Advanced Studies in Pure Mathematics

Triple covers of algebraic surfaces and a generalization of Zariski's example

Hirotaka Ishida and Hiro-o Tokunaga

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Let $B$ be a reduced sextic curve in $\mathbb{P}^2$. In the case when singularities of $B$ are only six cusps, Zariski proved that there exists a non-Galois triple cover branched at $B$ if and only if $B$ is given by an equation of the form $G_2^3 + G_3^2$, where $G_i$ denotes a homogeneous polynomial of degree $i$. In this article, we generalize Zariski's statement to any reduced sextic curve with at worst simple singularities. To this purpose, we give formulae for numerical invariants of non-Galois triple covers by using Tan's canonical resolution.

Article information

Singularities — Niigata–Toyama 2007, J.-P. Brasselet, S. Ishii, T. Suwa and M. Vaquie, eds. (Tokyo: Mathematical Society of Japan, 2009), 169-185

Received: 29 February 2008
Revised: 8 July 2008
First available in Project Euclid: 28 November 2018

Permanent link to this document euclid.aspm/1543448017

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14E20: Coverings [See also 14H30] 14J17: Singularities [See also 14B05, 14E15]

Triple cover cubic surface torus curve


Ishida, Hirotaka; Tokunaga, Hiro-o. Triple covers of algebraic surfaces and a generalization of Zariski's example. Singularities — Niigata–Toyama 2007, 169--185, Mathematical Society of Japan, Tokyo, Japan, 2009. doi:10.2969/aspm/05610169.

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