Configurations of \(-2\) rational curves on sectional surfaces of \(n\)-folds
type: Item
instance of: scholarly article
title: Configurations of \(-2\) rational curves on sectional surfaces of \(n\)-folds
zbMATH Open document ID: 0583.14003
DOI: 10.1007/BF01458465
published in: Mathematische Annalen
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164151
review text: Let X be an n-dimensional projective manifold. Let L be a line bundle on X which is ample and spanned by global sections. Assume that \(K_ X\otimes L^{n-1}\) is nef and big and that \(\kappa (X,K_ X\otimes L^{n-2})=n.\) It is well-known that there exists a reduction (X',L') of (X,L) such that \(K_{X'}\otimes L'{}^{n-2}\) is semiample. We classify the positive dimensional fibres of \(\phi_ n\), the morphism associated to the linear system \(| (K_{X'}\otimes L'{}^{n-2})^ N|\) for \(N\gg 0\). Such classification gives us information about the maximal connected reduced curves C on S' satisfying \(K_{S'}\cdot C=0\), where S' is the intersection of n-2 general members of \(| L'|\). We see that if dim \(X\geq 4\) then the only possible Dynkin diagram associated to C is \(A_ n\) with \(n=1, 2\).
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 14J15
Mathematics Subject Classification ID: 14H99
zbMATH DE Number: 3933211
zbMATH Keywords: rational curves on sectional surfaces
zbMATH Keywords: ample line bundle
zbMATH Keywords: morphism associated to linear system
zbMATH Keywords: Dynkin diagram
zbMATH Keywords: numerically effective divisor
zbMATH Keywords: Kodaira dimension
author: Maria Lucia Fania
MaRDI profile type: Publication
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OpenAlex ID: W2065380508
description: scientific article; zbMATH DE number 3933211