On non-rational numerical Del Pezzo surfaces
type: Item
instance of: scholarly article
title: On non-rational numerical Del Pezzo surfaces
zbMATH Open document ID: 0890.14023
author: Taro Fujisawa
published in: Osaka Journal of Mathematics
publication date: 1996-06-18T00:00:00Z
review text: In this paper the author studies the numerical Del Pezzo surfaces. By definition a numerical Del Pezzo surface is a compact complex Moishezon surface \(X\) whose anti-canonical (Weil) divisor \(-K_X\) satisfies the following two conditions: (i) \((-K_X)^2>0\), and (ii) \((-K_X.C)>0\) for every irreducible curve \(C\) on \(X\). -- This class of surfaces generalizes the class of classical Del Pezzo surfaces. By using an algebraicity criterion of L. Brenton, one sees imediately that every numerical Del Pezzo surface is projective algebraic. These surfaces have been considered in the early eighties by \textit{F. Sakai} [Math. Ann. 269, 389-410 (1984; Zbl 0533.14016)] and by \textit{L. Bădescu} [Ann. Univ. Ferrara, Nuova Ser., Sez. VII 29, 165-177 (1983; Zbl 0544.14023)], at least in the case when they have only \(\mathbb{Q}\)-Gorenstein singularities. The author first proves a slight improvement of a result of \textit{L. Bădescu} [which however also follows from the results of another paper by \textit{L. Bădescu}, J. Reine Angew. Math. 367, 76-89 (1986; Zbl 0604.14020)]. Then he defines a notion of minimal contraction of a ruled surface and proves a criterion for the minimal contraction of some ruled surfaces to be a numerical Del Pezzo surface. Finally he defines the notion of index of non-rational numerical Del Pezzo surfaces and studies its properties.
Mathematics Subject Classification ID: 14J26
Mathematics Subject Classification ID: 32J15
zbMATH DE Number: 871824
zbMATH Keywords: resolutions
zbMATH Keywords: minimal contraction of a ruled surface
zbMATH Keywords: index of non-rational numerical Del Pezzo surfaces
zbMATH Keywords: Moishezon surface
reviewed by: Lucian Bădescu
MaRDI profile type: Publication
Recommended article: On non-normal del Pezzo surfaces
Recommended article: De Pezzo surfaces with nonrational singularities
Recommended article: Arithmetic of Del Pezzo surfaces
Recommended article: Potentially Stably Rational Del Pezzo Surfaces over Nonclosed Fields
Recommended article: Rational cuspidal curves on del-Pezzo surfaces
Recommended article: Q3655917
Recommended article: RESENT PROGRESS ON THE QUANTITATIVE ARITHMETIC OF DEL PEZZO SURFACES
Recommended article: Rational curves on del Pezzo surfaces in positive characteristic
Recommended article: Arithmetic on Singular Del Pezzo Surfaces
Recommended article: A non-vanishing theorem of del Pezzo surfaces
description: scientific article; zbMATH DE number 871824