Volume 9, Issue 3

Latin trades can be interpreted geometrically as oriented surfaces. Those of genus zero are called spherical. The paper contains a construction that shows how all latin trades can be obtained from the spherical ones by a cut-and-paste method that raises the genus

We study configurations of 2-planes in that are combinatorially described by the Petersen graph. We discuss conditions for configurations to be locally Cohen–Macaulay and describe the Hilbert scheme of such arrangements. An analysis of the homogeneous ideals of these configurations leads, via linkage, to a class of smooth, general type surfaces in . We compute their numerical invariants and show that they have the unusual property that they admit (multiple) 7-secants. Finally, we demonstrate that the construction applied to Petersen arrangements with additional symmetry leads to surfaces with exceptional automorphism groups.

We adapt numerical continuation methods to compute all solutions of finite difference discretizations of nonlinear boundary value problems involving the Laplacian in two dimensions. New solutions on finer meshes are obtained from solutions on coarser meshes using a complex homotopy deformation. Two difficulties arise. First, the number of solutions typically grows with the number of mesh points and some form of filtering becomes necessary. Secondly, bifurcations may occur along homotopy paths of solutions and efficient methods to swap branches are developed when the mappings are analytic. For polynomial nonlinearities we generalize an earlier strategy for finding all solutions of two-point boundary value problems in one dimension and then introduce exclusion algorithms to extend the method to general nonlinearities.

Unstable minimal surfaces are the unstable stationary points of the Dirichlet integral. In order to obtain unstable solutions, the method of the gradient flow together with the minimax-principle is generally used, an application of which was presented in [Struwe, J. Reine Angew. Math. 349: 1–23, 1984] for minimal surfaces in Euclidean space. We extend this theory to obtain unstable minimal surfaces in Riemannian manifolds. In particular, we consider minimal surfaces of annulus type.

Izmestiev and Joswig described how to obtain a simplicial covering space (the partial unfolding ) of a given simplicial complex, thus obtaining a simplicial branched cover [Adv. Geom. 3:191–255, 2003]. We present a large class of branched covers which can be constructed via the partial unfolding. In particular, for d ≤ 4 every closed oriented PL d -manifold is the partial unfolding of some polytopal d -sphere.

Let X ⊂ be a smooth variety. The embedding in gives naturally rise to the notion of embedded tangent spaces. That is the locus spanned by tangent lines to a point x ∈ X . Generally the embedded tangent space intersects the variety X only at the point x . In this paper I am interested in those X for which this intersection, for x ∈ X general, is a positive dimensional subvariety. The results of this paper support the conjecture that these varieties are built out of some special varieties.