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The Hesse pencil of plane cubic curves. (English) Zbl 1192.14024
This paper deals with a well-known configuration of 9 points and 12 lines in \(\mathbb P^2(k)\), the Hesse configuration, in which each point lies on 4 lines and each line contains 3 points. Such points can be chosen as the nine inflection points of a nonsingular plane cubic curve, and they can be as well taken as common inflection points of the Hesse pencil \[ \lambda (x^3+y^3+z^3)+\mu xyz=0. \] The group of plane automorphisms preserving the Hesse pencil has order 216 and it is isomorphic to \((\mathbb Z/3\mathbb Z)SL)^2 \rtimes SL(2, \mathbb F_3)\). The algebra of invariant polynomials of one of its extensions to a subgroup of \(GL(3,\mathbb C)\) has a generator of degree 6 defining a plane sextic \(C_6\).
The double covers of \(\mathbb P^2\) branched over \(C_6\) and over the singular sextic \(C'_6\) with 8 cuspidal singularities are both \(K3\) surfaces and they are singular in the sense of Shioda, i.e. the subgroup of algebraic cycles in the second cohomology group is of maximal rank.
The authors compute the intersection form defined by the cup-product on these subgroups and describe the geometrical meaning of the set of intersection points \(C_6\) cuts each curve of the Hesse pencil at.

14H10 Families, moduli of curves (algebraic)
14H50 Plane and space curves
14J10 Families, moduli, classification: algebraic theory
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