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On the complement of a nef and big divisor on an algebraic variety

Published online by Cambridge University Press:  24 October 2008

Francesco Russo
Francesco Russo
Affiliation:
Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, 70700 Bucharest, Romania e-mail address: frusso@imar.ro
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Extract

Let X be an algebraic (complete) variety over a fixed algebraically closed field k. To every Cartier divisor D on X, we can associate the graded k-algebra . As is known, for a semi-ample divisor D, R(X, D) is a finitely generated k-algebra (see [21] or [9]), while this property is no longer true for arbitrary nef and big divisors (see [21]).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 120 , Issue 3 , October 1996 , pp. 411 - 422
Copyright
Copyright © Cambridge Philosophical Society 1996

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References

REFERENCES

[1]Artin, M.. Some numerical criteria for contractibility of curves on an algebraic surface. Amer. J. Math. 84 (1962), 485–492CrossRefGoogle Scholar
[2]Artin, M.. Algebraization of formal moduli: existence of modifications. Ann. of Math. 91 (1970), 88–136.CrossRefGoogle Scholar
[3]Baˇdescu, L.. Anticanonical models of ruled surfaces. Ann. Univ. Ferrara 29 (1983), 165–177.CrossRefGoogle Scholar
[4]Baˇdescu, L.. Algebra graduata asociata unui divisor pe o suprafaţâ neteda şi proiectiva, in Analiza complexa: aspecte clasice şi moderne, Editura Stiintifica şi enciclopedica Bucureşti, (1988), 295–337.Google Scholar
[5]Benveniste, X.. On the fixed part of certain linear systems on surfaces. Comp. Math. 51 (1984), 237–242.Google Scholar
[6]Borelli, M.. Affine complements of divisors. Pacif. J. Math. 31 (1969), 595–607.CrossRefGoogle Scholar
[7]Constantinescu, A.. Some remarks on the finite generation of graded subalgebras, to appear in Ann. Univ. Ferrara.Google Scholar
[8]Cutkosky, S. D. and Srinivas, V.. On a problem of Zariski on dimensions of linear systems. Ann. of Math. 137 (1993), 531–559.CrossRefGoogle Scholar
[9]Fujita, T.. Semipositive line bundles. J. Fac. Sci. Univ. Tokyo 30 (1983), 353–378.Google Scholar
[10]Goodman, J. E.. Affine open subsets of algebraic varieties and ample divisors. Ann. of Math. 89 (1969), 160–183.CrossRefGoogle Scholar
[11]Goodman, J. E. and Landman, A.. Varieties which are proper over an affine scheme. Inv. Math. 20 (1973), 267–312.CrossRefGoogle Scholar
[12]Grauert, H.. Uber modificazionen und exzeptionelle analytische Mengen. Math. Ann. 146 (1962), 331–368.CrossRefGoogle Scholar
[13]Hartshorne, R.. Ample subvarieties of algebraic varieties. Springer-Verlag, Lecture Notes Math. 156, 1970.CrossRefGoogle Scholar
[14]Iitaka, S.. Birational geometry of open varieties. (Press. Univ. Montreal), Sem. de Mat. Sup. 76, (1981).Google Scholar
[15]Mumford, D.. Hilbert's fourteenth problem - the finite generation of subrings such as rings of invariants. Proc. Symp. Pure Math. 28, Amer. Math. Soc., 1976, 431–44.CrossRefGoogle Scholar
[16]Nagata, M.. Imbedding of an abstract variety in a complete variety. J. Math. Kyoto Univ. 2 (1962), 1–10.Google Scholar
[17]Nagata, M.. A generalization of the imbedding problem of an abstract variety in a complete variety. J. Math. Kyoto Univ. 3 (1963), 89–102.Google Scholar
[18]Sakai, F.. Weil divisors on normal surfaces. Duke Math. J. 51 (1984), 877–888.CrossRefGoogle Scholar
[19]Wilson, P. M. H.. On the canonical ring of algebraic varieties. Comp. Math. 83 (1981), 365–385.Google Scholar
[20]Zariski, O.. Interpretation algebro-geometrique du quatorzieme problème de Hilbert. Bull. des Sciences Math. 78 (1954), 155–164.Google Scholar
[21]Zariski, O.. The theorem of Riemann-Roch for high multiples of an effective divisor on an algebraic surface. Ann. of Math. 16 (1962), 560–615.CrossRefGoogle Scholar
[22]Zariski, O. and Samuel, P.. Commutative algebra II (Springer Verlag, 1976).Google Scholar
[23]Zariski, O.. Applicazioni geometriche della teoria delle valuazioni. Rend. Mat. e Appl., serie V, 13 (1954), 1–37 (or Collected Papers, M.I.T. Press, 1973).Google Scholar