Ramification in the cohomology of algebraic surfaces arising from ordinary double point singularities
type: Item
instance of: scholarly article
title: Ramification in the cohomology of algebraic surfaces arising from ordinary double point singularities
zbMATH Open document ID: 1477.14054
published in: Journal of Number Theory
publication date: 2019-11-27T00:00:00Z
review text: A classical theorem in arithmetic geometry due to Serre-Tate (first proved by Néron-Ogg-Shafarevich in the curve case) states that an abelian variety has good reduction if and only if the action of the Inertia group on the first étale \(\ell\)-adic cohomology is trivial. One implication holding for all varieties alike, the converse is in general not true as can be seen already from curves of genus greater than one. There are, however, partial results, for instance for \(K3\) surfaces [\textit{C. Liedtke} and \textit{Y. Matsumoto}, Compos. Math. 154, No. 1, 1--35 (2018; Zbl 1386.14138); \textit{B. Chiarellotto} et al., Proc. Lond. Math. Soc. (3) 119, No. 2, 469--514 (2019; Zbl 1427.14075)] and for semistable surfaces in the realm of the weight-monodromy conjecture [\textit{M. Rapoport} and \textit{Th. Zink}, Invent. Math. 68, 21--101 (1982; Zbl 0498.14010)]. The present paper combines ideas from the previous two approaches to treat the following special situation: Let \(K\) be a complete field with respect to a discrete valuation such that the residue field \(k\) is algebraically closed of characteristric \(p>2\). Then there is a unique ramified quadratic extension \(L/K\); it corresponds to a unique quadratic character \(\psi\). Assume given a smooth surface \(X\) over \(K\) which possesses an integral model \(\mathcal X\) over the valuation ring \(R\) whose special fibre \(\mathcal X_k\) has at worst ordinary double points as singularities (the mildest singularity on a surface, given locally by \(xy=z^2\)). At a singular point \(P\in \mathcal X_k\), a formal neighbourhood in \(\mathcal X\) is isomorphic to \[ R[[x,y,z]/(z^2+xy+r) \] for some \(r\in \mathfrak p\), the maximal ideal of \(R\). Fix the \(\mathfrak p\)-adic order \(n_P\) of \(r\), i.e. \[ r\in \mathfrak p^{n_P}\setminus \mathfrak p^{n_P+1}. \] (There also is an intrinsic definition, so this is well-defined.) Let \(g\) be the number of singular points \(P\) such that \(n_P\) is odd: \[ g = \#\{P\in \math…
reviewed by: Matthias Schütt
Mathematics Subject Classification ID: 14J17
Mathematics Subject Classification ID: 14F20
zbMATH DE Number: 7137132
zbMATH Keywords: algebraic surfaces
zbMATH Keywords: ordinary double point singularity
zbMATH Keywords: local fields
zbMATH Keywords: integral model
zbMATH Keywords: weight-monodromy conjecture
author: Dohyeong Kim
MaRDI profile type: Publication
full work available at URL: j.jnt.2019.08.006
OpenAlex ID: W2973479980
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Wikidata QID: Q127226432
DOI: 10.1016/J.JNT.2019.08.006
description: scientific article; zbMATH DE number 7137132