Raynaud-Tamagawa theta divisors and new-ordinariness of ramified coverings of curves
type: Item
instance of: scholarly article
title: Raynaud-Tamagawa theta divisors and new-ordinariness of ramified coverings of curves
zbMATH Open document ID: 1484.14065
author: Yu Yang
published in: Journal of Algebra
publication date: 2021-09-16T00:00:00Z
review text: Let \((X,D_X)\) be a smooth pointed stable curve of genus \(g_X\) over an algebraic closed field \(k\) of characteristic \(p>0\). Define the Hasse-Witt invariant as \(\sigma_X:=\dim_{\mathbb{F}_p}(H_{et}^1(X,\mathbb{F}_p))\). We call the curve \(X\) \emph{ordinary} if \(g_X=\sigma_X\). Consider a multi-admissible cover \((Z,D_Z)\to (X,D_X)\) over \(k\) as in [\textit{Y. Yang}, Publ. Res. Inst. Math. Sci. 54, No. 3, 649--678 (2018; Zbl 1439.14102)]. The cover \((Z,D_Z)\) is called \emph{new-ordinary} if \(\sigma_Z-\sigma_X=g_Z-g_X\). In [\textit{S. Nakajima}, Adv. Stud. Pure Math. 2, 69--88 (1983; Zbl 0529.14016)], Nakajima consider the case when \(D_X=\emptyset\) and \(X\) is generic. He found that if the cover \(Z\) corresponds to a prime-to-\(p\) cyclic quotient of the tame fundamental group \(\Pi_X\) of \(X\) then \(Z\) is ordinary. The main result of the paper under review gives necessary and sufficient conditions for a cover \((Z,D_Z)\to (X,D_X)\) to be new-ordinary when \((X,D_X)\) is generic and the cover corresponds to a prime-to-\(p\) cyclic quotient of the tame fundamental group \(\Pi_{(X,D_X)}\). For the proof, the author relate new-ordinariness to the existence of Raynaud-Tamagawa theta divisors, see [\textit{M. Raynaud}, Bull. Soc. Math. Fr. 110, 103---125 (1982; Zbl 0505.14011)] and [\textit{A. Tamagawa}, Math. Sci. Res. Inst. Publ. 41, 47--105 (2003; Zbl 1073.14035)].
reviewed by: Rodolfo Aguilar Aguilar
Mathematics Subject Classification ID: 14H30
Mathematics Subject Classification ID: 14F35
Mathematics Subject Classification ID: 14G32
zbMATH DE Number: 7396018
zbMATH Keywords: pointed stable curve
zbMATH Keywords: admissible covering
zbMATH Keywords: generalized Hasse-Witt invariant
zbMATH Keywords: new-ordinary
zbMATH Keywords: Raynaud-Tamagawa theta divisor
zbMATH Keywords: positive characteristic
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2021.07.032
OpenAlex ID: W3193593766
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DOI: 10.1016/J.JALGEBRA.2021.07.032
description: scientific article; zbMATH DE number 7396018