Geometriae Dedicata

, Volume 153, Issue 1, pp 125–130 | Cite as

A note on adjoint linear systems

  • Mirel Caibăr
Original Paper


In dimension 2 there is a very useful criterion, due to Miles Reid, giving a necessary and sufficient condition for an adjoint linear system on a nonsingular complex projective surface X to be free at a point \({p\in X}\) , in terms of the coherent cohomology of certain divisors on the blowup of X at p. In this note we extend Miles Reid’s result to higher dimensions.


Linear systems Singularities Mixed Hodge structures 

Mathematics Subject Classification (2000)

14C20 14E30 32S35 


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  1. 1.
    Barth, W.P., Hulek, K., Peters, C.A.M., Van de Ven, A.: Compact Complex Surfaces, second edition, pp. xii +  436. Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer-Verlag, Berlin (2004)Google Scholar
  2. 2.
    Francia, P.: On the base points of the bicanonical system, Problems in the theory of surfaces and their classification (Cortona, 1988), Symposia Mathematica, XXXII, pp. 141–150. Academic Press, London (1991)Google Scholar
  3. 3.
    Ramanujam C.P.: Remarks on the Kodaira vanishing theorem. J. Indian Math. Soc. N.S. 36, 41–51 (1972)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Reid, M.: Chapters on algebraic surfaces. In: Kollár, J. (ed.) Complex Algebraic Geometry, IAS/Park City Math. Ser. 3, Amer. Math. Soc., pp. 3–159. Providence, RI, (1997)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Department of MathematicsThe Ohio State UniversityMansfieldUSA

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