Denote by \(\mathfrak{g}\simeq \mathfrak {gl}_n^{[F_v^+:\mathbf{Q}_p]}\) (respectively, \(\mathfrak{b}\)) the \(L\)-Lie algebra of \(G:=({\text{Res}}_{F^+_v/\mathbf{Q}_p}\text{GL}_{n/F_v^+})_L\) (respectively, of the Borel subgroup \(B\) of upper triangular matrices). The authors 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 \(\widetilde{\mathfrak{g}}\rightarrow \mathfrak{g}\) of singularities, where \(\widetilde{\mathfrak{g}}:= \{(gB,\psi)\in G/B\times \mathfrak{g}: \text{Ad}(g^{-1})\psi\in \mathfrak{b}\}\subseteq G/B\times \mathfrak{g}\).
The authors derive several local consequences at these points for the trianguline variety: local irreducibility, description of all local companion points in the crystalline case, and combinatorial description of the completed local rings of the fiber over the weight map. Taylor-Wiles assumptions are as follows: \(p>2\), the field \(F\) is unramified over \(F^+\), \(F\) does not contain a nontrivial root \(\sqrt[p]{1}\) of unity, and \(G\) is quasi-split at all finite places of \(F^+\), \(U_v\) is hyperspecial when the finite place \(v\) of \(F^+\) is inert in \(F\), and \(\overline{\rho}(\text{Gal}(\overline F/F(\sqrt[p]{1}))\) is adequate. Combined with the patched Hecke eigenvariety under the usual Taylor-Wiles assumptions, these results have global consequences, such as classicality of crystalline strictly dominant points on global Hecke eigenvarieties, existence of all expected companion constituents in the completed cohomology, and existence of singularities on global Hecke eigenvarieties.