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Pluricanonical systems for 3-folds and 4-folds of general type

Published online by Cambridge University Press:  19 October 2011

LORENZO DI BIAGIO
LORENZO DI BIAGIO*
Affiliation:
Dip. di Matematica, Sapienza - Università di Roma, Piazzale Aldo Moro 5, 00185, Roma, Italy. e-mail: dibiagio@mat.uniroma1.it
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Abstract

We explicitly find lower bounds on the volume of threefolds and fourfolds of general type in order to have non-vanishing of pluricanonical systems and birationality of pluricanonical maps. In the case of threefolds of large volume, we also give necessary and sufficient conditions for the fourth canonical map to be birational.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 152 , Issue 1 , January 2012 , pp. 9 - 34
Copyright
Copyright © Cambridge Philosophical Society 2011

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References

REFERENCES

[1]Angehrn, U. and Siu, Y. T.Effective freeness and point separation for adjoint bundles. Invent. Math., 122 (1995), pp. 291–308.CrossRefGoogle Scholar
[2]Barth, W. P., Hulek, K., Peters, C. A. M. and Van de Ven, A. Compact complex surfaces, vol. 4 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, second ed., 2004).CrossRefGoogle Scholar
[3]Bauer, I. C., Catanese, F. and Pignatelli, R. Complex surfaces of general type: some recent progress, in Global Aspects of Complex Geometry, (Springer, 2006), pp. 1–58.Google Scholar
[4]Bombieri, E. Canonical models of surfaces of general type. Inst. Hautes Études Sci. Publ. Math. (1973), pp. 171–219.Google Scholar
[5]Borrelli, G.The classification of surfaces of general type with nonbirational bicanonical map. J. Algebraic Geom. 16 (2007), pp. 625–669.CrossRefGoogle Scholar
[6]Broustet, A.Non-annulation effective et positivité locale des fibrés en droites amples adjoints. Math. Ann. 343 (2009), pp. 727–755.Google Scholar
[7]Chen, J. A. and Chen, M.Explicit birational geometry of 3-folds of general type, II. J. Differential Geom. 86 (2010), pp. 237–271.CrossRefGoogle Scholar
[8]Chen, M. and Zhang, D.-Q.Characterization of the 4-canonical birationality of algebraic threefolds. Math. Z., 258 (2008), pp. 565–585.CrossRefGoogle Scholar
[9]Debarre, O.Higher-Dimensional Algebraic Geometry. Universitext (Springer-Verlag, 2001).CrossRefGoogle Scholar
[10]Di Biagio, L. Pluricanonical systems for 3-folds, 4-folds and n-folds of general type. 2010. Ph.D. Thesis Sapienza Università di Roma. Available at http://www.mat.uniroma1.it/ricerca/dottorato/TESI/ARCHIVIO/dibiagiolorenzo.pdf.CrossRefGoogle Scholar
[11]Dong, J.-q.The 4-canonical maps of 3-folds of general type. Northeast. Math. J. 22 (2006), pp. 119–126.Google Scholar
[12]Ein, L., Lazarsfeld, R., Mustaţă, M., Nakamaye, M. and Popa, M.Asymptotic invariants of base loci. Ann. Inst. Fourier (Grenoble) 56 (2006), pp. 1701–1734.Google Scholar
[13]Hacon, C. D. and McKernan, J.Boundedness of pluricanonical maps of varieties of general type. Invent. Math. 166 (2006), pp. 1–25.Google Scholar
[14]Kawamata, Y.On the extension problem of pluricanonical forms, in Algebraic geometry: Hirzebruch 70 (Warsaw, 1998), vol. 241 of Contemp. Math., Amer. Math. Soc. (Providence, 1999), pp. 193–207.Google Scholar
[15]Kollár, J.Kodaira's canonical bundle formula and adjunction, in Flips for 3-folds and 4-folds, vol. 35 of Oxford Lecture Ser. Math. Appl. (Oxford University Press, 2007), pp. 134–162.Google Scholar
[16]Kollár, J. and Mori, S.Birational geometry of algebraic varieties, vol. 134 of Cambridge Tracts in Mathematics, (Cambridge University Press, 1998). With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original.CrossRefGoogle Scholar
[17]Lazarsfeld, R.Positivity in algebraic geometry. I, vol. 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, 2004). Classical setting: line bundles and linear series.Google Scholar
[18]Lazarsfeld, R.Positivity in algebraic geometry. II, vol. 49 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, 2004). Positivity for vector bundles, and multiplier ideals.Google Scholar
[19]McKernan, J. Boundedness of log terminal Fano pairs of bounded index. arXiv:math/0205214v1, (2002). Available at http://arxiv.org/abs/math/0205214.Google Scholar
[20]Takayama, S.Pluricanonical systems on algebraic varieties of general type. Invent. Math. 165 (2006), pp. 551–587.Google Scholar
[21]Todorov, G. T.Pluricanonical maps for threefolds of general type. Ann. Inst. Fourier (Grenoble) 57 (2007), pp. 1315–1330.Google Scholar