Remarks on Jordan algebras (dim 9, deg 3), cubic surfaces, and del Pezzo surfaces (deg 6)

by Markus Rost (Notes, December 1996, 5 pages)

We associate to a pair of separable degree 3 extensions a 9-dimensional Jordan algebra of degree 3 and consider related del Pezzo surfaces.

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The discriminant algebra of a cubic algebra

by Markus Rost (Notes, March 2002, 3 pages)

In this note we describe a construction of the discriminant algebra of a flat cubic algebra over any commutative ring.

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The construction is contained in the article "Discriminant Algebras of finite rank algebras and quadratic trace modules" by Ottmar Loos (see next item).

Notes on Morley's theorem

by Markus Rost (Notes, August 2003, 10 pages)

These notes evolved from a study of the fairly recent proof of Morley's theorem by Connes.

We briefly discuss the relation of Connes' point of view of affine transformations with triangles and quadrangles. Then we give a proof of Morley's theorem à la Connes. Finally we consider a purely group theoretic lemma which implies Connes' lemma on affine transformations.

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Some sources for Morley's theorem:

Notes on cubic equations

by Markus Rost (Notes, August 2003, 4 pages)

For a cubic element (or a triangle) we define its "basic line". It leads to the following normalization of cubic equations:

The method works in all characteristics for generic cubic elements. We describe for this 1-parameter family the discriminant extension and the corresponding variant of Cardano's formula:

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The holomorphic extension of triangle functions

by Markus Rost (Notes, February/April/May/August 2004, 21 pages)

Examples of triangle functions are given by the orthocenter, the circumcenter, the incenter, the excenters, the pedal points, the nine-point center, etc. Each of these functions has a "holomorphic extension" which is a complex function in four variables: the triangle points and the orthocenter.

For instance, for a triangle xyz with orthocenter t, the intersection of the lines xy and zt (a "pedal point" of xyz) is given by:

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Related texts

• Alain Connes, Symetries Galoisiennes et Renormalisation, Seminaire Poincare Octobre 2002, arXiv:math.QA/0211199

The variety of angles

by Markus Rost (Notes, March 2004, 6 pages)

This text grew out of an attempt to understand the derived triangle construction. We consider the variety of (algebraic) angles of an Euclidean quadrangle and an action of S6 on it. (More details can be found below in the diploma thesis by S. Glied.) This action is induced from an action on a 5-dimensional torus.

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by Svenja Glied (diploma thesis, Universität Bielefeld, 2007, 28 pages)

The major result of the thesis is the presentation of an S_6-symmetry of the set of similarity classes of (nondegenerate) Euclidean quadrangles. It is generated by the obvious S_4-symmetry (permutation of the points) and the so-called pedal triangle construction.

Furthermore, Lemma 1.18 describes the relations between the angles of a quadrangle. These are given by the obvious relations stemming from subtriangles (sum of angles = pi) but there is also a non-linear relation.

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The metric of a (n+2)-gon in affine n-space

by Markus Rost (Notes, August 2004, 12 pages)

For n+2 points in affine n-space in general position there is a canonical metric (unique up to a similarity factor) such that complementary faces of the (n+2)-gon are orthogonal.

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