5 research outputs found
Chain-order polytopes: toric degenerations, Young tableaux and monomial bases
Our first result realizes the toric variety of every marked chain-order
polytope (MCOP) of the Gelfand--Tsetlin poset as an explicit Gr\"obner (sagbi)
degeneration of the flag variety. This generalizes the
Sturmfels/Gonciulea--Lakshmibai/Kogan--Miller construction for the
Gelfand--Tsetlin degeneration to the MCOP setting. The key idea of our approach
is to use pipe dreams to define realizations of toric varieties in Pl\"ucker
coordinates. We then use this approach to generalize two more well-known
constructions to arbitrary MCOPs: standard monomial theories such as those
given by semistandard Young tableaux and PBW-monomial bases in irreducible
representations such as the FFLV bases. In an addendum we introduce the notion
of semi-infinite pipe dreams and use it to obtain an infinite family of poset
polytopes each providing a toric degeneration of the semi-infinite
Grassmannian
Resolving singularities of curves with one toric morphism
We give an explicit positive answer, in the case of reduced curve singularities, to a question of B. Teissier about the existence of a toric embedded resolution after reembedding. In the case of a curve singularity pC,Oq contained in a non singular surface S such a reembedding may be defined in terms of a sequence of maximal contact curves associated to C. We prove that there exists a toric modification, after reembedding, which provides an embedded resolution of C. We use properties of the semivaluation space of S at O to describe how the the dual graph of the minimal embedded resolution of C may be seen on the local tropicalization of S associated to this reembedding