Galois points for smooth hypersurfaces.
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 \(d-1\) for \(P\in V\) (of order \(d\) if \(P\not\in V\)). There are a variety of interesting results; two will sufficice to give some idea of them. Suppose that \(d\geq 5\), then the set of Galois points is independent, \(\delta(V)\leq m+1\) with equality if and only if \(V\) is projectively equivalent to the hypersurface with equation \(X_{m+1} X^{d-1}_0+\cdots+ X_{2m+1} X^{d-1}_m+ G= 0,\) where \(G\) is a form in \(k[X_{m+1},\dots, X_{n+1}]\) of degree \(d\). The number \(\delta(V^c)\) of outer Galois points satisfies \(\delta(V^c)\leq n+2\), with equality if and only if \(V\) is projectively equivalent to the Fermat variety \(F(n,d)\). More precisely, for a variety in \(\mathbb{P}^{n+1}\), \(\delta(V^c)= 0,1,2,\dots, n,n+2\). If \(\delta(V^c)= r+1\), then the defining equation of \(V\) has the form \(F= X^d_0+\cdots+ X^d_r+ G\), where \(G\) is a form of degree \(d\) in \(k[X_{r+1},\dots, X_{n+1}]\) and in particular if \(\delta(V^c)= n+2\), then \(F\) is the Fermat variety.
scientific article; zbMATH DE number 1956748