Unexpected properties of the Klein configuration of 60 points in \(\mathbb{P}^3\) (Q6590158)

The authors investigate intriguing properties of the Klein configuration, which consists of 60 points in projective space over the complex numbers. This configuration is notable for several reasons, including being the dual configuration to the 60 reflection planes of the group \( G_{31} \) in the Shephard-Todd list.\N\NThis paper contributes to the study of configurations with unexpected cones (initiated in [\textit{B. Harbourne} et al., Mich. Math. J. 70, No. 2, 301--339 (2021; Zbl 1469.14107)]), with particular emphasis on the surprising \textit{geproci} property -- a term introduced in this work, meaning that the general projection is a complete intersection. Non-trivial examples of geproci sets were previously explored in the appendix of \textit{L. Chiantini} and \textit{J. Migliore} [Trans. Am. Math. Soc. 374, No. 4, 2581--2607 (2021; Zbl 1457.14105)].\N\NIn Theorem 4.3, the authors provide the equation of a degree 6 cone containing the 60-point set \( Z_{60} \) and with vertex at a general point \( P \). This cone, which is unique and irreducible, is unexpected for the Klein configuration. Furthermore, the set \( Z_{60} \) lies on 10 lines, resulting in its general projection to \( \mathbb{P}^2 \) being a complete intersection of two curves of degrees 6 and 10, and making \( Z_{60} \) a geproci set (Theorem 5.5).