Adjunction and singular loci of hyperplane sections. II
type: Item
instance of: scholarly article
title: Adjunction and singular loci of hyperplane sections. II
zbMATH Open document ID: 1371.14009
publication date: 2014-10-02T00:00:00Z
review text: Let \((X,L)\) be a smooth complex polarized variety of dimension \(n\) such that there exists an effective irreducible divisor \(A\in |L|\), and let \(\Sigma:=\text{Sing}(A)\) be the singular locus of \(A\) with \(k=\dim \Sigma\) and \(c=\text{codim}_{X}(\Sigma)\). In [\textit{M. C. Beltrametti} et al., J. Math. Soc. Japan 67, No. 2, 861--875 (2015; Zbl 1319.14007)], the authors investigated positivity conditions for adjoint bundles \(K_{X}+tL\) with \(t\geq n-3\) under the assumption that (1) \(\Sigma\) consists of non-degenerate quadric singularities and (2) \(\Sigma\) is smooth. More precisely, the authors studied the following: (a) The existence of the first reduction of \((X,L)\) under the assumption that \(k\geq 2\) and \(c\geq 3\). (b) The existence of the second reduction of \((X,L)\) and the nefness and bigness of the third adjoint bundle under the assumption that \(k\geq 3\) and odd codimension \(c\geq 3\). In this paper, the authors point out that the same analysis also works without the above assumption (1) by using a result of [\textit{P. Aluffi}, Duke Math. J. 80, 325--351 (1995; Zbl 0876.14028)]. In any case, however, the assumption that \(K_{\Sigma}+(k-2)L_{\Sigma}\) is nef is crucial when we study (b) above. So in this paper, the authors study the positivity of adjoint bundles on \(\Sigma\) with odd codimension \(c\), and they describe all \((\Sigma, L_{\Sigma})\) which satisfy one of the following types: (i) \(k\geq 2\) and \(K_{\Sigma}+(k-1)L_{\Sigma}\) is not ample. (ii) \(k\geq 3\), \(K_{\Sigma}+(k-1)L_{\Sigma}\) is ample but \(K_{\Sigma}+(k-2)L_{\Sigma}\) is not nef. (iii) \(k\geq 3\), \(K_{\Sigma}+(k-2)L_{\Sigma}\) is nef but not big. Moreover the authors give several explicit examples, and these show that the lists are effective.
reviewed by: Yoshiaki Fukuma
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 14J17
Mathematics Subject Classification ID: 14J40
Mathematics Subject Classification ID: 14N30
zbMATH DE Number: 6351278
zbMATH Keywords: adjunction theory
zbMATH Keywords: special varieties
zbMATH Keywords: non-degenerate quadratic singularities
author: Mauro C. Beltrametti
author: Antonio Lanteri
author: Andrew J. Sommese
MaRDI profile type: Publication
full work available at URL: s12215-014-0155-9
OpenAlex ID: W2084578753
cites work: Singular schemes of hypersurfaces
cites work: The adjunction theory of complex projective varieties
cites work: Varieties with small dual varieties. I
cites work: Q3995221
cites work: Q3953893
DOI: 10.1007/S12215-014-0155-9
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published in: Rendiconti del Circolo Matematico di Palermo
description: scientific article; zbMATH DE number 6351278