An elliptic surface covered by Mumford's fake projective plane
type: Item
instance of: scholarly article
title: An elliptic surface covered by Mumford's fake projective plane
zbMATH Open document ID: 0694.14017
author: Masa-Nori Ishida
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1988-01-01T00:00:00Z
review text: An algebraic surface M of general type with \(K_{M^ 2}=9\) and \(p_ g=q=0\) has been studied by Mumford. This surface is called Mumford's fake projective plane, because it has the same Betti numbers as the complex plane. By its construction M has an unramified Galois covering \(V\to M\) of order 8. Moreover, a simple group G of order 168 acts on V and M is the quotient of V by a 2-Sylow subgroup of G. The quotient surface \(Y=V/G\) and its minimal desingularization \(\tilde Y\) is studied. (Thus M is a branched covering over Y of order 21.) It is shown that Y has only 4 isolated singularities as singularites, 3 of them are rational double points of type \(A_ 2\) and the 4-th singularity is the quotient singularity of type (7,3) by the cyclic group of order 7. Moreover, it is shown that \(\tilde Y\) is an elliptic surface with the following data: the Kodaira dimension 1, the Euler number 12, \(p_ g=q=0\), the second pluri- genus 1, the third pluri-genus 1, the combination of singular fibers \(_ 2I_ 0+_ 3I_ 0+I_ 3+I_ 3+I_ 3+I_ 3\). Since Mumford's paper is based on the theory of the 2-adic unit ball due to Mustafin and Kurihara, the theory of schemes over the ring \({\mathbb{Z}}_ 2\) of 2-adic integers is the main tool in this paper.
reviewed by: Tohsuke Urabe
Mathematics Subject Classification ID: 14J27
Mathematics Subject Classification ID: 14J10
Mathematics Subject Classification ID: 14J25
Mathematics Subject Classification ID: 14E20
zbMATH DE Number: 4136112
zbMATH Keywords: Mumford's fake projective plane
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178227980
OpenAlex ID: W2057032826
cites work: Q5749404
cites work: On compact complex analytic surfaces. I
cites work: On the Structure of Compact Complex Analytic Surfaces, I
cites work: Construction of p-Adic Unit Balls and the Hirzebruch Proportionality
cites work: An Algebraic Surface with K Ample, (K 2 ) = 9, p g = q = 0
DOI: 10.2748/TMJ/1178227980
description: scientific article; zbMATH DE number 4136112