Irene Polo-Blanco

Explicit Real Cubic Surfaces

The topological classification of smooth real cubic surfaces is recalled and compared to the classification in terms of the number of real lines and of real tritangent planes, as obtained by L. Schlafli in 1858. Using this, explicit examples of surfaces of every possible type are given.

A remark on parameterizing nonsingular cubic surfaces

Extending a geometric construction due to Sederberg and to Bajaj, Holt, and Netravali, an algorithm is presented for parameterizing a nonsingular cubic surface by polynomials of degree three. The fact that such a parametrization exists is classical. The present algorithm is, by its purely geometric nature, a very natural one. Moreover, it contains a practical way of finding all lines in an implicitly given cubic surface. Two explicit examples are presented, namely the classical Clebsch diagonal surface and the cubic Fermat surface.

Ruled quartic surfaces, models and classification

New historical aspects of the classification, by Cayley and Cremona, of ruled quartic surfaces and the relation to string models and plaster models are presented. In a ‘modern’ treatment of the classification of ruled quartic surfaces the classical one is corrected and completed. The string models of Series XIII of some ruled quartic surfaces (manufactured by L. Brill and by M. Schilling) are based on a result of Rohn concerning curves in

Algorithms for Del Pezzo surfaces of degree 5 (construction, parametrization)

{\mathbb{P}^1\times \mathbb{P}^1}

Alicia Boole Stott, a geometer in higher dimension

of bi-degree (2, 2). This is given here a conceptional proof.

A symbolic analysis of Vermeille and Borkowski polynomials for transforming 3D Cartesian to geodetic coordinates

We review different group based algorithms for matrix multiplication and discuss the relations between the combinatorial properties of the used group and the complexity of these algorithms. We introduce a variant of an algorithm based on the ideas exposed in [4] well-adapted for experimentation. Finally we show how this approach can also be used for matrix multiplication over a field with characteristic different from 2.

Four-Dimensional Polytopes: Alicia Boole Stott’s Algorithm

It is well known that every Del Pezzo surface of degree 5 defined over a field k is parametrizable over k. In this paper, we give an algorithm for parametrizing, as well as algorithms for constructing examples in every isomorphism class and for deciding equivalence.

A classical approach to the study of Archimedean four-dimensional polytopes

By a criterion of Swinnerton-Dyer it is known that a smooth cubic surface S defined over Q is birationally trivial over Q if and only if S ( Q ) ź ź and S contains a Gal ( Q ź , Q ) -stable set of 2, 3 or 6 skew lines. In this text we describe birationally trivial smooth cubic surfaces over Q and provide algorithms to construct explicit cubic surfaces for each of the cases in Swinnerton-Dyers criterion.