Second sectional classes of polarized three-folds
type: Item
instance of: scholarly article
title: Second sectional classes of polarized three-folds
zbMATH Open document ID: 1391.14014
author: Yoshiaki Fukuma
published in: Tokyo Journal of Mathematics
publication date: 2018-04-04T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.tjm/1515466836
review text: Let \(X\) be a smooth complex projective variety of dimension \(n\), polarized by an ample line bundle \(L\). When \(L\) is very ample one can consider \(X\) embedded in \(\mathbb P^N\) by \(|L|\) and in these case the class of \(X\), as well as those of its general linear sections (the ranks in Severi's terminology), are well known since longtime and have been largely investigated by many authors. A natural generalization of these invariants to polarized manifolds where \(L\) is merely an ample line bundle is provided by the notion of \(i\)th sectional class of \((X,L)\) (\(i=0, \dots, n\)), defined by the author in terms of other sectional invariants in [\textit{Y. Fukuma}, Riv. Mat. Univ. Parma (N.S.) 6, No. 2, 215--240 (2015; Zbl 1346.14018)]. In fact, it can be defined as \(c_i(X,L) = c_i(J_1(L)) L^{n-i}\), where \(J_1(L)\) is the first jet bundle of \(L\). In the paper under review the author investigates the case \((n,i)=(3,2)\), proving that \(c_2(X,L) \geq L^3-1\) for any polarized threefold \((X,L)\), characterizing the pairs for which equality holds as well as those with \(c_2(X,L)=L^3\), and showing that \(1, 2\) and \(3\) are gap values for \(c_2(X,L)-L^3\). As a consequence he obtains the list of polarized threefolds \((X,L)\) with \(c_2(X,L) \leq 4\). Finally, he classifies the pairs for which \(c_2(X,L) \leq 2L^3\). A key role is played by the classical special varieties of adjunction theory.
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 14C30
Mathematics Subject Classification ID: 14C17
zbMATH DE Number: 6855945
zbMATH Keywords: polarized manifold
zbMATH Keywords: sectional class
zbMATH Keywords: adjunction theory
reviewed by: Antonio Lanteri
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 6855945