On a conjecture of W. Veys
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.
scientific article; zbMATH DE number 1488551