Deformations of some algebraic surfaces with \(q=0\) and \(p_ g=1\)
type: Item
instance of: scholarly article
title: Deformations of some algebraic surfaces with \(q=0\) and \(p_ g=1\)
zbMATH Open document ID: 0661.14008
author: Naoki Hashimoto
published in: Kodai Mathematical Journal
publication date: 1988-01-01T00:00:00Z
review text: Let \(M^ a\) be an affine algebraic surface in \({\mathbb{C}}^ 3\) defined by \(h(w)=1+w^ 8_ 1+w^ 2_ 1W^ 4_ 3+W^ 3_ 1W^ 2_ 2=0\). Then \(M^ a\) admits a smooth compactification M, embedded in a 3- dimensional toric variety W. There are 27 additional integral points \(A_ 1\) on the simplex \(\Delta\) spanned by (8,0,0), (2,0,4), (3,2,0), and they determine a family \(h(w)+\sum t_ iw^{A_ i} =0\) of deformations of \(M_ 0\). They can be compactified to a family \(M_ t\) of deformations of \(M=M_ 0\) in W. The author studies how this is related to all the small deformations of M: The family generates a subspace of codimension 2 in the infinitesimal deformation space \(H^ 1(M,\Theta)\). The surface M itself is an elliptic surface with \(p_ g=1\), \(q=0\) and \(\pi_ 1(M)={\mathbb{Z}}/2{\mathbb{Z}}\).
Mathematics Subject Classification ID: 14D15
Mathematics Subject Classification ID: 14J25
zbMATH DE Number: 4079589
zbMATH Keywords: family of deformations
zbMATH Keywords: toric variety
zbMATH Keywords: small deformations
zbMATH Keywords: elliptic surface
reviewed by: Eiji Horikawa
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138038820
OpenAlex ID: W2004805290
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DOI: 10.2996/KMJ/1138038820
description: scientific article; zbMATH DE number 4079589