Asymptotic behaviour of numerical invariants of algebraic varieties
type: Item
instance of: scholarly article
title: Asymptotic behaviour of numerical invariants of algebraic varieties
zbMATH Open document ID: 1250.14007
DOI: 10.4171/JEMS/301
author: F. L. Zak
published in: Journal of the European Mathematical Society (JEMS)
publication date: 2012-01-09T00:00:00Z
review text: When classifying projective varieties, a first approach is to fix certain numerical invariants and then study the set of varieties that have these numerical invariants. Typical such invariants are Chern and Hodge numbers. In this paper, the author addresses the following three questions, for a given numerical invariant: 1) Find upper bounds for the invariant in terms of dimension, codimension and degree of the variety; 2) describe the varieties for which the invariant attains maximal value; 3) study the relations between various numerical invariants. In the case of curves, Castelnuovo answered 1) and 2) for the genus, and this was generalized to higher dimensional varieties for the geometric genus by \textit{J. Harris} [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 8, 35--68 (1981; Zbl 0467.14005)]. For 3), work was done by \textit{U. Persson} [in: Algebraic geometry, Proc. Summer Res. Inst., Brunswick/Maine 1985, part 1, Proc. Symp. Pure Math. 46, 195--218 (1987; Zbl 0656.14020)] on the ``geography'' of surfaces. In the present paper the author solves what he calls the ``asymptotic version'' of the above three questions, with respect to Chern and Hodge numbers. More precisely, he shows that for varieties of high degree, those for which any one of the most important invariants -- such as class, Betti number, Chern number, middle Hodge number -- is maximal, leads to the same class of extremal varieties. Typically, as in Castelnuovo's case of curves, these varieties are hypersurfaces in a variety of minimal degree in the ambient projective space. The results in this paper builds on, and extends, some of those in [\textit{F. L. Zak}, Castelnuovo bounds for higher dimensional varieties. Compos. Math. 148 (2012) (to appear)].
reviewed by: Ragni Piene
Mathematics Subject Classification ID: 14C17
Mathematics Subject Classification ID: 14C30
Mathematics Subject Classification ID: 14F25
Mathematics Subject Classification ID: 14F45
Mathematics Subject Classification ID: 14J40
Mathematics Subject Classification ID: 14M99
Mathematics Subject Classification ID: 14N05
Mathematics Subject Classification ID: 14N15
Mathematics Subject Classification ID: 14N25
Mathematics Subject Classification ID: 19L10
Mathematics Subject Classification ID: 32Q55
zbMATH DE Number: 5995803
zbMATH Keywords: asymptotic bound
zbMATH Keywords: Castelnuovo theory
zbMATH Keywords: Betti number
zbMATH Keywords: Hodge number
zbMATH Keywords: Chern number
zbMATH Keywords: variety of minimal degree
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.4171/jems/301
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description: scientific article; zbMATH DE number 5995803