On the adjunction theoretic classification of projective varieties
type: Item
instance of: scholarly article
title: On the adjunction theoretic classification of projective varieties
zbMATH Open document ID: 0712.14029
DOI: 10.1007/BF01459237
published in: Mathematische Annalen
publication date: 1991-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164810
review text: Let \(X^{\wedge}\) be a connected projective submanifold of \({\mathbb{P}}_{{\mathbb{C}}}\) of dimension \(n\geq 3\), and let \(L^{\wedge}={\mathcal O}_{{\mathbb{P}}_{{\mathbb{C}}}}(1)_{X^{\wedge}}\). In this article we give a precise biregular structure theory for those pairs \((X^{\wedge},L^{\wedge})\) with \(n\geq 6\), such that \(| m(K_{X^{\wedge}}+(n-3)L^{\wedge})|\) does not give a birational map for any positive \(m.\) For \(n=3, 4, 5\), we give partial results, i.e. we give a precise biregular structure theory for those pairs \((X^{\wedge},L^{\wedge})\) such that \(| m(K_{X^{\wedge}}+tL^{\wedge})|\) does not give a birational map for any positive m with mt integral and \(t=2/3, 3/2\), and 9/4 for \(n=3, 4, 5\) respectively. In the case of \(n=4\) these results were obtained by \textit{M. L. Fania} and the reviewer [Ark. Mat. 27, No.2, 245-256 (1989; Zbl 0703.14005)]. Some simple applications of these results are given. Let \((X^{\wedge},L^{\wedge})\) be such that \(n\geq 6\), and assume that the degree of \(X^{\wedge}\) embedded by \(| L^{\wedge}|\) is \(>g(L^{\wedge})-1,\) where \(g(L^{\wedge})\) is the genus of a curve section of \(X^{\wedge}\). Then \(| m(K_{X^{\wedge}}+(n- 3)L^{\wedge})|\) does not give a birational map for any positive \(m,\) and we can apply our result to get precise information on the biregular structure for these pairs. Some applications are also made to pairs \((X^{\wedge},L^{\wedge})\) with the discriminant locus, \({\mathcal D}\), not a divisor on \(| L^{\wedge}|\).
Mathematics Subject Classification ID: 14N05
Mathematics Subject Classification ID: 14M99
zbMATH DE Number: 4171104
zbMATH Keywords: adjunction
zbMATH Keywords: classification of projective varieties
author: Mauro C. Beltrametti
author: Maria Lucia Fania
author: Andrew J. Sommese
MaRDI profile type: Publication
cites work: On ample divisors
cites work: Projective n-Folds of Log-General Type. I
cites work: On the discriminant variety of a projective manifold
cites work: Varieties with small dual varieties. I
cites work: Varieties with small dual varieties. II
cites work: Configurations of \(-2\) rational curves on sectional surfaces of \(n\)-folds
cites work: Q3726257
cites work: On the projective classification of smooth n-folds with n even
cites work: Q5185171
cites work: Remarks on quasi-polarized varieties
cites work: Q3995221
cites work: Q3328651
cites work: Q3825176
cites work: Q3789686
cites work: Hyperplane sections of projective surfaces. I: The adjunction mapping
cites work: On the minimality of hyperplane sections of projective threefolds.
cites work: Q3674815
cites work: On the nonemptiness of the adjoint linear system of a hyperplane section of a threefold.
cites work: On the adjunction mapping
cites work: The theorem of Riemann-Roch for high multiples of an effective divisor on an algebraic surface
description: scientific article; zbMATH DE number 4171104