27 retas na superfície cúbica

Abstract

In this dissertation we present the idea of proof of the theorem that on a non-singular cubic surface in P3 contains 27 straight lines. Also, they are shown as these lines intersect, Which planes form, which are sextuples of straight lines that do not intersect (Sextuplets of Schlaefli). An explicit example of the surface defined on Q such that all its lines are defined on Q is treated. Finally, it is shown that there is an isomorphism between a cubic surface and a 6-point swollen plane. This is a classic subject of nineteenth-century research, but it has development to this day. As an introduction, the dissertation contains the definition of the affine space, projective space, its subspaces, Grassmannian varieties of the subspaces and especially the G(2, 4) variety of the straight lines in P3. Then the initial construction of a straight line at P3 and P5 which cross it is treated. It is shown that there is one and only one cubic surface containing these 6 lines. Using the theorem that 4 straight in space have 2 secant lines, it is possible to construct all 27 straight lines on this surface. Also, its intersection matrix is found. This gives us solutions to various combinatorial problems related to this configuration of the lines. Projection of the surface of two straight lines allows to show that there is an isomorphism between the surface and a plane swollen in 6 points. In his turn, this isomorphism makes it easier to obtain the matrix of the intersection of the lines. Finally, explicit calculations for a simple configuration of the 6 straight lines are made. As a result, we obtain a surface such that all its 27 straight lines have rational coordinates.