Birational sequences for the Grassmannian Gr(3,n)
Joaquin Torres Henestroza
TL;DR
The paper develops a framework to construct toric degenerations of the Grassmannian $Gr(3,n)$ via Gröbner degenerations coming from valuations induced by iterated birational sequences. It introduces lowest-term valuations $ u_S$ associated to sequences $S$ built from PBW data, producing weighting matrices $M_S$ that determine initial ideals $ ext{in}_{M_S}( mathcal{I}_{3,n})$, and proves that for iterated sequences these initial forms of Plücker relations are binomial. Focusing on $Gr(3,6)$, the authors compute $240$ weight vectors up to $S_6$-equivalence, organizing them into four orbits, each corresponding to a toric degeneration; they also show that not every tropical maximal cone arises from an iterated sequence. The work links tropical geometry, birational sequences, and explicit toric degenerations, providing both a constructive method and a classification in the $n=6$ case, with insights into the geometry of the tropical Grassmannian and connections to plabic graphs.
Abstract
Following the ideas of Bossinger and Fang, Fourier, and Littelman, we study iterated sequences for the Grassmannian $\operatorname{Gr} (3, n)$ as a special class of birational sequences. For each iterated sequence $S$, there is a weighting matrix $M_{S}$ corresponding to a valuation on the rational coordinate ring and we show that the initial form of a Plücker relation $\operatorname{in}_{M_S} (R_{I,J} )$ is binomial. We show that, in some cases, the cones $C_S$ in the tropical Grassmannian that satisfy $\operatorname{in}_{M_S} (\mathcal{I}_{3,n}) = \operatorname{in}_{C_S} (\mathcal{I}_{3,n})$ only depend on the first two indices used in each iteration. In the case of $\operatorname{Gr} (3, 6)$, these cones are obtained computationally and are classified up to automorphism induced by the symmetric group $S_6$.