On blowing-up of polarized surfaces
type: Item
instance of: scholarly article
title: On blowing-up of polarized surfaces
zbMATH Open document ID: 0968.14003
author: Koji Yokoyama
published in: Journal of the Mathematical Society of Japan
publication date: 2001-09-13T00:00:00Z
review text: Let \(X\) be a smooth projective variety defined over the complex number field, and let \(L\) be an ample line bundle on \(X\). Let \(p_{1},\dots , p_{r}\) be \(r\) points on \(X\) and let \(\pi : X''\to X\) be the blowing up at these \(r\) points. We put \(L''=\pi^{*}(L)-\sum_{i}a_{i}E_{i}\), where \(a_{i}\) is an integer and \(E_{i}\) is an exceptional divisor with \(\pi(E_{i})=p_{i}\) for each \(i\). In this situation, there is the following classical question: Is the line bundle \(L''\) ample if \(p_{1}, \dots , p_{r}\) are in general positions? By considering this problem, it is possible to make a new example of polarized varieties and so this problem is very important to study a classification of polarized varieties. Recently a study of this problem has been developed. First we assume that \(\dim X=2\). In Math. Ann. 304, 151-155 (1996; Zbl 0834.14024), \textit{O. Küchle} proved that \(L\) is ample if \((L'')^{2}>0\), \(L=mH\) for some ample line bundle \(H\) on \(X\), \(m\) an integer with \(m\geq 3\) and \(a_{i}=1\) for any \(i\). In Manuscr. Math. 86, 195-197 (1995; Zbl 0836.14004), \textit{G. Xu} has obtained the same result if \(X\) is a 2-dimensional projective space. In J. Reine Angew. Math. 469, 199-209 (1995; Zbl 0833.14028), \textit{G. Xu} also studied the case where \(X\) is a 2-dimensional projective space and \(a_{i}\geq 1\) for \(1\leq i\leq r\). In Rev. Roum. Math. Pures Appl. 42, 371-373 (1997; Zbl 0929.14001), \textit{E. Ballico} considered the case in which \(L\) is a \(\mathbb{Q}\)-divisor, and obtained a criterion of ampleness of \(L''\) under some conditions. In Proc. Am. Math. Soc. 127, 2527-2528 (1999; Zbl 0918.14002), \textit{E. Ballico} treated the case where \(X\) is an \(n\)-dimensional projective space \(L=\mathcal{O}(d)\) with \(d\geq 2\), and \(a_{i}=1\) for \(1\leq i\leq r\). In the paper under review, the author proves the following theorem: Let \(X\), \(L\), \(X''\), \(L''\), \(p_{i}\), \(E_{i}\), and \(a_{i}\) be as above. Ass…
reviewed by: Yoshiaki Fukuma
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 14J25
Mathematics Subject Classification ID: 14J10
zbMATH DE Number: 1347246
zbMATH Keywords: polarized surface
zbMATH Keywords: blowing up
zbMATH Keywords: ampleness
zbMATH Keywords: classification of polarized varieties
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/05130523
OpenAlex ID: W2094774449
DOI: 10.2969/JMSJ/05130523
description: scientific article; zbMATH DE number 1347246