Singularities of configuration and graph hypersurfaces

Abstract

I show that the singularities of a configuration hypersurface correspond to the points whose corank is at least two. The result generalizes to compute multiplicities at higher corank loci in configuration hypersurfaces. Though the result is similar to the case of a generic determinantal variety, the proof proceeds from a different perspective. The key is simple linear algebra: the rank of a quadratic form is determined by its ranks when restricted to each hyperplane in a complete set of hyperplanes.

I review the construction of the first and second graph hypersurfaces as examples of configuration hypersurfaces. In particular, the singularities of the first and second graph hypersurfaces are corank at least two loci. As an application, the singularities of the first graph hypersurface are contained in the second graph hypersurface, and the singularities of the second graph hypersurface are contained in the first hypersurface.

I show that the map to which the graph hypersurface is dual is not an embedding in general. Therefore, the incidence variety may not resolve the singularities of the graph hypersurface.

I present a formula that describes the graph polynomial of a graph produced by a specific gluing-deleting operation on a pair of graphs. The operation produces log-divergent graphs from log-divergent graphs, so it is useful for generating examples relevant for physics. A motivic understanding of the operation is still missing.