Configurations of an Articulated Arm and Singularities of Special Multi-Flags

P. Mormul has classified the singularities of special multi-flags in terms of “EKR class” encoded by sequences $j_1,\dots, j_k$ of integers (see [Singularity Theory Seminar, Warsaw University of Technology, Vol. 8, 2003, 87–100] and [Banach Center Publ., Vol. 65, Polish Acad. Sci., Warsaw, 2004, 157–178]). However, A.L. Castro and R. Montgomery have proposed in [Israel J. Math. 192 (2012), 381–427] a codification of singularities of multi-flags by RC and RVT codes. The main results of this paper describe a decomposition of each “EKR” set of depth $1$ in terms of RVT codes as well as characterize such a set in terms of configurations of an articulated arm. Indeed, an analogue description for some “EKR” sets of depth $2$ is provided. All these results give rise to a complete characterization of all “EKR” sets for $1\leq k\leq 4$.