Surfaces with \(p_g=q=1\), \(K^2=6\) and non-birational bicanonical maps
type: Item
instance of: scholarly article
title: Surfaces with \(p_g=q=1\), \(K^2=6\) and non-birational bicanonical maps
zbMATH Open document ID: 1409.14070
author: Yong Hu
author: Lei Zhang
publication date: 2019-03-27T00:00:00Z
review text: Thanks to the efforts of several authors the behaviour of the bicanonical map of surfaces of general type is well understood, apart from a very few exceptions of surfaces with small numerical invariants. The present paper addresses one of these exceptions: minimal surfaces of general type with $p_g=q=1$ and $K^2=6$. It is shown that the bicanonical map of such surfaces is either birational or has degree 2. Note that examples for this last case were constructed by \textit{C. Rito} [Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 6, No. 1, 81--102 (2007; Zbl 1180.14040)] and as such this bound is effective. \par By Reider's theorem it is known that the bicanonical map of these surfaces is a morphism and as such the possible degrees of the bicanonical map would be 1, 2, 3 or 4. The result is obtained by a long and detailed analysis excluding the possibility of degree bigger than 2.
reviewed by: Margarida Mendes Lopes
Mathematics Subject Classification ID: 14J29
Mathematics Subject Classification ID: 14J10
zbMATH DE Number: 7043606
zbMATH Keywords: algebraic surface
zbMATH Keywords: bicanonical map
zbMATH Keywords: surfaces with $p_g=q=1$
zbMATH Keywords: Albanese map
MaRDI profile type: Publication
full work available at URL: s10114-018-7262-z
OpenAlex ID: W2805398216
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DOI: 10.1007/S10114-018-7262-Z
published in: Acta Mathematica Sinica, English Series
description: scientific article; zbMATH DE number 7043606