Endomorphisms of logarithmic schemes
type: Item
instance of: scholarly article
title: Endomorphisms of logarithmic schemes
zbMATH Open document ID: 1051.14004
author: Mark Kisin
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 2004-09-07T00:00:00Z
full work available at URL: https://eudml.org/doc/108603
review text: Consider a log. smooth log. scheme \((X,M)\), defined over a discrete valuation ring \(V\), with residue field \(k\) of characteristic \(p\). Fix a uniformizer \(\pi\) of \(V\). The author studies the extensions of automorphisms \(\sigma\) of the special fiber \(X_s\) of \(X\) to the whole family \(X\). Assuming that the smooth locus is dense in \(X_s\), the author proves the existence of an integer \(m\) (actually computable) such that \(\sigma\) lifts to the log. structure \((X_s,M_s)\) over the special fiber if and only it lifts locally to the infinitesimal neighbourhood \(X/\pi^m\). This explains in greater detail the known fact that sheaves of nearby cycles to the special fiber depend only on the completion of \(X\). In the case of (relative) curves, the author shows a trace formula for the number of points collapsing to fixed points of \(\sigma\), in its lifting to \(X\).
reviewed by: Luca Chiantini
Mathematics Subject Classification ID: 14A15
Mathematics Subject Classification ID: 14F20
Mathematics Subject Classification ID: 14G20
zbMATH DE Number: 2099242
zbMATH Keywords: endomorphisms
zbMATH Keywords: valuation ring
zbMATH Keywords: logarithmic schemes
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2099242