On the local negativity of surfaces with numerically trivial canonical class
type: Item
instance of: scholarly article
title: On the local negativity of surfaces with numerically trivial canonical class
zbMATH Open document ID: 1395.14004
DOI: 10.4171/RLM/803
author: Roberto Laface
author: Piotr Pokora
published in: Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Serie IX. Rendiconti Lincei. Matematica e Applicazioni
publication date: 2018-07-02T00:00:00Z
review text: Let \(X\) be a smooth complex projective surface. One of the most challenging problems in the study of negative curves on \(X\) is the bounded negativity conjecture, which claims that there exists an integer \(b(X)\) such that \(C^2 \geq - b(X)\) for every reduced curve \(C\) on \(X\). The number \(b(X)\) can be arbitrarily large depending on \(X\), as one can easily see by taking the blow-up of \(\mathbb P^2\) at \(s>>0\) distinct points. To approach the problem more effectively, the authors look at the so-called local Harbourne constant, a variation of the Harbourne constant considered by \textit{T. Bauer} et al. [Int. Math. Res. Not. 2015, No. 19, 9456--9471 (2015; Zbl 1330.14007)]. Given a configuration \(C: = \sum_{i=1}^n C_i\), where \(C_i\) are \(n\) mutually transverse smooth irreducible curves on \(X\), giving rise to \(s\) singular points, consider the blow-up \(Y \to X\) at these points, and let \(\widetilde{C}=\sum_{i=1}^n \widetilde{C_i}\), where \(\widetilde{C_i}\) is the proper transform of \(C_i\) in \(Y\). The local Harbourne constant of \(C \subset X\) is the rational number \(h(C):= {\widetilde{C}^2}/{s}\). The authors focus on the case of configurations \(C\) of \(n\) smooth rational curves with \(s\) singular points on surfaces \(X\) having numerically trivial canonical bundle. In this setting they prove that \(h(C) \geq - 4 + \frac{1}{ s}\big(2n+t-3c_2(X)\big)\), where \(t\) is the number of double points among the \(s\) singular points of \(C\). Then they produce several interesting examples of configurations \(C\) of smooth rational curves on \(K3\) and Enriques surfaces, showing that the bound above is not far from the precise value of \(h(C)\) they compute explicitly.
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 14J70
Mathematics Subject Classification ID: 14J28
zbMATH DE Number: 6898066
zbMATH Keywords: rational curve configuration
zbMATH Keywords: algebraic surface
zbMATH Keywords: Miyaoka inequality
zbMATH Keywords: negative curve
zbMATH Keywords: bounded negativity conjecture
zbMATH Keywords: Harbourne constant
reviewed by: Antonio Lanteri
MaRDI profile type: Publication
Wikidata QID: Q129920199
description: scientific article; zbMATH DE number 6898066