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Field theory for function fields of singular plane quartic curves. (English) Zbl 0986.14016

We work over an algebraically closed field of characteristic zero. Let \(C\subset\mathbb{P}^2: =\mathbb{P}^2_k\) be a plane projective curve of degree \(d\), \(K=k(C)\) the field of rational functions of \(C\), \(p:X\to C\) the normalization (or desingularization) of \(C\). (Then, \(k(X)= k(C)=K\).) If \(P\in \mathbb{P}^2\), there is a natural rational map from \(C\) to \(\mathbb{P}^1\) (regarded as the space of lines through \(P)\), namely sending a general point \(Q\) of \(C\) to the line joining \(P\) and \(Q\). Composing with \(p\), this extends to a morphism \(\pi_P: X\to\mathbb{P}^1\); this induces a homomorphism of fields \(\pi_P^*: k(\mathbb{P}^1)\to K\) or, taking the image \(K_P\) of \(\pi_P^*\), an inclusion of fields \(K_P\subset K\). Let \(L_P\) be be the Galois closure of \(K/K_P\), \(G_P\) the Galois group of \(L_P\) over \(K_P\) and \(g(P)\) the genus of a non-singular model of the field \(L_P\). A point \(P\in \mathbb{P}^2\) is called a Galois point if \(K=L_P\).
In this paper, the author investigates properties of the field extensions \(K_P\subset K\subset L_P\), the group \(G_P\), the number \(g(P)\) and the set of Galois points, in the case where \(d=\deg(C)=4\) and \(C\) is singular. The regular case had been studied by K. Miura and H. Yoshihara [J. Algebra 226, 283-294 (2000; Zbl 0983.11067)]. Now the author obtains, among other, the following results:
(1) If \(C\) has a triple point and \(P\in C\), then \(g(P)=0\) or 1.
(2) If the only singularities of \(C\) are double points an \(P\in C\), then \(g(P)=g\) or \(3g+1-a\), where \(0\leq a \leq g+1\). If P is a general point of \(C\), then \(G_P\) is isomorphic to the symmetric group \(S_3\) and \(g(P)=3g+1\).
(3) If \(P\) is not on \(C\), then \(G_P\) is isomorphic to either \(S_4\), or to the alternating group \(A_4\), or to the dihedral group of order 8, or to the cyclic group of order 4, or to Klein’s 4-group.
(4) For a general point \(P\in\mathbb{P}^2\) there are no fields between \(K\) and \(K_P\).

MSC:

14H05 Algebraic functions and function fields in algebraic geometry
14H20 Singularities of curves, local rings
12F20 Transcendental field extensions

Citations:

Zbl 0983.11067
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References:

[1] Tokunaga, J. Math. Kyota Univ. 31 pp 359– (1991)
[2] Serre, Topics in Galois theory (1992)
[3] Namba, Branched coverings and algebraic functions (1987)
[4] Iitaka, Descartes no seishin to Daisûkika (1980)
[5] DOI: 10.1080/00927870008826940 · Zbl 0978.14024
[6] DOI: 10.1006/jabr.1999.8173 · Zbl 0983.11067
[7] DOI: 10.2307/2323198 · Zbl 0702.11075
[8] Namba, Geometry of projective algebraic curves (1984) · Zbl 0556.14012
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