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Geometriae Dedicata

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

A note on adjoint linear systems

  • Mirel Caibăr
Original Paper
  • 97 Downloads

Abstract

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.

Keywords

Linear systems Singularities Mixed Hodge structures 

Mathematics Subject Classification (2000)

14C20 14E30 32S35 

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References

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    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
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    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
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    Ramanujam C.P.: Remarks on the Kodaira vanishing theorem. J. Indian Math. Soc. N.S. 36, 41–51 (1972)zbMATHMathSciNetGoogle Scholar
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    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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