Galois points for smooth hypersurfaces.
type: Item
instance of: scholarly article
title: Galois points for smooth hypersurfaces.
zbMATH Open document ID: 1048.14025
DOI: 10.1016/S0021-8693(03)00235-7
author: Hisao Yoshihara
published in: Journal of Algebra
publication date: 2003-07-30T00:00:00Z
review text: Denote by \(k\) the field of complex numbers and by \(V\) a smooth hypersurface of degree \(d\) in the projective space \(\mathbb{P}^{n+1}\), whose function field is \(K= k(V)\); let \(P\) be a point in \(\mathbb{P}^{n+1}\) and denote by \(\pi_P:V\to H\) a projection with centre \(P\) to a hyperplane \(H\), not containing \(P\). That projection induces an extension of function fields \(k(V)/k(H)\), denoted by \(\pi^*_P\). The structure of that extension does not depend on \(H\) but it does depend on \(P\) and the author introduces the notation \(K_P\) in place of \(k(H)\) to emphasize that fact; \([K: K_P]= d-1\) if \(P\in V\), or \(d\) if \(P\not\in V\). (It is assumed, unless otherwise stated, that \(d\geq 4\).) The point \(P\) is called a Galois point if \(K/K_P\) is a Galois extension and the author distinguishes between inner and outer points in the cases \(P\in V\), \(P\not\in V\), respectively. The number of inner or of outer Galois points is denoted by \(\delta(V)\) or \(\delta(V^c)\), respectively. The author's main purpose is to obtain estimates of the number of Galois points and results concerning their distribution, and, in particular, to obtain the defining equation for the variety \(V\) with the maximal number of Galois points. More precisely, if \(L_P\) denotes the Galois closure of \(K/K_P\) and if \(G_P= \text{Gal}(L_P/K_P)\), then the following three problems are addressed: (a) find all the Galois points and investigate rules for the distribution of the points; (b) determine the structure of \(G_P\) at each \(P\in \mathbb{P}^{n+1}\); (c) find the structure of a non-singular projective model of \(L_P\). [The motivation for those problems is to be found in \textit{H. Yoshihara's} paper: ``Function field theory of plane curves by dual curves'', J. Algebra 239, 340--355 (2001; Zbl 1064.14023).] In general, for \(d\geq 4\), the hypersurface \(V\) has no Galois points. If for a hypersurface, \(V\), \(P\) is a Galois point, then \(G_P\) is cyclic of order…
Mathematics Subject Classification ID: 14J70
Mathematics Subject Classification ID: 14G32
Mathematics Subject Classification ID: 14J25
zbMATH DE Number: 1956748
zbMATH Keywords: hypersurface
zbMATH Keywords: Galois group
zbMATH Keywords: inner Galois points
zbMATH Keywords: outer Galois points
zbMATH Keywords: number of Galois points
zbMATH Keywords: Fermat variety
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1956748