Monodromies of hyperelliptic families of genus three curves
type: Item
instance of: scholarly article
title: Monodromies of hyperelliptic families of genus three curves
zbMATH Open document ID: 1073.14019
author: Mizuho Ishizaka
published in: Tôhoku Mathematical Journal. Second Series
publication date: 2005-04-18T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.tmj/1113246379
review text: Let \(\phi :S\rightarrow \Delta\) be a proper surjective holomorphic map from a non-singular complex surface \(S\) to a small disk \(\Delta :=\{ t\in {\mathbb C}\mid | t| <\varepsilon \}\) such that \(\phi ^{-1}(t)\) is a non-singular curve of genus \(g\geq 2\) for each \(t\in \Delta ^*:=\Delta \backslash \{ 0\}\). Call \((\phi ,S,\Delta )\) a degeneration of curves of genus \(g\). If the curve \(\phi ^{-1}(t)\) is hyperelliptic for \(t\in \Delta ^*\), then \((\phi ,S,\Delta )\) is called a hyperelliptic family. In the paper the complete list of monodromies not realizable by hyperelliptic families of genus three is given. It is proved that all other monodromies of genus three are realized as monodromies of certain hyperelliptic families. In the proof an operation called ``the inverse of Horikawa's canonical resolution'' is used.
Mathematics Subject Classification ID: 14D06
Mathematics Subject Classification ID: 14H45
Mathematics Subject Classification ID: 14H15
Mathematics Subject Classification ID: 57M99
Mathematics Subject Classification ID: 30F99
zbMATH DE Number: 2158072
zbMATH Keywords: Horikawa's canonical resolution
reviewed by: Vladimir P. Kostov
MaRDI profile type: Publication
OpenAlex ID: W2145059414
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DOI: 10.2748/TMJ/1113246379
description: scientific article; zbMATH DE number 2158072