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The 21 reducible polars of Klein’s quartic. (English) Zbl 1460.14071
The paper under review deals with the study of arrangements of lines, with an eye towards extension to higher degree curves, with particular reference to the Klein quartic curve \(C\) and its rich geometry. The curve \(C\) is defined by the homogeneous equation \(x^3y + y^3z + z^3x=0\) in the complex projective plane, and its relevant properties derive from the fact that its automorphism group has the maximal possible order, according to Hurzwitz’s bound. Moreover, it is realized by a subgroup \(G \subset \mathrm{PGL}(3, \mathbb C)\). There exists a unique arrangement of \(21\) lines invariant under the action of \(G\), discovered by Klein, which is particularly symmetric and extremal in many senses. The authors study an arrangement of lines and conics closely related to it. Every sufficiently general smooth plane quartic has 21 reducible polars and, in the case of \(C\), each of them splits as the union of one of the lines in Klein’s arrangement plus a smooth conic. The arrangement \(\mathcal K\) the authors study is that consisting of these 21 lines and 21 conics, taken together. All singularities of \(\mathcal K\) are described, in particular their position with respect to relevant curves associated with \(C\), as well as their number according to the various multiplicities. Further specific properties of \(\mathcal K\) are discussed in connection with problems in the study of arrangements of plane curves addressed in recent years by several authors.

14H50 Plane and space curves
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
14J26 Rational and ruled surfaces
14C20 Divisors, linear systems, invertible sheaves
14N05 Projective techniques in algebraic geometry
14E05 Rational and birational maps
13A15 Ideals and multiplicative ideal theory in commutative rings
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
Full Text: DOI
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