Miura, Kei Field theory for function fields of singular plane quartic curves. (English) Zbl 0986.14016 Bull. Aust. Math. Soc. 62, No. 2, 193-204 (2000). 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 \(K_P\). Reviewer: Augusto Nobile (Baton Rouge) Cited in 2 ReviewsCited in 10 Documents MSC: 14H05 Algebraic functions and function fields in algebraic geometry 14H20 Singularities of curves, local rings 12F20 Transcendental field extensions Keywords:singular plane quartic curve; function field; Galois group; Galois point Citations:Zbl 0983.11067 PDFBibTeX XMLCite \textit{K. Miura}, Bull. Aust. Math. Soc. 62, No. 2, 193--204 (2000; Zbl 0986.14016) Full Text: DOI 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.