Unramified double coverings of hyperelliptic surfaces. II. (English) Zbl 0634.30040
In 1976 H. M. Farkas [J. Anal. Math. 30, 150-155 (1976; Zbl 0348.32006)] proved that among the \(2^{2g}-1\) unramified double coverings of a hyperelliptic surface S of genus g there are \(\left( \begin{matrix} 2g+2\\ 2\end{matrix} \right)\) hyperelliptic surfaces. The reviewer [Arch. Math. 47, 93-96 (1986; Zbl 0599.30070)] classified all the unramified double coverings of a hyperelliptic surface using the representation of compact Riemann surfaces as quotient spaces of Fuchsian groups. In this paper a new proof of this result is obtained by simply lifting the hyperelliptic involution from S to the unramified double covering and counting its fixed points.
Reviewer: E.Bujalance
References:
| [1] | E. Bujalance, A classification of unramified double coverings of hyperelliptic Riemann surfaces, Arch. Math. (Basel) 47 (1986), no. 1, 93 – 96. · Zbl 0599.30070 · doi:10.1007/BF01202505 |
| [2] | H. M. Farkas, Unramified double coverings of hyperelliptic surfaces, J. Analyse Math. 30 (1976), 150 – 155. · Zbl 0348.32006 · doi:10.1007/BF02786710 |
| [3] | -, Unramified coverings of hyperelliptic Riemann surfaces (Preprint). · Zbl 0625.30044 |
| [4] | Hershel M. Farkas and Irwin Kra, Riemann surfaces, Graduate Texts in Mathematics, vol. 71, Springer-Verlag, New York-Berlin, 1980. · Zbl 0764.30001 |
| [5] | C. Maclachlan, Smooth coverings of hyperelliptic surfaces, Quart. J. Math. Oxford Ser. (2) 22 (1971), 117 – 123. · Zbl 0208.10101 · doi:10.1093/qmath/22.1.117 |
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