Abstract
Let X be a smooth projective variety of dimension 5 and L be an ample line bundle on X such that \(L^5>7^5\) and \(L^d\cdot Z\ge 7^d\) for any subvariety Z of dimension \(1\le d\le 4\). We show that \(\mathcal {O}_X(K_X+L)\) is globally generated.
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Angehrn, U., Siu, Y.T.: Effective freeness and point separation for adjoint bundles. Invent. Math. 122(2), 291–308 (1995)
Bogomolov, F.A.: Holomorphic symmetric tensors on projective surfaces. Uspekhi Mat. Nauk 33(5(203)), 171–172 (1978)
Bombieri, E.: Canonical models of surfaces of general type. Publications Mathmatiques de l’IHS 42, 171–219 (1973)
Demailly, J.-P.: A numerical criterion for very ample line bundles. J. Differ. Geom. 37(2), 323–374 (1993)
Ein, L.: Multiplier ideals, vanishing theorems and applications. In: Algebraic Geometry—Santa Cruz 1995, vol. 62 of Proceedings of Symposium on Pure Mathematics, pp. 203–219. American Mathematical Society Providence, RI (1997)
Ein, L., Lazarsfeld, R.: Global generation of pluricanonical and adjoint linear series on smooth projective threefolds. J. Am. Math. Soc. 6(4), 875–903 (1993)
Fujino, O., Gongyo, Y.: On canonical bundle formulas and subadjunctions. Mich. Math. J. 61(2), 255–264 (2012)
Fujita, T.: Contribution to birational geometry of algebraic varieties: open problems. In the 23rd International Symposium of the Division of Mathematics of the Taniguchi Foundation, Katata (1988)
Fujita, T.: Remarks on ein-lazarsfeld criterion of spannedness of adjoint bundles of polarized threefolds. alg-geom/9311013 (1993)
Heier, G.: Effective freeness of adjoint line bundles. Doc. Math. 7, 31–42 (2002)
Helmke, S.: On Fujita’s conjecture. Duke Math. J. 88(2), 201–216 (1997)
Helmke, S.: On global generation of adjoint linear systems. Math. Ann. 313(4), 635–652 (1999)
Kawamata, Y.: On the finiteness of generators of a pluricanonical ring for a \(3\)-fold of general type. Am. J. Math. 106(6), 1503–1512 (1984)
Kawamata, Y.: On Fujita’s freeness conjecture for \(3\)-folds and \(4\)-folds. Math. Ann. 308(3), 491–505 (1997)
Kodaira, K.: Pluricanonical systems on algebraic surfaces of general type. J. Math. Soc. Jpn. 20, 170–192 (1968)
Kollár, J.: Effective base point freeness. Math. Ann. 296(4), 595–605 (1993)
Kollár, J.: Singularities of pairs. In: Algebraic geometry–Santa Cruz 1995, vol. 62 of Proceedings of Symposium Pure Mathematics, pp. 221–287, American Mathematical Society, Providence, RI (1997)
Lazarsfeld, R.: Positivity in algebraic geometry. I, volume 48 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer-Verlag, Berlin, 2004. Classical setting: line bundles and linear series
Lee, S.: Remarks on the pluricanonical and the adjoint linear series on projective threefolds. Commun. Algebra 27(9), 4459–4476 (1999)
Reid, M.: Projective morphisms according to Kawamata. Unpublished manuscript (1983)
Reider, I.: Vector bundles of rank 2 and linear systems on algebraic surfaces. Ann. Math. 127(2), 309–316 (1988)
Shokurov, V.V.: A nonvanishing theorem. Izv. Akad. Nauk SSSR Ser. Mat. 49(3), 635–651 (1985)
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Ye, F., Zhu, Z. Global generation of adjoint line bundles on projective 5-folds. manuscripta math. 153, 545–562 (2017). https://doi.org/10.1007/s00229-016-0897-0
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DOI: https://doi.org/10.1007/s00229-016-0897-0