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The fundamental group of the complement of the Hesse configuration

Department of Mathematical Sciences, University of the Ryukyus, Nishihara, Okinawa 903-0213, Japan

E-mail Address: kaneko@math.u-ryukyu.ac.jp

https://doi.org/10.1142/S0129167X23500143Cited by:0 (Source: Crossref)

Abstract

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 $S$ $S$ be the union of the nodal cubic curve and its three inflectional tangents in the complex projective plane $ℙ^{2}$ . 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 $X = ℙ^{2} ∖ S$ [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 $ℙ^{2}$ . Hence one can apply the method of Reidemeister–Schreier to derive a finite presentation of $π_{1} (\tilde{X})$ , which we carry out in this note.

Communicated by Tomohide Terasoma

Keywords:

AMSC: 14F35, 57M05, 57M10

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