The singularities of Feynman diagrams in the coordinate space and the $\alpha$-representation

The $\alpha$-representation is used to prove restrictions on the wave front of Feynman diagrams in the coordinate space, and also to establish the form of a singularity lying on the surface $\operatorname{det}S=0$ ($S$ is the matrix composed of the elements $s_{jj'}=(x_j-x_{j'})^2$) for the simplest diagram with four external vertices.