Redundant blow-ups of rational surfaces with big anticanonical divisor
type: Item
instance of: scholarly article
title: Redundant blow-ups of rational surfaces with big anticanonical divisor
zbMATH Open document ID: 1327.14176
author: Dongseon Hwang
author: Jinhyung Park
published in: Journal of Pure and Applied Algebra
publication date: 2015-08-31T00:00:00Z
full work available at URL: https://arxiv.org/abs/1411.3891
review text: The authors prove the existence and provide a complete classification of the so-called redundant blow-ups of rational surfaces having big anticanonical divisor. This classification is motivated by a paper of \textit{F. Sakai} [Math. Ann. 269, 389--410 (1984; Zbl 0533.14016)], where the author shows that the anticanonical morphism \(f: S \rightarrow \overline{S}\) factors through the minimal resolution of a del Pezzo surface with only rational singularities followed by a sequence of redundant blow-ups. For a normal projective surface \(S\) with \(-K_{S}\) pseudoeffective we denote by \(-K_{S} = P + N\) its Zariski decomposition. We say that \(p \in S\) is a redundant point if \(\mathrm{mult}_{p}(N) \geq 1\) and we called the blow up \(f: \tilde{S} \rightarrow S\) at a redundant point \(p\) as a redundant blow-up. In order to classify all such redundant blow-ups one needs only to read off the information about redundant points on the surface \(S\). The first result of this note is the following. Theorem 1. Let \(\overline{S}\) be a normal projective surface with nef anticanonical \(\mathbb{Q}\)-Cartier divisors and let \(\tilde{S}\) be a surface obtained by a sequence of redundant blow-ups from the minimal resolution \(S\) of \(\overline{S}\). We have the following. {\parindent=6mm \begin{itemize} \item[1)] The number of surfaces obtained by a sequence of redundant blow-ups from \(S\) is finite if and only if \(\overline{S}\) contains at worst log terminal singularities. \item [2)] If \(\overline{S}\) contains at worst log terminal singularities, then every redundant point on \(\tilde{S}\) lies on the intersection points of the two curves contracted by the morphism \(h: \tilde{S} \rightarrow S\). 3) If \(\overline{S}\) contains a non-log terminal singularity, then there is a curve in \(\tilde{S}\) contracted by the morphism \(h: \tilde{S} \rightarrow \overline{S}\) such that every point lying on this curve is a redundant point. \end{itemize}} The second result of this…
reviewed by: Piotr Pokora
Mathematics Subject Classification ID: 14J26
Mathematics Subject Classification ID: 14J17
zbMATH DE Number: 6476967
zbMATH Keywords: rational surfaces
zbMATH Keywords: blow-ups
zbMATH Keywords: redundant points and blow-ups
zbMATH Keywords: Zariski decompositions
zbMATH Keywords: singularities
Wikidata QID: Q112882029
MaRDI profile type: Publication
OpenAlex ID: W2963902569
arXiv ID: 1411.3891
cites work: Q4280955
cites work: Cox rings of surfaces and the anticanonical Iitaka dimension
cites work: Algebraic surfaces. Transl. from the Romanian by V. Maşek
cites work: Rationale Singularitäten komplexer Flächen
cites work: The maximum number of singular points on rational homology projective planes
cites work: Algebraic Montgomery-Yang problem: the non-rational surface case
cites work: Q4212603
cites work: On extremal rational elliptic surfaces
cites work: Anticanonical models of rational surfaces
cites work: Big rational surfaces
DOI: 10.1016/J.JPAA.2015.05.015
description: scientific article