  We describe the completed local rings of the trianguline variety at certain
points of integral weights in terms of completed local rings of algebraic
varieties related to Grothendieck's simultaneous resolution of singularities.
We derive several local consequences at these points for the trianguline
variety: local irreducibility, description of all local companion points in the
crystalline case, combinatorial description of the completed local rings of the
fiber over the weight map, etc. Combined with the patched Hecke eigenvariety
(under the usual Taylor-Wiles assumptions), these results in turn have several
global consequences: classicality of crystalline strictly dominant points on
global Hecke eigenvarieties, existence of all expected companion constituents
in the completed cohomology, existence of singularities on global Hecke
eigenvarieties.
