Narrow Results

This note is of supplementary nature to our previous paper [J. Kaneko, K. Matsumoto and K. Ohara, The structure of a local system associated with a hypergeometric system of rank 9, Int. J. Math. 31 (2020) 2050021]. Let $S$ be the union of the nodal cubic curve and its three inflectional tangents in the complex projective plane ${\mathbb{P}}^2$. Such $S$ has appeared as the singular locus of certain hypergeometric system introduced in [J. Kaneko, K. Matsumoto and K. Ohara, A system of hypergeometric differential system in two variables of rank 9,Int. J. Math. 28 (2017) 1750015], and we have given generators and defining relations of the fundamental group of $X={\mathbb{P}}^2\setminus S$ [J. Kaneko, K. Matsumoto and K. Ohara, The structure of a local system associated with a hypergeometric system of rank 9, Int. J. Math. 31 (2020) 2050021]. $X$ has a 9-fold Galois covering space $\tilde{X}$ given by the complement of the Hesse configuration of 12 lines in ${\mathbb{P}}^2$. Hence one can apply the method of Reidemeister–Schreier to derive a finite presentation of ${\pi }_1(\tilde{X})$, which we carry out in this note.