On complex projective surfaces with trigonal hyperplane sections
type: Item
instance of: scholarly article
title: On complex projective surfaces with trigonal hyperplane sections
zbMATH Open document ID: 0707.14031
DOI: 10.1007/BF01168368
author: Sonia Brivio
author: Antonio Lanteri
published in: Manuscripta Mathematica
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/155427
review text: Let \((S,L)\) be a polarized surface, i.e., \(S\) is a complex smooth projective surface and \(L\) is an ample line bundle on it. Assume that \(L\) is generated everywhere by its global sections. A main result of the present article is that, with the additional hypothesis \((L^ 2)\geq 10\), we have \((K_ S+L\cdot K_ S)\leq -3\) and \(| L|\) contains a trigonal curve \(C\) of genus \(\geq 4\) if and only if either \((S,L)\) is a rational surface ruled by cubics or the \(g^ 1_ 3\) of \(C\) is cut out by \(| -K_ S|\). If \(L\) is very ample and \(S\) is regular this is a result of \textit{F. Serrano} [``Surfaces having a hyperplane section with a special pencil'' (preprint)]; the hypothesis \((L^ 2)\geq 10\) is not necessary in Serrano's result.
Mathematics Subject Classification ID: 14J10
Mathematics Subject Classification ID: 14C17
Mathematics Subject Classification ID: 14C20
zbMATH DE Number: 4160932
zbMATH Keywords: ruled rational surface
zbMATH Keywords: hyperplane section
zbMATH Keywords: scroll
zbMATH Keywords: conic bundle
zbMATH Keywords: polarized surface
zbMATH Keywords: trigonal curve
reviewed by: M. Miyanishi
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4160932