We compute toric degenerations arising from the tropicalization of the full flag varieties Fℓ 4 and Fℓ 5 embedded in a product of Grassmannians. For Fℓ 4 and Fℓ 5 we compare toric degenerations arising from string polytopes and the FFLV polytope with those obtained from the tropicalization of the flag varieties. We also present a general procedure to find toric degenerations in the cases where the initial ideal arising from a cone of the tropicalization of a variety is not prime.

We establish an explicit bijection between the toric degenerations of the Grassmannian Gr(2, n) arising from maximal cones in tropical Grassmannians and the ones coming from plabic graphs corresponding to Gr(2, n). We show that a similar statement does not hold for Gr (3, 6).

We introduce the notion of a $Y$-pattern with coefficients and its geometric counterpart: an $\mathcal {X}$-cluster variety with coefficients. We use these constructions to build a flat degeneration of every skew-symmetrizable specially completed $\mathcal {X}$-cluster variety $\widehat {\mathcal {X} }$ to the toric variety associated to its g-fan. Moreover, we show that the fibers of this family are stratified in a natural way, with strata the specially completed $\mathcal {X}$-varieties encoded by $\operatorname {Star}(\tau )$ for each cone $\tau$ of the $\mathbf {g}$-fan. These strata degenerate to the associated toric strata of the central fiber. We further show that the family is cluster dual to $\mathcal {A}_{\mathrm {prin}}$ of Gross, Hacking, Keel and Kontsevich [Canonical bases for cluster algebras, J. Amer. Math. Soc. 31 (2018), 497–608], and the fibers cluster dual to $\mathcal {A} _t$. Finally, we give two applications. First, we use our construction to identify the toric degeneration of Grassmannians from Rietsch and Williams [Newton-Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians, Duke Math. J. 168 (2019), 3437–3527] with the Gross–Hacking–Keel–Kontsevich degeneration in the case of $\operatorname {Gr}_2(\mathbb {C} ^{5})$. Next, we use it to link cluster duality to Batyrev–Borisov duality of Gorenstein toric Fanos in the context of mirror symmetry.

Let V be the weighted projective variety defined by a weighted homogeneous ideal J and C a maximal cone in the Gröbner fan of J with m rays. We construct a flat family over A m that assembles the Gröbner degenerations of V associated with all faces of C. This is a multi-parameter generalization of the classical one-parameter Gröbner degeneration associated to a weight. We explain how our family can be constructed from Kaveh-Manon's recent work on the classification of toric flat families over toric varieties: it is the pull-back of a toric family defined by a Rees algebra with base X C (the toric variety associated to C) along the universal torsor A m → X C . We apply this construction to the Grassmannians Gr(2, C n ) with their Plücker embeddings and the Grassmannian Gr 3, C 6 with its cluster embedding. In each case, there exists a unique maximal Gröbner cone whose associated initial ideal is the Stanley-Reisner ideal of the cluster complex. We show that the corresponding cluster algebra with universal coefficients arises as the algebra defining the flat family associated to this cone. Further, for Gr(2, C n ) we show how Escobar-Harada's mutation of Newton-Okounkov bodies can be recovered as tropicalized cluster mutation.

We study the combinatorics of pseudoline arrangements and their relation to the geometry of flag and Schubert varieties. We associate to each pseudoline arrangement two polyhedral cones, defined in a dual manner. We prove that one of them is the weighted string cone by Littelmann and Berenstein-Zelevinsky. For the other we show how it arises in the framework of cluster varieties and mirror symmetry by Gross-Hacking-Keel-Kontsevich: for the flag variety the cone is the tropicalization of their superpotential while for Schubert varieties a restriction of the superpotential is necessary.We prove that the two cones are unimodularly equivalent. As a corollary of our combinatorial result we realize Caldero's toric degenerations of Schubert varieties as GHKKdegeneration using cluster theory.1 Littelmann showed in [Lit98] that the Gelfand-Tsetlin polytope is unimodular equivalent to a certain string polytope. STRING CONE AND SUPERPOTENTIAL 3The cone S w 0 appears in the framework of mirror symmetry for cluster varieties [GHKK18]: denote by B − ⊂ SL n the Borel subgroup of lower triangular matrices and by U ⊂ B (resp. U − ⊂ B − ) the unipotent radical with all diagonal entries being 1. The double Bruhat cell G e,w 0 = B − ∩ Bw 0 B is an A-cluster variety (see [BFZ05]) and can be identified with an open subset of Bw 0 B/U . Let X the be dual of the A-cluster variety G e,w 0 and let s 0 = sŵ 0 be the seed of the cluster algebra C[G e,w 0 ] corresponding to the reduced expressionŵ 0 = s 1 s 2 s 1 · · · s n−1 · · · s 2 s 1 . Let W be the superpotential defined by the sum of the ϑ-functions for frozen variables in s 0 as introduced in [GHKK18]. Then W trop denotes the tropicalization of the superpotential. Magee has shown in [Mag15] (see also Goncharov-Shen in [GS15]) thatWe show that mutation of the pseudoline arrangement and hence of the cone S w 0 , is compatible with mutation of the superpotential [GHK15] by introducing mutation of GP-paths. We obtain the following result 2 (see Corollary 2):Theorem 2. Let w 0 be an arbitrary reduced expression of w 0 ∈ S n and s w 0 be the seed corresponding to the pseudoline arrangement, X sw 0 the toric chart of the seed s w 0 . Thenthe polyhedral cone defined by the tropicalization of W expressed in the seed s w 0 .Consider w ∈ S n arbitrary and w a reduced expression of w. Let W be as above and consider its restriction res w (W | Xs w 0 ) to the mirror dual of the A-cluster variety G e,w . Let s w be the corresponding seed in the cluster algebra. Then the tropicalization of the restriction yields again a cone Ξ sw . The last result establishes an answer to the question above for Schubert varieties (see Theorem 8).Theorem 3. Let w ∈ S n , and fix w 0 = ws i ℓ(w)+1 . . . s i N a reduced expression of w 0 ∈ S n . Let s w resp. s w 0 be the corresponding seeds, thenThe article is structured as follows: after introducing relevant notation, we recall pseudoline arrangements and define the two collections of polyhedral objects and unimodular equivalences among them in §3. In §4 we show that ...