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The sheaf of relative differentials of a fibred surface

Published online by Cambridge University Press:  24 October 2008

Fernando Serrano
Fernando Serrano
Affiliation:
Departament d'Àlgebra i Geometria, Facultat de Matemàtiques, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain
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Abstract

Let Φ: S → C denote a fibration from a smooth projective surface onto a smooth curve, with fibres of genus ≥2. The double dual of the sheaf of relative differentials has been studied by F. Serrano [14]. There, it was proved that dim grows asymptotically as the square of n in case Φ is not isotrivial (i.e. fibres vary in modulus), and the converse holds true in most cases, in a way that can be made precise. In the non-isotrivial case, the present paper provides further information about by analysing the linear systems for large n. If P denotes the positive part of in its Zariski decomposition, then it is shown that |rP| is eventually base-point free for some r > 0. Furthermore, Proj is a normal projective surface, fibred over C, birational to S, and with only rational singularities.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 114 , Issue 3 , November 1993 , pp. 461 - 470
Copyright
Copyright © Cambridge Philosophical Society 1993

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References

REFERENCES

[1]Arakelov, S.. Families of algebraic curves with fixed degeneracy. In Russian: Izv. Akad. Nauk. SSSR Ser. Mat. 35 (1971), 1269–1293.Google Scholar
Arakelov, S.. Families of algebraic curves with fixed degeneracy. In English: In English: Math. USSR-Izv. 5 (1971), 1277–1302.CrossRefGoogle Scholar
[2]Barth, W., Peters, C. and Van de Ven, A.. Compact Complex Surfaces. Ergeb. Math. Grenzgeb (3) 4 (Springer-Verlag, 1984).CrossRefGoogle Scholar
[3]Beauville, A.. L'inégalité p 0 ≥ 2q−4 pour les surfaces de type general. Bull. Soc.Math. France 110 (1982), 343–346.Google Scholar
[4]Benveniste, X.. On the fixed part of certain linear systems on surfaces. Compositio Math. 51 (1984), 237–242.Google Scholar
[5]Fujita, T.. On the Zariski problem. Proc. Japan Acad., Ser. A. Math. Sci. 55 (1979), 106–110.Google Scholar
[6]Fujita, T.. Semipositive line bundles. J. Fac. Sci. Univ. Tokyo 30 (1983), 353–378.Google Scholar
[7]Kunz, E.. Kähler Differentials. Viehweg Adv. Lect. Math. (Friedr. Vieweg & Sohn, 1986).CrossRefGoogle Scholar
[8]Miyanishi, M. and Tsunoda, S.. Open algebraic surfaces with Kodaira dimension −∞. Alg. Geom. Bowdoin 1985. Proc. Symp. Pure Math. vol. 46, Part 1 (1987), 435–450.CrossRefGoogle Scholar
[9]Reid, M.. Canonical 3-folds. Journées Géom. Alg. d'angers (Sijthoff and Noordhoff, Alphen aan den Rijn (1980)), 273–310.Google Scholar
[10]Reid, M.. Young person' guide to canonical singularities. Alg. Geom. Bowdoin 1985. Proc. Symp. Pure Math. vol. 46, Part 1 (1987), 345–414.CrossRefGoogle Scholar
[11]Sakai, F.. Weil divisors on normal surfaces. Duke Math. J. 51 (1984), 877–887.CrossRefGoogle Scholar
[12]Sakai, F.. Anticanonical models of rational surfaces. Math. Ann. 269 (1984), 389–410.CrossRefGoogle Scholar
[13]Sakai, F.. Ample Cartier divisors on normal surfaces. J. reine angew. Math. 366 (1986), 121–128.Google Scholar
[14]Serrano, F.. Fibred surfaces and moduli. Duke Math. J. 67 (1992), 407–421.CrossRefGoogle Scholar
[15]Sepiro, L.. Propriétés numériques du faisceaux dualisant relatif. Astérisque 86 (1981), 44–78.Google Scholar
[16]Zariski, O.. The theorem of Riemann-Roch for high multiples of an effective divisor on an algebraic surface. Ann. of Math. 76 (1962), 560–615.CrossRefGoogle Scholar