Logarithmic transformations of rigid analytic elliptic surfaces
type: Item
instance of: scholarly article
title: Logarithmic transformations of rigid analytic elliptic surfaces
zbMATH Open document ID: 1285.14028
author: Kentaro Mitsui
published in: Mathematische Annalen
publication date: 2013-04-03T00:00:00Z
review text: Mitsui introduces a non-archimedean analogue of logarithmic transformations for elliptic fibrations over the field of complex numbers, working within the framework of rigid-analytic geometry. His surgery constructions apply to Tate elliptic fibrations over algebraically closed non-archimedean fields, and they can be used to construct interesting new examples of elliptic surfaces in positive characteristic: on the one hand, Mitsui is able to construct new types of algebraic elliptic surfaces, and on the other hand, he finds rigid-analytic elliptic surfaces whose combinations of invariants appear neither in the algebraic nor in the complex-analytic case. Let us be more precise about the content of the paper. The first section is introductory: the author summarizes the structure of his manuscript, and he outlines his main results. Moreover, he briefly mentions various standard techniques which can be used to construct elliptic fibrations with multiple fibers over fields of positive characteristic. In Section 2, the author sets up some notation, and he recalls basic facts from rigid geometry. Furthermore, he quotes results from \textit{E. Bombieri} and \textit{D. Mumford} [Complex Anal. Algebr. Geom., Collect. Pap. dedic. K. Kodaira, 23--42 (1977; Zbl 0348.14021), p.\ 27] regarding algebraic elliptic surfaces, most notably the canonical bundle formula. He shows (cf.\ his Proposition 2) that the total space of an elliptic fibration has trivial Euler characteristic if and only if every reduced fiber is an elliptic curve. The proof of this statement is reduced to the case where the fibration has a section; this reduction is achieved by passing to the relative Jacobian of the fibration and by using results from \textit{Q. Liu, D. Lorenzini} and \textit{M. Raynaud} [Invent. Math. 157, No. 3, 455--518 (2004; Zbl 1060.14037)]. In the third section, which is the heart of the paper, the author defines his non-archimedean logarithmic transformations. Let \(K\) be an algebraically…
reviewed by: Christian Kappen
Mathematics Subject Classification ID: 14G22
Mathematics Subject Classification ID: 14J27
Mathematics Subject Classification ID: 14C30
Mathematics Subject Classification ID: 14J10
zbMATH DE Number: 6149474
MaRDI profile type: Publication
full work available at URL: s00208-012-0813-1
OpenAlex ID: W2063298434
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DOI: 10.1007/S00208-012-0813-1
description: scientific article; zbMATH DE number 6149474