Hyperelliptic curves among cyclic coverings of the projective line. I. (English) Zbl 1287.14013
This paper under review gives a necessary and sufficient condition for a \(d\)-cyclic cover of \(\mathbb{P}^1({\mathbb C})\) with \(3\) branch points being hyperelliptic or elliptic. More precisely, a smooth curve \(V\) is a \(d\)–cyclic cover of \(\mathbb{P}^1({\mathbb C})\) with \(3\) branch points if and only if \(d\geq3\) and \(V\) has a (possibly singular) plane model \(y^d=x(x-1)\) or \(y^d=x(x-1)^{\frac{d-2}{2}}\) (\(d\) is even and \(d\geq6\)).
Reviewer: Jie Wang (Athens)
References:
| [1] | H. M. Farkas and I. Kra, Riemann surfaces, Graduate Texts in Mathematics, vol. 71, second edition, Springer-Verlag, New York, 1992. · Zbl 0764.30001 |
| [2] | Kallel S., Sjerve D.: On the group of automorphisms of cyclic covers of the Riemann sphere. Math. Proc. Cambridge Philos. Soc. 138, 267-287 (2005) · Zbl 1064.30038 · doi:10.1017/S0305004104008096 |
| [3] | Moh T. T., Heinzer W. J.: A generalized Lüroth theorem for curves. J. Math. Soc. Japan 31, 85-86 (1979) · Zbl 0402.14011 · doi:10.2969/jmsj/03110085 |
| [4] | Namba M.: Equivalence problem and automorphism groups of certain compact Riemann surfaces. Tsukuba J. Math. 5, 319-338 (1981) · Zbl 0499.30031 |
| [5] | F. Sakai, The gonality of singular plane curves II, Affine algebraic geometry, Proceedings of the conference, 243-266, World Scientific Publishing, Singapore, 2013. · Zbl 1279.14038 |
| [6] | Yoshida K.: Automorphisms with fixed points and Weierstrass points of compact Riemann surfaces. Tsukuba J. Math. 17, 221-249 (1993) · Zbl 0790.30030 |
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