## Advanced Studies in Pure Mathematics

### A proof of a conjecture of Degtyarev on non-torus plane sextics

#### Abstract

A $\mathbb{D}_{10}$-sextic is an irreducible sextic $C \subset \mathbb{CP}^2$ with simple singularities such that the fundamental group $\pi_1 (\mathbb{CP}^2 \setminus C)$ factors to the dihedral group $\mathbb{D}_{10}$. A $\mathbb{D}_{10}$-sextic is not of torus type. In this paper, we show that if $C$ is a $\mathbb{D}_{10}$-sextic with the set of singularities $4\mathbf{A}_4$ or $4\mathbf{A}_4 \oplus \mathbf{A}_1$, then $\pi_1 (\mathbb{CP}^2 \setminus C)$ is isomorphic to $\mathbb{D}_{10} \times \mathbb{Z}/3\mathbb{Z}$. This positively answers a conjecture by Degtyarev.

#### Article information

Dates
Revised: 30 May 2008
First available in Project Euclid: 28 November 2018

https://projecteuclid.org/ euclid.aspm/1543448014

Digital Object Identifier
doi:10.2969/aspm/05610109

Mathematical Reviews number (MathSciNet)
MR2604079

Zentralblatt MATH identifier
1193.14039

#### Citation

Eyral, Christophe; Oka, Mutsuo. A proof of a conjecture of Degtyarev on non-torus plane sextics. Singularities — Niigata–Toyama 2007, 109--131, Mathematical Society of Japan, Tokyo, Japan, 2009. doi:10.2969/aspm/05610109. https://projecteuclid.org/euclid.aspm/1543448014