The fundamental group of the complement of a resultant hypersurface.
type: Item
instance of: scholarly article
title: The fundamental group of the complement of a resultant hypersurface.
zbMATH Open document ID: 1074.14017
author: Ichiro Shimada
published in: Pacific Journal of Mathematics
publication date: 2004-09-16T00:00:00Z
review text: Let \(X\) be a smooth irreducible complex projective variety of dimension \(n\), and consider \(n+1\) very ample line bundles \(L_0, \ldots, L_n\) on \(X\). Let \(V:=\bigoplus_i H^0(L_i)\). The resultant variety \(R \subset V\) is the locus of the elements \((f_0,\ldots,f_n)\) such that the \(f_i\)' s have common zeroes. It is known [\textit{I. M. Gelfand, M. M. Kapranov, A. V. Zelevinsky}, ``Discriminants, resultants, and multidimensional determinants'' (1994; Zbl 0827.14036)] that \(R\) is an irreducible hypersurface (called the resultant hypersurface). In this paper it is proved that the fundamental group of the complement of a resultant hypersurface is isomorphic to the group of the integers \(\mathbb{Z}\).
reviewed by: Roberto Pignatelli
Mathematics Subject Classification ID: 14F35
Mathematics Subject Classification ID: 14J70
zbMATH DE Number: 2101093
MaRDI profile type: Publication
full work available at URL: pjm.2003.210.351
OpenAlex ID: W2054613839
DOI: 10.2140/PJM.2003.210.351
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description: scientific article; zbMATH DE number 2101093