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A bound on certain local cohomology modules and application to ample divisors

Published online by Cambridge University Press:  22 January 2016

Claudia Albertini  and
Markus Brodmann
Claudia Albertini
Affiliation:
Kantonsschule Zuercher Oberland, Fachkreis Mathematik, Fachkreis Mathematik 8620 Wetzikon, Switzerland
Markus Brodmann
Affiliation:
Institute of Mathematics, University of Zurich, Winterthurerstrasse 190, 8057 Züurich, Switzerland, [email protected]
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Abstract

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We consider a positively graded noetherian domain R = ⊕n∈NoRn for which R0 is essentially of finite type over a perfect field K of positive characteristic and we assume that the generic fibre of the natural morphism π: Y = Proj(R) → Y0 = Spec(R0) is geometrically connected, geometrically normal and of dimension > 1. Then we give bounds on the “ranks” of the n-th homogeneous part H2(R)n of the second local cohomology module of R with respect to R+:= ⊕m>0Rm for n < 0. If Y is in addition normal, we shall see that the R0-modules H2R+ (R)n are torsion-free for all n < 0 and in this case our bounds on the ranks furnish a vanishing result. From these results we get bounds on the first cohomology of ample invertible sheaves in positive characteristic.

Keywords

13D0314B1514F1514F05
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 163 , September 2001 , pp. 87 - 106
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2000

References

[A] Albertini, C., Schranken für die Kohomologie ampler Divisoren über normalen projektiven Varietäten in positiver Charakteristik, Dissertation, Universität Zürich, 1996.Google Scholar
[B1] Brodmann, M., Bounds on the cohomological Hilbert functions of a projective variety, Journal of Algebra, 109 (1987), 352–380.CrossRefGoogle Scholar
[B2] Brodmann, M., Cohomology of certain projective surfaces with low sectional genus and degree, Commutative Algebra, Algebraic Geometry and Computational Methods (Eisenbud, D., ed.), Springer, New York (1999), pp. 173–200.Google Scholar
[B3] Brodmann, M., Cohomology of surfaces X ⊆ Pr with degree ≤ 2r - 2, Commutative Algebra and Algebraic Geometry (van Oystaeyen, F., ed.), M. Dekker Lecture Notes 206, M. Dekker (1999), pp. 15–33.Google Scholar
[B-N] Brodmann, M. and Nagel, U., Bounding cohomological Hilbert functions by hyperplane sections, Journal of Algebra, 174 (1995), 323–348.Google Scholar
[B-S] Brodmann, M. and Sharp, R. Y., Local cohomology - An algebraic introduction with geometric applications, Cambridge Studies in Advanced Mathematics 60, Cambridge University Press, 1998.Google Scholar
[B-V] Brodmann, M. and Vogel, W., Bounds for the cohomology and the Calstelnuovo regularity of certain surfaces, Nagoya Math. Journal, 131 (1993), 109–126.Google Scholar
[D-I] Deligne, P. and Illusie, L., Relèvements modulo p2 et décomposition du complexe de Rham, Invent. Math., 89 (1987), 247–270.Google Scholar
[E-V] Esnault, H. and Viehweg, E., Lectures on vanishing theorems, DMV Seminar, Birkhäuser, Basel, 1992.Google Scholar
[F] Fujita, , Classification theories of polarized varieties, LMS lecture notes 155, Cambridge University Press, 1990.Google Scholar
[G] Grothendieck, A., Eléments de géometrie algébrique III, Publ. Math. IHES 11, 1961.Google Scholar
[H1] Hartshorne, R., Ample subvarieties of algebraic varieties, Lecture Notes in Mathematics 156, Springer, Berlin, 1970.Google Scholar
[H2] Hartshorne, R., Algebraic geometry, Graduate Text in Mathematics 52, Springer, New York, 1977.Google Scholar
[K] Kodaira, K., On a differential geometric method in the theory of analytic stacks, Proc. Nat. Acad. Sci. USA, 39 (1953), 1268–1273.Google Scholar
[Ku] Kunz, E., Characterizations of regular local rings of characteristic p, American Journal of Mathematics, 91 (1969), 772–784.Google Scholar
[L-R] Lauritzen, N. and Rao, A. P., Elementary counterexamples to Kodaira vanishing in prime characteristic, Proc. Indian Acad. Sci. Math. Sci., 107 (1997), no. 1, 21–25.Google Scholar
[M] Matsumura, H., Commutative ring theory, Cambridge Studies in Advanced Mathematics 8, Cambridge University Press, 1989.Google Scholar
[Me-Ram] Mehta, V. B. and Ramanathan, A., Frobenius splitting and cohomology vanishing for Schubert varieties, Ann. Math., 122 (1985), 27–40.Google Scholar
[Mu] Mumford, D., Pathologies III, American Journal of Mathematics, 89 (1967), 96–104.CrossRefGoogle Scholar
[Ra] Raynaud, M., Contre-example au “vanishing theorem” en characteristique p > 0, C. P. Ramanujam – a tribute, TIFR Studies in Mathematics 8, Springer, Oxford (1978), pp. 273–278.Google Scholar
[S] Smith, K., Fujita’ freeness conjecture in terms of local cohomology, Journal of Algebraic Geometry, 6 (1997), 417–429.Google Scholar