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Unformatted text preview: THE HESSE PENCIL OF PLANE CUBIC CURVES MICHELA ARTEBANI AND IGOR DOLGACHEV Abstract. This is a survey of the classical geometry of the Hesse configuration of 12 lines in the projective plane related to inflection points of a plane cubic curve. We also study two K3 surfaces with Picard number 20 which arise naturally in connection with the configuration. 1. Introduction In this paper we discuss some old and new results about the widely known Hesse configuration of 9 points and 12 lines in the projective plane P 2 ( k ): each point lies on 4 lines and each line contains 3 points, giving an abstract configuration (12 3 , 9 4 ). Through most of the paper we will assume that k is the field of complex numbers C although the configuration can be defined over any field containing three cubic roots of unity. The Hesse configuration can be realized by the 9 inflection points of a nonsingular projective plane curve of degree 3. This discovery is attributed to Colin Maclaurin (1698-1746) (see [44], p. 384), however the configuration is named after Otto Hesse who was the first to study its properties in [24], [25] 1 . In particular, he proved that the nine inflection points of a plane cubic curve form one orbit with respect to the projective group of the plane and can be taken as common inflection points of a pencil of cubic curves generated by the curve and its Hessian curve. In appropriate projective coordinates the Hesse pencil is given by the equation (1) λ ( x 3 + y 3 + z 3 ) + μxyz = 0 . The pencil was classically known as the syzygetic pencil 2 of cubic curves (see [9], p. 230 or [16], p. 274), the name attributed to L. Cremona. We do not know who is responsible for the renaming the pencil, but apparently the new terminology is widely accepted in modern literature (see, for example, [4]). Recently Hesse pencils have become popular among number-theorists in connec- tion with computational problems in arithmetic of elliptic curves (see, for exam- ple, [49]), and also among theoretical physicists, for example in connection with homological mirror symmetry for elliptic curves (see [54]). The group of projective automorphisms which transform the Hesse pencil to itself is a group G 216 of order 216 isomorphic to the group of affine transformations with determinant 1 of the plane over the field F 3 . This group was discovered by C. Jordan in 1878 [30] who called it the Hessian group . Its invariants were described in 1889 by H. Maschke in [35]. A detailed historical account and the first figure of the Hesse pencil can be found in [21]. The first author was supported in part by: PRIN 2005: Spazi di moduli e teoria di Lie , Indam (GNSAGA) and NSERC Discovery Grant of Noriko Yui at Queen’s University, Canada....
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This note was uploaded on 02/24/2012 for the course MATH 285 taught by Professor Igordolgachev during the Fall '04 term at University of Michigan-Dearborn.

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