Curves with low Harbourne constants on Kummer and abelian surfaces
type: Item
instance of: scholarly article
title: Curves with low Harbourne constants on Kummer and abelian surfaces
zbMATH Open document ID: 1404.14044
author: Xavier Roulleau
publication date: 2018-12-12T00:00:00Z
full work available at URL: https://arxiv.org/abs/1711.06449
review text: In the paper under review, the author studies the so-called Harbourne constants which measure the local negativity property. Let us recall that for a smooth complex projective surface \(X\) it is conjectured that there exists an integer \(b(X)\) such that for all reduced curves (not necessarily irreducible) one has \(C^2\geq -b(X)\). This conjecture is called the bounded negativity conjecture and it is widely open. In the paper under review, the author presents interesting curve configurations in abelian or Kummer surfaces having low Harbourne constants. In order to formulate the main results, let us recall the notion of mentioned Harbourne constants. Definition. Let \(X\) be a smooth surface and \(C \subset X\) a reduced curve. Let \(\mathcal{P} \subset X\) be a finite non-empty set of points and denote by \(\pi: \tilde{X} \rightarrow X\) be the blowing up along \(\mathcal{P}\). The number \[ H(X,\mathcal{P}) = \frac{ \tilde{C}^{2}}{\# \mathcal{P}}, \] where \(\tilde{C}\) is the strict transform of \(C\), is called the local Harbourne constant. Then the Harbourne constant of \(C\) is defined as \[ H(C) = \text{inf}_{\mathcal{P}} H(C,\mathcal{P}), \] where \(\mathcal{P}\) varies among all finite non-empty subsets of points in \(X\). The global Harbourne constant of \(X\) is defined by \[ H(X) = \text{inf}_{C} H(C), \] where the infimum is taken over all reduced curves \(C \subset X\). Theorem A. Let \(A\) be a simple complex abelian surface. There exists a sequence of curves \(\{R_{n}\}_{n \in \mathbb{N}}\) such that \(H(R_{n}) = - 4\). If \(A\) is the Jacobian of a smooth genus \(2\) curve, then \(R_{1}\) can be chosen either as the union of \(16\) specific smooth curves or as a single irreducible curve. It is worth emphasizing that curves \(R_{n}\) are constructed with use of the multiplication map \(n \in \mathbb{Z}\). Theorem B. Let \(X\) be a Jacobian-Kummer complex surface. For any \(n>1\), there are configurations \(C_{n}\) of genus \(g > 1\) curves such that…
reviewed by: Piotr Pokora
Mathematics Subject Classification ID: 14J28
zbMATH DE Number: 6992783
zbMATH Keywords: bounded negativity conjecture
zbMATH Keywords: Harbourne constants
zbMATH Keywords: abelian surfaces
zbMATH Keywords: Kummer surfaces
zbMATH Keywords: Kummer configuration
MaRDI profile type: Publication
OpenAlex ID: W2963824060
arXiv ID: 1711.06449
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DOI: 10.1007/S12215-017-0328-4
published in: Rendiconti del Circolo Matematico di Palermo
description: scientific article