Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-26T09:41:44.907Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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