Fibrations by nonsmooth genus three curves in characteristic three
type: Item
instance of: scholarly article
title: Fibrations by nonsmooth genus three curves in characteristic three
zbMATH Open document ID: 1218.14006
author: Rodrigo Salomão
published in: Journal of Pure and Applied Algebra
publication date: 2011-05-10T00:00:00Z
review text: The classification of complete, geometrically integral, regular and nonsmooth curves \(C\) over a field \(K\) is equivalent to the classification of nonconservative separably generated function fields, whose field of coefficients is algebraically closed in it. A function field \(K(C)/K\) is nonconservative if the genus decreases when we pass to \(\overline{K}(C)/\overline{K}\). By a theorem of \textit{J. Tate} [Proc. Am. Math. Soc. 3, 400--406 (1952; Zbl 0047.03901)], there is an upper bound for the characteristic \(p\) of the base field \(K\), in terms of its geometric genus \(g\), namely \(p\leq 2g+1\). Nonconservative function fields of genus one and two were classified by \textit{C. S. Queen} [Arch. Math. 22, 612--623 (1971; Zbl 0251.14010)] and \textit{H. Borges Neto} [Change of genus and classification of fields of genus two. Doctoral thesis, IMPA (1979)]. The case \(g=3\) and \(p=5,7\) was solved by \textit{H. Stichtenoth} [J. Reine Angew. Math. 301, 30--45 (1978; Zbl 0375.12014)] and \textit{K. O. Stöhr} and \textit{M. L. Torres Villela} [Manuscr. Math. 66, No. 1, 61--71 (1989; Zbl 0699.14031)]. The paper under review starts the classification of nonconservative function fields of genus \(g=3\) in characteristic \(p=3\). There is a close relation between nonsmoothness of an algebraic curve and nonregularity of its image under the relative Frobenius morphism. Assuming a certain good behaviour with respect to the Frobenius pullback normalization, the main theorem states that any complete, geometrically integral, regular and nonsmooth curve \(C\) of genus \(3\) over a field \(K\) of characteristic \(3\) is isomorphic to a plane quartic defined by the homogeneous polynomial \(ZY^3-aZ^4-bZX^3-X^4\), where \(a\in K\setminus K^3\) and \(b\in K\). The paper focuses on nonsmooth fibrations. In analogy to the Kodaira-Néron classification of special fibers of minimal fibrations by elliptic curves, the author constructs the minimal proper regular model of some fibration…
Mathematics Subject Classification ID: 14D06
Mathematics Subject Classification ID: 14H05
Mathematics Subject Classification ID: 14H10
Mathematics Subject Classification ID: 14H20
Mathematics Subject Classification ID: 14H50
zbMATH DE Number: 5886353
zbMATH Keywords: fibration
zbMATH Keywords: nonsmooth curve
zbMATH Keywords: nonconservative function field
zbMATH Keywords: genus three
reviewed by: Enric Nart
MaRDI profile type: Publication
full work available at URL: j.jpaa.2010.11.007
OpenAlex ID: W2108433821
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DOI: 10.1016/J.JPAA.2010.11.007
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description: scientific article; zbMATH DE number 5886353