Desmic Systems and Desmic Surfaces

In this thesis we are mainly interested in desmic systems and desmic surfaces. A desmic system is the set of three distinct tetrahedra, such related that any edge of one tetrahedron intersects two opposite edges of each of the two other tetrahedra. A desmic surface is the zero set of a linear combination of the defining equations of three tetrahedra in a desmic system, and each desmic system gives a pencil of desmic surfaces. Since all desmic systems are shown to be projectively equivalent, we can restrict to a particularly nice subset of desmic systems to study. We choose the desmic systems strictly invariant under H, the quotient of the level (2,2)-Heisenberg group by its commutator subgroup, for which the parameter space, the projective four space, have a particularly nice symmetry. We find that the generic desmic surface has twelve ordinary double points and no other singularities. The singularities are the corners of three tetrahedra in a desmic system, and lie by three on 16 lines contained in the surface in such a way that each singularity is contained in four of the lines. We also present two modes of classification for the surfaces of interest. The first is due to the orbits under H of the singular locus of a surface, and the second is due to the ramification locus of the linear projection from a singularity to a plane