Surfaces of general type whose canonical map is composed of a pencil of genus 3 with small invariants
type: Item
instance of: scholarly article
title: Surfaces of general type whose canonical map is composed of a pencil of genus 3 with small invariants
zbMATH Open document ID: 0958.14030
author: Jingen Yang
author: M. Miyanishi
published in: Journal of Mathematics of Kyoto University
publication date: 1999-09-02T00:00:00Z
review text: For a surface of general type with canonical system composed with a pencil of genus \(b\), if the curves of the pencil have genus \(g\) at least \(3\), \(K^2 \geq 4 p_g + 4 (b-1)\) with few exceptions [cf. \textit{F. Catanese}, in: Algebraic Geometry, Proc. Summer Res. Inst., Brunswick 1985, part I, Proc. Symp. Pure Math. 46, 175-194 (1997; Zbl 0656.14021) and \textit{A. Beauville}, Invent. Math. 55, 121-140 (1979; Zbl 0403.14006)]. In this paper the authors prove that, if \(b=0\) and \(p_g \geq 3\) and the inequality does not hold, then \(K^2=7\) and \(p_g=3\), excluding the other unknown cases \(p_g=3\), \(K^2=6\) and \(p_g=4, K^2=9\). Interesting is the example with \(K^2=7\) and \(p_g=3\), which is given in three different ways: as hypersurface in a suitable \({\mathbb P}^2\)-bundle over \({\mathbb P}^1\); as Galois triple cover of a suitable blow up of the Hirzebruch surface \(F_3\); as a minimal resolution of a sextic with three singular points, one with local equation \(x^2+y^3+z^6=0\), two with local equation \(x^2+y^6+z^6=0\). All the constructions are explicit. The non-existence theorems are obtained carrying out a very detailed analysis of the possible behaviour of the relative canonical map.
Mathematics Subject Classification ID: 14J29
zbMATH DE Number: 1295045
zbMATH Keywords: surfaces of general type
zbMATH Keywords: relative canonical map
zbMATH Keywords: fibrations
zbMATH Keywords: canonical system
reviewed by: Roberto Pignatelli
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250518163
OpenAlex ID: W1927748280
DOI: 10.1215/KJM/1250518163
description: scientific article; zbMATH DE number 1295045