 (1991) · Zbl 0766.14034
[21] Reid, M.: Projective morphisms according to Kawamata. University of Warwick (Preprint 1983)
[22] Reider, I.: Vector bundles of rank 2 and linear systems on algebraic surfaces. Ann. Math.127, 309-316 (1988) · Zbl 0663.14010 · doi:10.2307/2007055
[23] Shokurov, V.: The nonvanishing theorem. Izv. Akad. Nauk SSSR, Ser. Mat.49, 635-651 (1985); Math. USSR. Izv.19, 591-604 (1985)
[24] Thom, R.: Sur l’homologie des variet?s alg?briques r?elles. In: Differential and combinatorial topology, pp. 255-265. Princeton: Princeton University Press 1965
[25] Wilson, P.M.H.: On complex algebraic varieties of general type. In: Int. Symp. on Algebraic Geometry. (Symp. Math., vol. 24, pp. 65-74) London New York: Academic Press 1981 · Zbl 0462.14010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.