Minimal algebraic surfaces of general type with \(c_1^2=3\), \(p_g=1\) and \(q=0\), which have non-trivial 3-torsion divisors
type: Item
instance of: scholarly article
title: Minimal algebraic surfaces of general type with \(c_1^2=3\), \(p_g=1\) and \(q=0\), which have non-trivial 3-torsion divisors
zbMATH Open document ID: 1061.14035
author: Masaaki Murakami
published in: Journal of Mathematics of Kyoto University
publication date: 2004-12-29T00:00:00Z
review text: The torsion group of minimal surfaces \(X\) of general type over \({\mathbb C}\) satisfying \(c_1^2(X)< 2\chi({\mathcal {O}}_X)\) is finite. The bounds for the order of the torsion group are well studied for some numerical invariants and very often the existence of torsion allows the classification or construction of examples of such surfaces. Instances of this can be seen in \textit{Y. Miyaoka} [Invent. Math. 34, 99--111 (1976; Zbl 0337.14010)] and \textit{M. Reid} [J. Fac. Sci., Univ. Tokyo, Sect. I A 25, 75--92 (1978; Zbl 0399.14025)] for the case \(p_g=0\), \(c_1^2(X)=1\), \textit{A. Todorov} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 13, No. 1, 1--21 (1980; Zbl 0478.14030)] for the case \(p_g=1\), \(c_1^2(X)=1\). Also the case \( p_g=3\), \(c_1^2(X)=6\) having non-trivial torsion was studied by \textit{A. Bartalesi} and \textit{F. Catanese} [in: Algebraic varieties of small dimension, Proc. Int. Conf., Turin/Italy 1985, Rend. Semin. Mat., Torino, Fasc. Spec., 91--110 (1986; Zbl 0623.14015)], whilst the case \( p_g=1\), \(c_1^2(X)=2\) was studied by \textit{O. Debarre} and \textit{F. Catanese} [J. Reine Angew. Math. 395, 1--55 (1989; Zbl 0658.14016)] and, using these results, \textit{C. Ciliberto} and the reviewer [Geom. Dedicata 66, No. 3, 313--329 (1997; Zbl 0905.14021)] characterized minimal surfaces \(X\) of general type with \(c_1^2(X)= 2\chi({\mathcal {O}}_X)-2\) and non-trivial torsion. The present paper studies minimal algebraic complex surfaces \(X\) of general type satisfying \(c_1(X)^2 = 3,\) \( p_g(X) =1,\) and having non trivial torsion, focusing in particular the case \({\mathbb Z}_3\subseteq\text{Tors}(X)\). It is shown that for minimal surfaces of general type with \(c_1(X)^2 = 3,\) \( p_g(X) =1,\) \( q(X) = 0\), card \( \text{Tors}(X)\leq 4\). Furthermore the case in which the order of the torsion group is \(3\) (i.e \( \text{Tors}(X)= \mathbb{Z}_3 \)) is described by giving a very explicit formulation of the étale triple cover \(Y\to X\) associated …
reviewed by: Margarida Mendes Lopes
Mathematics Subject Classification ID: 14J29
Mathematics Subject Classification ID: 14J10
zbMATH DE Number: 2124035
zbMATH Keywords: small invariants
zbMATH Keywords: étale covers
zbMATH Keywords: torsion
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250283747
OpenAlex ID: W1567366364
DOI: 10.1215/KJM/1250283747
description: scientific article; zbMATH DE number 2124035