Divisors on the blow-up of the projective plane
type: Item
instance of: scholarly article
title: Divisors on the blow-up of the projective plane
zbMATH Open document ID: 0836.14004
DOI: 10.1007/BF02567988
author: Geng Xu
published in: Manuscripta Mathematica
publication date: 1996-05-02T00:00:00Z
full work available at URL: https://eudml.org/doc/156048
review text: Let \(\mathbb{P}^2\) be the complex projective plane over the field of complex numbers, and \(\pi : S \to \mathbb{P}^2\) be the blow-up of \(\mathbb{P}^2\) at \(k\) generic points \(P_1, P_2, \dots, P_k\) with exceptional divisors \(E_1, E_2, \dots, E_k\). Then the Picard group of \(S\) is generated by \(H = \pi^* {\mathcal O} (1)\), \(E_1, \dots, E_k\). The author shows that if \(k\) and \(p > 2\) are positive integers, then the divisor \(pH - \sum^k_{i = 1} E_i\) is ample on \(S\) if and only if \(p^2 > k\). This gives an affirmative answer to a question of \textit{T. Fujita}.
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 14C22
zbMATH DE Number: 733201
zbMATH Keywords: blow-up of the projective plane
zbMATH Keywords: ampleness of divisor
zbMATH Keywords: Picard group
MaRDI profile type: Publication
cites work: Symplectic packings and algebraic geometry
cites work: Curves in \(P^ 2\) and symplectic packings
OpenAlex ID: W2010847190
description: scientific article; zbMATH DE number 733201