The 21 Reducible Polars of Klein’s Quartic

Abstract

We describe the singularities and related properties of the arrangement of 21 reducible polars of Klein’s quartic, containing Klein’s well-known arrangement of 21 lines.

KEYWORDS:

• Klein quartic curve
• line arrangements
• plane curve arrangements
• freeness
• symbolic and ordinary powers of ideals

AMS Subject Classification:

• 14H50; 14C17, 14J26, 14C20, 14N05, 14E05, 13A15, 13F20

Acknowledgments

We want to thank I. Dolgachev for sharing with us facts and references on the Steinerian curve, which were essential for completing this work. We would also like to thank an anonymous referee for crucial comments that allowed to improve Section 5 devoted to the freeness of arrangements, and for suggesting Remarks 4.2 and 4.3.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 Not to be confused with the nodal sextic usually associated with Wiman, whose group of automorphisms is isomorphic to S₅.

Additional information

Funding

This work was begun during the first author’s visit at Universitat Autònoma de Barcelona, under the financial support of the Spanish MINECO grant MTM2016-75980-P, which also supports the second author. During the project Piotr Pokora was supported by the program for young researchers at Institute of Mathematics Polish Academy of Sciences. In the first phase of the project Piotr Pokora was a member of Institute of Algebraische Geometrie at Leibniz Universität Hannover.