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Castelnuovo bounds for higher-dimensional varieties

Part of: Projective and enumerative geometry (Co)homology theory Analytic and descriptive geometry Special varieties Differential topology Surfaces and higher-dimensional varieties Complex manifolds

Published online by Cambridge University Press:  09 July 2012

F. L. Zak
F. L. Zak*
Affiliation:
CEMI (Central Economics and Mathematics Institute of the Russian Academy of Sciences), Nakhimovskiĭ av. 47, Moscow 117418, Russia (email: zak@mccme.ru)
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Abstract

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We give bounds for the Betti numbers of projective algebraic varieties in terms of their classes (degrees of dual varieties of successive hyperplane sections). We also give bounds for classes in terms of ramification volumes (mixed ramification degrees), sectional genus and, eventually, in terms of dimension, codimension and degree. For varieties whose degree is large with respect to codimension, we give sharp bounds for the above invariants and classify the varieties on the boundary, thus obtaining a generalization of Castelnuovo’s theory for curves to varieties of higher dimension.

Keywords

Castelnuovo bounddual varietyclassramificationBetti numbervariety of minimal degreeLefschetz theory

MSC classification

Secondary: 14F25: Classical real and complex (co)homology 14F45: Topological properties 14J40: $n$-folds ($n>4$) 14M99: None of the above, but in this section 14N15: Classical problems, Schubert calculus 14P25: Topology of real algebraic varieties 32Q55: Topological aspects of complex manifolds 51N15: Projective analytic geometry 51N35: Questions of classical algebraic geometry 57R19: Algebraic topology on manifolds 57R40: Embeddings
Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 4 , July 2012 , pp. 1085 - 1132
Copyright
Copyright © Foundation Compositio Mathematica 2012

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