On a conjecture of W. Veys
type: Item
instance of: scholarly article
title: On a conjecture of W. Veys
zbMATH Open document ID: 0969.14030
author: A. Melle-Hernández
author: Enrique Artal Bartolo
author: Ignacio Luengo
published in: Mathematische Annalen
publication date: 2000-11-22T00:00:00Z
full work available at URL: 2000Veys-Conj2.pdf
review text: The paper is dedicated to disproving the following conjecture, stated by \textit{W. Veys} [Math. Ann. 312, No. 3, 527-548 (1998; Zbl 0913.14009)]: Let \(C_i\), \(1\leq i\leq r\), be irreducible hypersurfaces in \({\mathbb{P}}^n({\mathbb{C}})\). If \((-1)^n\chi ({\mathbb{P}}^n-\bigcup ^r_{i=1}C_i) \leq 0\), then all \(C_i\) have Kodaira dimension \(-\infty \). The conjecture is true for \(n=2\), but in this article a surface in \({\mathbb{P}}^3\) which disproves it is constructed. The idea is as follows: Consider the sextic plane curve \(C\) given by the union of two cubics with one node each which meet in a point with a common tangent; let \(X\) be the double covering of \({\mathbb{P}}^2\) branched along \(C\), and \(\widetilde X\) its canonical resolution. \(\widetilde X\) is a \(K3\) surface, with \(\kappa(\widetilde X)=0\), hence also \(\kappa(X)=0\). In the paper the equation of a sextic surface \(S\) in \({\mathbb{P}}^3\) is given, which is birational to \(X\) and such that \(\chi ({\mathbb{P}}^3-S) = 0\). The authors ask which could be the bound for \(\kappa(C_i)\) under the topological hypothesis of the conjecture.
reviewed by: Alessandro Gimigliano
Mathematics Subject Classification ID: 14J70
Mathematics Subject Classification ID: 14J28
zbMATH DE Number: 1488551
zbMATH Keywords: \(K3\) surface
zbMATH Keywords: Kodaira dimension
zbMATH Keywords: Euler characteristic
Wikidata QID: Q122957382
MaRDI profile type: Publication
OpenAlex ID: W2084422924
DOI: 10.1007/S002080000099
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description: scientific article; zbMATH DE number 1488551