0000000000764427
AUTHOR
Oliver Labs
Illustrating the classification of real cubic surfaces
Knorrer and Miller classified the real projective cubic surfaces in P(R) with respect to their topological type. For each of their 45 types containing only rational double points we give an affine equation, s.t. none of the singularities and none of the lines are at infinity. These equations were found using classical methods together with our new visualization tool surfex. This tool also enables us to give one image for each of the topological types showing all the singularities and lines.
Eine K3-Fläche - oder: Eine Anmerkung zum Umschlag
K3-Flachen bekamen ihren Namen im 20. Jahrhundert vom franzosischen Mathematiker Andre Weil; er schreibt: Im zweiten Teil meines Berichts geht es um kahlersche Varietaten, K3 genannt, zu Ehren von Kummer, Kodaira, Kahler und des Berges K2 im Kaschmir-Gebirge. Dies sind komplexe Flachen, deren exakte Definition zwar recht technisch ist, von denen man jedoch einen guten Eindruck bekommt, wenn man Flachen vom Grad 4 im komplexen Dreiraum betrachtet, die namlich grostenteils K3Flachen sind.
Real Line Arrangements and Surfaces with Many Real Nodes
A long standing question is if the maximum number μ(d) of nodes on a surface of degree d in P( ) can be achieved by a surface defined over the reals which has only real singularities. The currently best known asymptotic lower bound, μ(d) 5 12 d, is provided by Chmutov’s construction from 1992 which gives surfaces whose nodes have non-real coordinates. Using explicit constructions of certain real line arrangements we show that Chmutov’s construction can be adapted to give only real singularities. All currently best known constructions which exceed Chmutov’s lower bound (i.e., for d = 3, 4, . . . , 8, 10, 12) can also be realized with only real singularities. Thus, our result shows that, up t…
A Visual Introduction to Cubic Surfaces Using the Computer Software Spicy
At the end of the 19th century geometers like Clebsch, Klein and Rodenberg constructed plaster models in order to get a visual impression of their surfaces, which are so beautiful from an abstract point of view. But these were static visualizations. Using the computer program Spicy 1, which was written by the second author, one can now draw algebraic curves and surfaces depending on parameters interactively.