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.