On Enriques surfaces in characteristic \(p\). II
type: Item
instance of: scholarly article
title: On Enriques surfaces in characteristic \(p\). II
zbMATH Open document ID: 0708.14026
DOI: 10.1007/BF01456844
author: William E. Lang
published in: Mathematische Annalen
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164444
review text: From the introduction: This paper is a sequel to part I [ibid. 265, 45--65 (1983; Zbl 0575.14032).] Let \(X\) be a classical Enriques surface in characteristic two. Recall that an Enriques surface is classical if and only if the canonical line bundle is a non-trivial element of order two in \(\mathrm{Pic}(X)\). Thus, the canonical line bundle gives us a \(\mu_2\)-double cover \(\tilde Y\) of \(X\). It is easy to see that \(\tilde Y\) is birationally isomorphic to a K3 surface if and only if \(\tilde Y\) has only rational double points as singularities, and in this case, the minimal nonsingular model \(Y\) of \(\tilde Y\) is in fact a supersingular K3 surface. It is natural to ask the following questions: (1) Assuming the singularities of \(\tilde Y\) are rational double points, what kind of singularities occur ? (2) What is the Artin invariant \(\sigma_0\) of \(Y\) ? If \(X\) is suitably generic, then \(Y\) has 12 rational double points of type \(A_1\), and \(\sigma_0(Y)=10\). Thus, the Picard group of \(Y\) is as small as possible, given that \(Y\) is a supersingular K3 surface. We have added two appendices to the paper: Appendix A deals with consequences of the fact that if \(X\) is any Enriques surface (classical or not), then \(\mathrm{Num}(X)\) is isomorphic to the root lattice \(E_{10}\), and in particular, \(\mathrm{Num}(X)\) is hyperbolic. The rest of Appendix A consists of complements to the important paper by \textit{F. R. Cossec} [Math. Ann. 271, 577--600 (1985; Zbl 0541.14031)]. In particular, we extend Cossec's theorem on Enriques surfaces of special type to characteristic two in this appendix. Appendix B is an extension to the characteristic \(p\ne 2\) of the theorem of Burns-Wahl that if \(X\) is a surface with smooth local moduli, then any \(-2\) curve on \(X\) can be removed by a first-order deformation.
Mathematics Subject Classification ID: 14J28
Mathematics Subject Classification ID: 14G15
Mathematics Subject Classification ID: 14J17
Mathematics Subject Classification ID: 14J10
zbMATH DE Number: 4163064
zbMATH Keywords: classical Enriques surface in characteristic two
zbMATH Keywords: canonical line bundle
zbMATH Keywords: K3 surface
zbMATH Keywords: rational double points
zbMATH Keywords: Artin invariant
zbMATH Keywords: Picard group
MaRDI profile type: Publication
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OpenAlex ID: W4233531923
description: scientific article; zbMATH DE number 4163064