Families of Gr\"obner Degenerations, Grassmannians and Universal Cluster Algebras

Let $V$ be the weighted projective variety defined by a weighted homogeneous ideal $J$ and $C$ a maximal cone in the Gr\"obner fan of $J$ with $m$ rays. We construct a flat family over $\mathbb A^m$ that assembles the Gr\"obner degenerations of $V$ associated with all faces of $C$. This is a multi-parameter generalization of the classical one-parameter Gr\"obner 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 $\mathbb A^m \to X_C$. We apply this construction to the Grassmannians ${\rm Gr}(2,\mathbb C^n)$ with their Pl\"ucker embeddings and the Grassmannian ${\rm Gr}\big(3,\mathbb C^6\big)$ with its cluster embedding. In each case, there exists a unique maximal Gr\"obner 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 ${\rm Gr}(2,\mathbb C^n)$ we show how Escobar-Harada's mutation of Newton-Okounkov bodies can be recovered as tropicalized cluster mutation.


Introduction
The theory of Gröbner fans, introduced by Mora and Robbiano [40], is a popular tool in commutative algebra, and one of its modern applications is degenerating ideals into simpler ones such as monomial, binomial or toric ideals. More precisely, let K be an algebraically closed field and J ⊆ K[x 1 , . . . , x n ] a d-weighted homogeneous ideal for some d ∈ Z n >0 . The Gröbner fan of J is a complete fan in R n whose elements represent weight vectors on the variables x 1 , . . . , x n . Two weight vectors lie in the same open cone if and only if they give rise to the same initial ideal of J, see Definition 2.3. The ideal J defines a weighted projective variety V inside the weighted projective space P(d). Every open cone in the Gröbner fan gives rise to a one-parameter flat family degenerating V to the variety defined by the associated initial ideal of J. This construction is realized by choosing a weight in the relative interior of the cone.
We modify the classical one-parameter construction as follows: for a maximal cone C in the Gröbner fan of J we choose integral generators of its rays r 1 , . . . , r m and denote by r the (m × n)-matrix whose rows are r 1 , . . . , r m . For an element f = α∈Z n ≥0 c α x α of J, we define µ(f ) ∈ Z m as the vector whose i-th entry is min cα =0 {r i · α}. Then the lift of f is where t a for a ∈ Z m denotes the monomial m i=1 t a i i . The lifted idealJ ⊆ K[t 1 , . . . , t m ][x 1 , . . . , x n ] is the ideal generated by the lifts of all polynomials in J. We prove thatJ is generated by the lifts of elements of the reduced Gröbner basis for J and C, see Proposition 3.9. The lifts of these elements are independent of the choice of r and homogeneous with respect to the d-grading on x i 's, see Proposition 3.3. Consequently,J is independent of the choice of r and it defines a variety inside P(d) × A m . Our first main result is the following. Theorem 1.1 (Theorem 3.14). Let J be a weighted homogeneous ideal, C a maximal cone in the Gröbner fan of J and r an (m × n)-matrix whose rows are integral ray generators of C. Then the algebraÃ := K[t 1 , . . . , t m ][x 1 , . . . , x n ]/J is a free K[t 1 , . . . , t m ]-algebra. It defines a flat family such that for every face τ of C there exists a τ ∈ A m with fiber π −1 (a τ ) isomorphic to the variety defined by the initial ideal associated to τ . In particular, generic fibers are isomorphic to Proj(A), where A = K[x 1 , . . . , x n ]/J, and there exist special fibers for every proper face τ ⊂ C.
Next, we explain how the algebraÃ arises in Kaveh-Manon's recent work on the classification of affine toric flat families of finite type over toric varieties [33]. Consider a fan Σ defining a toric variety X Σ which contains a dense torus T . Then a toric family is a T -equivariant flat sheaf A of positively graded algebras of finite type over X Σ such that: (i) the relative spectrum of A has reduced fibers and (ii) its generic fibers are isomorphic to the spectrum of some positively graded K-algebra A. Such families are classified by so-called PL-quasivaluations on A whose codomain is the semifield of piecewise linear functions on the intersection of Σ with the cocharacter lattice of T (see [33,Section 1.1] or Section 3.1 below). Given a PL-quasivaluation, Kaveh-Manon construct a sheaf of Rees algebras on X Σ that is a toric family. In the special case when Σ is a cone, their construction yields a single Rees algebra rather than a sheaf. We prove the following result: Theorem 1.2 (Theorem 3. 19). Let J ⊆ K[x 1 , . . . , x n ] be a weighted homogeneous ideal and C a maximal cone in the Gröbner fan of J. Let R C (A) be the Rees algebra associated to the PL-quasivaluation on A = K[x 1 , . . . , x n ]/J defined by C and let ψ : Spec(R C (A)) → X C be the corresponding toric family. The morphism π : Spec Ã → A m fits into a pull-back diagram as follows: Here p C : A m → X C is the universal torsor of X C (see (3.10) for details) and m is the number of rays of C. Both flat families, the one defined by ψ and the one defined by π, are direct sums of line bundles indexed by the standard monomial basis associated to the maximal cone C.
Toric degenerations are a particular class of flat families with a single special fiber that is a toric variety. They are of significant interest and have been widely studied in recent years. Among the faces of the maximal cone C one may find binomial prime initial ideals that hence define toric Gröbner degenerations of V . Such faces lie in the tropicalization of J denoted Trop(J), which by definition is the subfan of the Gröbner fan consisting of cones whose associated initial ideal does not contain monomials. Let Σ be the intersection of C with the tropicalization of J. Then the affine space A m contains the universal torsor T Σ of the toric variety X Σ and all fibers of π −1 (T Σ ) → T Σ correspond to initial ideals of cones in Σ. In Corollary 3.20 we show how the family defined by π| π −1 (T Σ ) also arises as a pull-back from a toric family defined by a sheaf of Rees algebras on X Σ .
Grassmannians and cluster algebras. The interest of studying toric degenerations in the context of cluster algebras has grown in the last years (see for example [9,28]). Therefore, we would like to understand the framework introduced above from the perspective of cluster algebras. As a first step in this direction, we analyze in depth the situation for the Grassmannians Gr(2, C n ) and Gr 3, C 6 whose coordinate rings are cluster algebras.
For Gr(2, C n ) we choose its Plücker embedding and obtain the homogeneous coordinate ring A 2,n as a quotient of the polynomial ring in Plücker variables C[p ij : 1 ≤ i < j ≤ n] by the Plücker ideal I 2,n . It was shown in [23] that A 2,n is a cluster algebra in which the cluster variables are in one-to-one correspondence with Plücker coordinatesp ij ∈ A 2,n and the seeds are in one-to-one correspondence with triangulations of the n-gon. Every triangulation of the n-gon gives rise to a toric degeneration of Gr(2, C n ) obtained by adding principal coefficients to A 2,n at the corresponding seed [28]. Principal coefficients were introduced by Fomin and Zelevinsky in [25] and are a core concept in the theory of cluster algebras. A central piece of the construction is endowing every cluster variable (i.e., every Plücker coordinate) with a so-called g-vector depending on the fixed seed. We want to understand the toric degenerations coming from principal coefficients in the context of Gröbner theory: to achieve this first fix a triangulation T . We use g-vectors for Plücker coordinates to construct a weight vector w T , see Definition 4. 35. In Proposition 4.36 we prove that the initial ideal with respect to w T is binomial and prime, hence w T lies in the tropicalization of I 2,n . Moreover, the central fiber of the Gröbner degeneration induced by w T is isomorphic to the central fiber of the toric degeneration induced by endowing A 2,n with principal coefficients at the seed determined by T , see Corollary 4.45. There is a single object in cluster theory that simultaneously encodes principal coefficients at all seeds and thus the g-vectors of all Plücker coordinates with respect to all triangulations. Namely, the cluster algebra with universal coefficients A univ 2,n associated to A 2,n , see Definition 4.24. In spirit, this algebra is very similar to the algebra defining the flat family associated to a maximal cone in the Gröbner fan of I 2,n as it encodes various toric degenerations of Gr(2, C n ) at the same time. It is therefore natural to ask if the cluster algebra with universal coefficients fits into the above framework. The following result answers this question for Gr(2, C n ): Theorem 1.3. There exists a maximal cone C in the Gröbner fan of I 2,n whose rays are in bijection with diagonals of the n-gon. Moreover, the unique cone C has the following properties: (i) The standard monomial basis for A 2,n associated with C coincides with the basis of cluster monomials (Proposition 4.43).
(ii) For every triangulation T of the n-gon the weight vector w T lies in the boundary of C and the intersection of C with the tropicalization of I 2,n is the totally positive part of Trop(I 2,n ) (Proposition 4.44 and Theorem 4.46).
(iii) LetĨ 2,n be the lift of I 2,n with respect to C and denote byÃ 2,n the quotient C[t ij : 2 ≤ i + 1 < j ≤ n][p ij : 1 ≤ i < j ≤ n]/Ĩ 2,n . Then there exists a canonical isomorphism between the cluster algebra with universal coefficients A univ 2,n andÃ 2,n (Theorem 4.50). (iv) The monomial initial ideal of I 2,n with respect to C is squarefree and coincides with the Stanley-Reisner ideal of the cluster complex (Corollary 4.59).
Similarly, we describe the cluster algebra with universal coefficients for Gr 3, C 6 from the viewpoint of Gröbner theory. In this case, we fix its cluster embedding obtained as follows: consider the Plücker embedding and let A 3,6 be the homogeneous coordinate ring. As a cluster algebra of type D 4 , A 3,6 has 22 cluster variables, 20 of which are the Plücker coordinates. The additional two cluster variables are homogeneous binomials in Plücker coordinates of degree 2. Hence, we can present A 3,6 as the quotient of a polynomial ring in 22 variables by a weighted homogeneous ideal denoted by I ex . This yields an embedding of Gr 3, C 6 in the weighted projective space P(1, . . . , 1, 2, 2), where the Plücker coordinates correspond to the coordinates of weight one, and the additional cluster variables to those of weight two, see Section 4.4. Our main result is the following: Theorem 1.4. There exists a unique maximal cone C in the Gröbner fan of I ex such that (i) The algebraÃ 3,6 is canonically isomorphic to the cluster algebra with universal coefficients A univ 3,6 , where rays of C are identified with mutable cluster variables of A 3,6 (Theorem 4.53(ii)).
(ii) For every seed of A 3,6 there exists a face of C whose associated initial ideal is a totally positive binomial prime ideal. More precisely, the intersection of C with Trop(I ex ) is the totally positive part of Trop(I ex ) (Theorem 4.53(iii)).
(iii) The monomial initial ideal in C (I ex ) associated to C is squarefree and coincides with the Stanley-Reisner ideal of the cluster complex. In particular, the basis of standard monomials associated to C coincides with the basis of cluster monomials (Corollary 4.59).
As mentioned before, the toric fibers of the above families are of particular interest. They arise from maximal cones in the tropicalization whose initial ideal is binomial and prime; such cones are called maximal prime cones. The corresponding projective toric varieties (respectively, their normalizations) have associated polytopes. Following [34] these polytopes can be realized as Newton-Okounkov bodies of full rank valuations constructed from maximal prime cones. In the recent preprint [19] Escobar and Harada study wall-crossing phenomena of Newton-Okounkov bodies associated to maximal prime cones that intersect in a facet. They give piecewise linear maps called flip and shift that relate the Newton-Okounkov bodies. For cluster algebras like A 3,6 and A 2,n it has been shown in several cases that Newton-Okounkov bodies can equivalently be obtained from the cluster structure (see, e.g., [6,8,26,46]). Hence, one might wonder how Escobar-Harada's wall-crossing formulas can be understood in the context of cluster algebras. In the particular case of A 2,n each Newton-Okounkov body arising from Trop(I 2,n ) is unimodularly equivalent to the convex hull of g-vectors of Plücker coordinates for some triangulation. More precisely, we obtain the following result: Corollary 1.5 (Corollary 4.64). Let σ 1 and σ 2 be two maximal prime cones in Trop(I 2,n ) that intersect in a facet. Then their associated Newton-Okounkov bodies are (up to unimodular equivalence) related by a shear map obtained from tropicalizing a cluster mutation.
In combination with Escobar-Harada's results about the flip map for Gr(2, C n ) this corollary implies that it is of cluster nature (for details see Section 4.6).
We would like to remark that this paper is not the first to make a connection between cluster algebras and Gröbner theory. In [41] Muller, Rajchgot and Zykoski obtained presentations for lower bounds of cluster algebras using Gröbner theory. Further, it is worth noticing that the theory of universal coefficients for cluster algebras is particularly well-developed for finite and surface type cluster algebras, see [44]. We believe that the above results can be extended to projective varieties containing a cluster variety of finite type. It is further an interesting and challenging problem to extend the results of this paper to cluster algebras of (infinite) surface type as it would involve Gröbner theory of non-Noetherian algebras.
Structure of the paper. In Section 2 we recall the background on weighted projective varieties and Gröbner basis theory. In Section 3 we introduce the construction of the flat families and prove the main theorem. We relate to Kaveh-Manon's work in Section 3.1. In Section 4 we turn to the Grassmannians Gr(2, C n ) and Gr 3, C 6 . We recall the background on cluster algebras and universal coefficients in Section 4.1. We explain in detail how toric degenerations from cluster algebras arise as Gröbner degenerations for Gr(2, C n ) in Section 4.2. Then we apply the main construction to Gr(2, C n ) in Section 4.3 and afterwards to Gr 3, C 6 in Section 4.4. We explore further connections to Escobar-Harada's work in Section 4.6. Finally, in the Appendix A we present computational results used for the application to Gr 3, C 6 .

Preliminaries
We first fix our notation throughout the note. Let K be an algebraically closed field. We are mainly interested in the case when K = C. In the polynomial ring K[x 1 , . . . , x n ], we fix the notation x α with α = (a 1 , . . . , a n ) ∈ Z n ≥0 denoting the monomial x a 1 1 . . . x an n . Throughout when we write f = α∈Z n ≥0 c α x α for f ∈ K[x 1 , . . . , x n ] we refer to the expression of f in the basis of monomials.
Definition 2.1. For c ∈ K * and α = (a 1 , . . . , a n ) ∈ Z n ≥0 the weight of the monomial Assume d ∈ Z n >0 . In this case K d [x 1 , . . . , x n ] is Z-graded and we can define the weighted projective space P(d 1 , . . . , d n ) as the quotient of K n under the equivalence relation For K = C it is well-known that P(d 1 , . . . , d n ) is a projective toric variety [15]. Moreover, as a scheme Conversely, given a subset X ⊆ P(d 1 , . . . , d n ), its associated ideal I(X) is generated by polynomials: Subsets of P(d) of the form V (J) for some weighted projective ideal J are called weighted projective varieties. These sets are the closed sets of a Zariski-type topology on P(d). The theory of weighted projective varieties is very similar to the theory of usual projective varieties and background on this theory can be found for example in [17,30,45]. In particular, the projective space P n−1 can be realized as P n−1 = P(1), where 1 = (1, . . . , 1) ∈ Z n . So every projective variety can be considered as a weighted projective variety of P(1).
Definition 2.2. The weighted homogeneous coordinate ring of a weighted projective variety X ⊆ P(d) is defined as By construction, S(X) is a positively graded ring. Moreover, the weighted homogeneous coordinate ring of a projective variety (considered as a weighted projective variety) coincides with its homogeneous coordinate ring. There is a natural notion of morphism between weighted projective varieties which we will not need in this work. It will be sufficient to recall that if X ⊆ P(d 1 , . . . , d n ) and Y ⊆ P(d 1 , . . . , d m ) are weighted projective varieties, then an isomorphism between the graded rings S(X) and S(Y ) induces an isomorphism of the weighted projective varieties X and Y .

Gröbner basis theory
We first review some results in Gröbner basis theory in order to fix our notation and keep the paper self-contained. Most of the material here is well-known and we refer to Buchberger's thesis [11] and standard books (e.g., [1,29,37,51]) for proofs and more details.
. Given a weight vector w ∈ R n the initial form of f with respect to w is defined as where a = min{w · α : c α = 0}. For an ideal J ⊆ K[x 1 , . . . , x n ] its initial ideal with respect to w is defined as in w (J) : = in w (f ) : f ∈ J . A finite set G = {g 1 , . . . , g r } ⊆ J is called a Gröbner basis for J with respect to w if in w (J) = in w (g 1 ), . . . , in w (g r ) .
Definition 2.4. A monomial term order on K[x 1 , . . . , x n ] is a total order < on the set of monic monomials in K[x 1 , . . . , x n ] such that for every α, β, γ in Z n ≥0 we have that The initial monomial of an element f = Recall that given an ideal J ⊆ K[x 1 , . . . , x n ] and a monomial term order < there always exists a weight vector −w ∈ Z n ≥0 such that in w (J) = in < (J), see, e.g., [29, Theorem 3.1.2] (Note the sign switch here, this is due to our min-convention for initial ideals with respect to weight vectors and max-convention for initial ideals with respect to monomial term orders.) On the other hand, if in w (J) is generated by monomials and there exists a monomial term order < such that in w (J) = in < (J), then we say that w is compatible with <.
Definition 2.7. Let < be a monomial term order on K[x 1 , . . . , x n ] and G = {g 1 , . . . , g s } a finite generating set of an ideal J. Then the S-polynomial of g i and g j is defined as We say that Buchberger's criterion holds if for all 1 ≤ i < j ≤ s, the S-pairs reduce to zero with respect to {g 1 , . . . , g s }. If Buchberger's criterion holds, then G forms a Gröbner basis for J with respect to <. Moreover, a Gröbner basis G for J with respect to < is reduced if A reduced Gröbner basis for J with respect to < always exists and is unique, see, e.g., [29,Theorem 2.2.7]. We let G < (J) denote the reduced Gröbner basis for J with respect to <.
Studying all possible initial ideals of a given ideal leads to the notion of Gröbner fan (see, e.g., [51,Proposition 2.4]): . , x n ] be an ideal. The Gröbner region of J denoted by GR(J) is the set of w ∈ R n such that there exists a monomial term order < with in < (in w (J)) = in < (J). The Gröbner fan of J denoted by GF(J) is a fan with support GR(J) in which a pair of elements u, w ∈ R n lie in the relative interior of the same cone C ⊂ R n (denoted by C • ) if and only if in u (J) = in w (J). We introduce the notation in C (J) := in w (J) for any w ∈ C • . By definition of GR(J), for every full-dimensional cone C, there exists a monomial term order < such that C is the topological closure of {w ∈ R n : in w (J) = in < (J)}. Moreover, we define the lineality space L(J) as the linear subspace of GF(J) that contains all elements u for which in u (J) = J.
Integral weight vectors in GF(J) can be seen as points in a lattice N = Z n whose dual lattice M = (Z n ) * contains exponent vectors of monomials in K[x 1 , . . . , x n ]. Consequently, for w ∈ N and α ∈ M we denote by w · α the pairing between the two lattices. Remark 2.9. Proposition 15.16 in [18] gives a criterion for whether a weight vector is compatible with a monomial term order < for J, or not. Namely, a weight w is compatible with a monomial term order < if and only if in w (g) = in < (g) for every element of G < (J). Let C be the topological closure of the corresponding Gröbner cone of <, then w ∈ C if and only if in < (g) = in < (in w (g)) for every g ∈ G < (J). Lemma 2.11. Fix an arbitrary monomial term order <. Then for every ∈ L(J) and every g ∈ G < (J) we have in (g) = g. In particular, if J is d-homogeneous then so is every g ∈ G < (J).
Proof . Consider a Gröbner basis F := {f 1 , . . . , f t } of J with respect to a fixed element ∈ L(J). Then J = in (J) is generated by in(F ) := {in (f 1 ), . . . , in (f t )}. In particular, we have in (in (f )) = in (f ) for all f ∈ F . From the set F we can construct the reduced Gröbner basis G < (J) by doing the following three steps: First, extend F to a Gröbner basis H with respect to < by adding S-pairs using Buchberger's criterion. Then if necessary eliminate elements from H until the outcome is a minimal Gröbner basis G (see [1,Corollary 1.8.3]). Finally, reduce all elements in G to obtains G < (J) (see [1,Corollary 1.8.6]). We invite the reader to verify that the property in (g) = g holds for all g ∈ H, hence for all g ∈ G and finally for all g ∈ G < (J).
Recall from [40,Corollary 5.7] that for a d-homogeneous ideal J ⊆ K d [x 1 , . . . , x n ] the support of its Gröbner fan, i.e., GR(J), is R n . In other words, there exists a compatible monomial term order for every maximal cone C ∈ GF(J).
. , x n ] be a d-homogeneous ideal, C ∈ GF(J) a maximal cone, and < a compatible monomial term order of C. Consider v ∈ C \ C • and u ∈ R n such that w := u + v ∈ C • . Then for every g ∈ G < (J), we have in < (g) = in w (g) = in u (in v (g)).
Proof . Consider an element g in G < (J). Since w ∈ C • we have in < (g) = in w (g). On the other hand, since v ∈ C, by Remark 2.9 we have that in < (g) = in < (in v (g)). This implies that in w (g) is a refinement of in v (g). In other words, in v (g) contains the monomial in w (g) with possibly some extra terms which will disappear after taking its initial form with respect to u.
We note that Lemma 2.12 does not hold for arbitrary elements of J. See Example 4.55.
Proof . As we have chosen J to be d-homogeneous every cone in GF(J) is a face of a fulldimensional cone. Hence, we may assume without loss of generality that C is a cone of dimension n. Let < be the corresponding monomial term order and g ∈ G < (J). Let v and −v be in C \ C • and choose u, u ∈ R n such that u + v, u − v ∈ C • . By Lemma 2.12 we have that in < (g) = in u (in v (g)) = in u (in −v (g)). However, in < (g) cannot simultaneously be a monomial in in v (g) and in −v (g) unless v ∈ L(J).   Due to [50] Trop + (J) is a closed subfan of Trop(J) (that may be empty).

Families of Gröbner degenerations
In this section, we introduce the main construction of the paper. Let J ⊆ K d [x 1 , . . . , x n ] be a d-homogeneous ideal, C a maximal cone in GF(J) and A the quotient K d [x 1 , . . . , x n ]/J. Our aim is to construct a flat family of degenerations of Proj(A) that contains as fibers the varieties corresponding to the various initial ideals associated to the interior of the faces of C. To achieve this we generalize the following classical construction: take w ∈ C • and consider the ideal (3.1) ) whose fiber over the closed point (t) is isomorphic to Spec(K[x 1 , . . . , x n ]/ in w (J)) and the fiber over any non-zero closed point (t − a) is isomorphic to Spec(A), see [18,Theorem 15.17]. In fact, (3.1) and [18,Theorem 15.17] hold more generally for arbitrary cones in GF(J). However, for the following generalization we focus on maximal cones for simplicity.
To generalize the construction (3.1) we fix vectors r 1 , . . . , r m ∈ C such that {r 1 , . . . , r m } is the set of primitive ray generators for C, which is possible due to Lemma 2.13. Using Lemma 2.10 we may assume if necessary that r 1 , . . . , r m are non-negative or positive vectors. We denote by r the (m × n)-matrix whose rows are r 1 , . . . , r m . Additionally, we write < for a monomial term order compatible with C and denote by G the associated reduced Gröbner basis.
that is we think of µ r (f ) as a column vector with m entries. We define the lift of f to be the polynomialf r ∈ K[t 1 , . . . , t m ][x 1 , . . . , x n ] given by the following formulã Similarly, we define the lifted ideal as We proceed by showing that the lifted algebraÃ r is independent of the choice of vectors r 1 , . . . , r m ∈ C that represent the primitive ray generators r 1 , . . . , r m of C. For this, we identify generators of the idealJ r which is a crucial matter for applications. Proposition 3.3. Suppose r 1 , . . . , r m ∈ C are such that r i = r i in R n /L(J). Let r be the (m × n)-matrix whose rows are r 1 , . . . , r m . Then for g ∈ G we have thatg r =g r .
Proof . We have that r i = r i + l i for some l i ∈ L(J). Let L be the (m × n)-matrix whose rows are l 1 , . . . , l m . In particular, r = r + L. Write g = α∈Z n ≥0 c α x α . Since g ∈ G we have by Lemma 2.11 that the value l i ·α is the same for all α with c α = 0. Let a i be this common value. Observe that min cα =0 {r i · α} = min cα =0 {r i · α} + a i . In particular, we have that the column vector (a 1 , . . . , a m ) is equal to L · α for all α with c α = 0 and therefore µ r (f ) = µ r (f ) + L · α. Finally, we computẽ c α x α t r·α+L·α−(µr(g)+L·α) =g r .
In the following, when there is no risk of confusion, we writeJ forJ r ,f forf r andÃ forÃ r . Recall the isomorphism K[t 1 , . . . , t m ]/(t − a) → K that sends an elementf to f | t=a . It extends to an isomorphismÃ| t=a ∼ = K[x 1 , . . . , x n ]/J| t=a , which explains our choice of notation.
This implies the first claim. The second claim is immediate asJ = f : f ∈ J .
Remark 3.6. In Lemma 3.5 more generally we can Lemma 3.7. Let h ∈J be w -homogeneous with w as in Lemma 3.5. Then h = t uf for some f ∈ J and u ∈ Z m ≥0 .
c α x α ∈ J with a unique monomial c β x β ∈ in < (J). That is, for every α = β with c α = 0, there exists no g ∈ G such that x α is divisible by in < (g ). Theñ contains a unique monomial with coefficient in K. In particular, this is the case for elements of the reduced Gröbner basis G.
The assumption that only the monomial c β x β of f is contained in in < (J) ensures that c β x β is also a monomial appearing in in r i (f ) for all i. To see this, assume there is a ray r i such that c β x β is not a monomial in in r i (f ) and let w = j =i r j . By [38, Lemma 2.4.5] we can find an so that in w +r i (f ) = in w (in r i (f )). As we assumed c β x β is not a monomial in in r i (f ), it is also not a monomial in in w +r i (f ). But w +r i ∈ C • and so in w +r i (f ) ∈ in w +r i (J) = in < (J). This however is a contradiction as we assumed that c β x β is the only monomial in f that is contained in in < (J). Therefore, r i · β = min cα =0 {r i · α} for all i and r · β = µ r (f ). Assume on the contrary that there exists another monomial c γ x γ in f with r · γ = r · β. Then c γ x γ is also a monomial in in r i (f ) for all i. Moreover, for w = r 1 + · · · + r m ∈ C • the initial form in w (f ) contains both monomials c β x β and c γ x γ . But as in w (J) = in < (J) this implies that c γ x γ ∈ in < (J), a contradiction. Hence, c β x β is the unique monomial in f with µ r (f ) = r · β and we obtain (3.3) as r · (α − β) = 0 for all α = β. This completes the proof of the first claim.
To prove the second part, assume that g ∈ G has a monomial term c γ x γ ∈ in < (J) with c γ x γ = in < (g). Then by the definition of the reduced Gröbner basis, c γ x γ is not divisible by in < (g ) for any g ∈ G \ {g}. Hence, it should be divisible by in < (g), but this cannot happen as g is d-homogeneous by Lemma 2.11.
We extend the monomial term order < on K[x 1 , . . . , x n ] to a monomial term order on the polynomial ring K[t 1 , . . . , t m , x 1 , . . . , x n ] in such a way that t i x j for all 1 ≤ i ≤ m and 1 ≤ j ≤ n, and Proposition 3.9. The setG = {g : g ∈ G} is a Gröbner basis forJ with respect to . In particular,G is a generating set forJ.
Proof . SinceJ is w -homogeneous by Lemma 3.5, it is enough to show that for every whomogeneous polynomial h ∈J there exists someg i ∈G whose initial term divides in (h).
where g is a suitable element of the Gröbner basis G for J and v ∈ Z m ≥0 . Asg ∈G, this shows thatG is a Gröbner basis forJ. Hence, by [29, Theorem 2.1.8]G is a generating set forJ.
As a direct consequence of Proposition 3.3 and Proposition 3.9 we obtain that the lifted algebra is independent of the choice of vectors r 1 , . . . , r m . The algebrasÃ| t=a constitute the fibers of a flat family introduced below. The following result leads to fibers over different points being isomorphic if they have zero entries in the same positions.
, so the claim follows by Notation 3.4.
Remark 3.12. For computational reasons it might be desirable to work with ray generators for C that are not representatives of primitive ray generators for C. Let r and r be two choices of ray matrices whose rows satisfy r j = q j r j with q j ∈ Q for all j. Then we still have an isomorphism betweenÃ r | t=a andÃ r | t=a for all a ∈ K m . For the proof it is necessary to extend the polynomial ring K[t 1 , . . . , t m ] to a ring K t Q 1 , . . . , t Q m , where the t i 's are allowed to have rational exponents. The automorphism with the property h f r =f r for all d-homogeneous polynomials f ∈ J. The rest follows from Proposition 3.11.
Before presenting our main result we explain how the idealJ r is related to the idealĴ w in (3.1). Proposition 3.13. Consider a face τ of C spanned by a subset {r i 1 , . . . , r is } of {r 1 , . . . , r m } and the lineality space L(J). We define t τ ∈ K[t] m by Then for w τ = r i 1 + · · · + r is we haveJ| t=tτ =Ĵ wτ .
Proof . Let < be the monomial term order compatible with C. Consider an element g = Since the vectors w τ , w := w τ − r is and r is are all in C, by Remark 2.9: This implies that the initial forms of in wτ (g), in w (g) and in r is (g) contain c β x β . In other words, Therefore, {r is · α}.
Using the same argument multiple times we obtain that Now the claim follows by comparing the generating sets ofĴ wτ andJ| t=tτ .
We are now prepared to present our main theorem: . , x n ]/J, C a maximal cone in GF(J) with compatible monomial term order < and r an (m × n)-matrix whose rows are representatives of the primitive ray generators of C ⊂ R n /L(J). Then: (i) The algebraÃ r is a free K[t 1 , . . . , t m ]-module with basis B < , the standard monomial basis of A with respect to in < (J). In particular, we have a flat family In particular, generic fibers are isomorphic to Proj(A) and there exist special fibers for every proper face τ ⊂ C.
Proof . (i) We extend < to a monomial term order on K[t 1 , . . . , t m , x 1 , . . . , x n ] as in (3.4). Let B be the standard monomial basis forÃ r induced by in J r . Then B is a basis forÃ r as a K-vector space and a generating set forÃ r as a K[t 1 , . . . , t m ]-module. By Proposition 3.9 and [29, Proposition 1.1.5] we have Observe that B can be reduced to a K[t 1 , . . . , t m ]-basis ofÃ r by defining where γ < β means that γ i ≤ β i for all i and γ j < β j for some j. Note that B < ⊆ B . Now assume there is a monomialt βxα ∈ B \ B < . By Proposition 3.9 we have that in For the second claim, let . , x n ] denote the polynomial ring in x 1 , . . . , x n with coefficients in K[t 1 , . . . , t m ] and grading induced by d. Then by Lemma 3.8 and Proposi- A m induces the flat morphism π : Proj(Ã) → A m as A has a K[t 1 , . . . , t m ]-basis by the first claim.
(ii) Every face τ of C induces a face τ of C with primitive ray generators r i 1 , . . . , r is for some Then by Proposition 3.13 we haveJ| t=aτ =Ĵ wτ | t=0 which is equal to in τ (J) by Remark 3.1.

Torus equivariant families
We explain how the above results are related to Kaveh-Manon's recent work [33] on the classification of torus equivariant families. Consider the lattice N = Z n , its dual lattice M = N * and a fan Σ ⊂ N ⊗ Z R. We write X Σ for the toric variety associated to Σ. Furthermore, we define O N to be the semifield of piecewise linear functions on N and O Σ the semifield of piecewise linear functions on |Σ| ∩ N . For a, b ∈ O Σ we have a ⊗ b := a + b and a ⊕ b := min{a, b}, where the minimum is taken pointwise.
where the minimum is taken pointwise in O Σ . If (i) is an equality, v is called a PL-valuation. If A is a graded algebra A = n∈Z A n then v is called homogeneous if it is compatible with the grading, i.e., for f = n∈Z c n g n with g n ∈ A n and coefficients c n we have v(f ) = v(g k ) for the smallest k with c k = 0.
Given a sheaf of algebras A on X Σ its relative spectrum denoted by Spec(A) is the scheme obtained from gluing affine schemes Spec(A(U i )), where i U i is an open cover of X Σ and A(U i ) is the corresponding section of A. Consider a presentation of A, i.e., a surjection pr : K[x 1 , . . . , x n ] → A whose kernel is a weighted homogeneous prime ideal J and so it induces an isomorphism of graded algebras A ∼ = K[x 1 , . . . , x n ]/J. Take a cone σ in GF(J). Then σ defines PL-quasivaluation v σ : A → O σ as follows: let C be a maximal cone in GF(J) with face σ and denote by B < the standard monomial basis associated to C (here < is the compatible monomial term order). If in σ (J) is prime, then v σ is a PL-valuation. Then forx α ∈ B < we have v σ (x α ) := − · α : σ → Z, where − · − is the pairing between the lattices N and M , as above. For an arbitrary element where we think of t m as a character of the torus More generally, a PL-quasivaluation can be obtained from a collection of equidimensional cones σ 1 , . . . , σ k in Trop(J) which are faces of the same maximal cone C in GF(J). Let Σ ⊂ Trop(J) be the subfan whose maximal cones are σ 1 , . . . , σ k and v Σ : A → O N be the corresponding PL-valuation. By Kaveh-Manon's classification of toric families this yields a flat family . Moreover, ψ Σ has a special fiber over every torus fixed point of X. More precisely, let p σ i be the torus fixed point corresponding to the (maximal) cone σ i ∈ Σ. Then Note that we can apply the construction of Definition 3.2 to the ideal J and the maximal cone C ∈ GF(J). In what follows we explore the relation between the flat families from (3.5) and Theorem 3.14. Before stating our results (Theorem 3.19 and Corollary 3.20), we recall necessary background from [33]. We fix a maximal cone C in GF(J), denote by G the associated reduced Gröbner basis and let B < be the standard monomial basis for the compatible monomial term order <. [33] the standard monomial basis B < is an adapted basis for v C . Similarly, the monomial basis of K[x 1 , . . . , x n ] is adapted to w C . Hence, R C (A) is a free K[S C ]-algebra with basis B < and R C (K[x 1 , . . . , x n ]) is a free K[S C ]-algebra whose basis is the monomial basis, where S C := −C ∨ ∩ Z n due to our min-convention. In particular: (3.7) Manon explained the proof of the following proposition to us. It follows from results in [33].
Proposition 3.18. For any maximal cone C ∈ GF(J), the Rees algebra R C (A) has an explicit presentation of form K[S C ][x 1 , . . . , x n ]/Ĵ. More precisely, the idealĴ is generated by homogenizations of elements in the reduced Gröbner basis G: let g = x γ + c α x α ∈ G and x γ = in C (g), then the homogenization of g iŝ is the image of a monomial in F m (w C ). Hence, pr m is surjective and we obtain: By (3.7) this implies the first claim. Now letĴ be the kernel of pr. ThenĴ = m∈MĴ m , whereĴ m = ker(pr m ) ⊂ F m (w C ). Now consider an element g ∈ G of form x γ + c α x α with x γ = in C (g). We want to show that the elementĝ defined in (3.8) lies inĴ for all g ∈ G. Recall that by the proof of Proposition 3.13, γ is such that w · γ ≤ w · α for all w ∈ C and all α with Lastly, we need to show thatĴ is generated by In particular, similar to the above the exponent of in < (x µ g) = x µ+γ has the property that w·(µ+γ) ≤ w·(µ+α) for all w ∈ C and all α with c α = 0 in g. So, h − x µ g ∈ F m (w C ). In particular, the division algorithm with respect to G takes place inside F m (w C ) and yields an expression of h in terms of {ĝ : g ∈ G}.
We now recall the Cox construction [14] for toric varieties and refer to [15, Section 5] for more details. Let Σ = C ∩ Trop(J) and l denote the dimension of L(J). Fix an (m × n)-matrix r whose rows r 1 , . . . , r m are representatives of the primitive ray generators of C ⊂ R n /L(J). The rays of Σ, denoted by Σ(1), form a subset of {r 1 , . . . , r m }. Recall the definition of the lifted algebraÃ r from (3.2). We can apply the Cox construction in two ways, to X C and to X Σ : First, let X C be the affine toric variety associated to C. Recall that a quasitorus is a product of a torus with a finite abelian group. Then X C is isomorphic to the almost geometric quotient A m //G (see, e.g., [ One can easily check that if p, q ∈ A m lie in the same G-orbit, then π −1 (p) ∼ = π −1 (q) by Proposition 3.11. Hence, the flat family π : Spec(Ã r ) → A m induces the commutative diagram If p, q ∈ X C lie in the same torus orbit, then (p C • π) −1 (p) ∼ = (p C • π) −1 (q) by Proposition 3.11. Note that p C : A m → X C is indeed a morphism. Namely, it is the universal torsor for X C .
Similarly, we can apply the Cox construction to the toric variety X Σ . For simplicity we assume that Σ(1) = {r 1 , . . . , r m }. In this case the construction has two steps: first we remove the locus of A m that does not correspond to torus orbits in X Σ , denoted by Z(Σ). Recall that C ⊂ {r 1 , . . . , r m } is a primitive collection for Σ, if (i) C ⊂ σ(1) for all σ ∈ Σ, and (ii) for every C C there exists σ ∈ Σ with C ⊂ σ(1). Then by [15, Proposition 5.1.6] Now X Σ is isomorphic to the almost geometric quotient (A m − Z(Σ))//G, where G is as in (3.9) (see, e.g., [15,Theorem 5.1.11]). In contrast to the morphism p C : A m → X C from (3.10), we obtain a rational map Similarly to the first case, this gives the diagram: which is commutative. The fibers of the family defined by In the following, we explain how the flat family of Theorem 3.14 is related to Kaveh-Manon's flat family associated to a maximal cone in the Gröbner fan.
. , x n ] be a d-homogeneous ideal and C a maximal cone in GF(J). Let r be an (m × n)-matrix whose rows are representatives of primitive ray generators for C. Then the morphism π : Spec Ã r → A m fits into a pull-back diagram Here p C : A m → X C is the universal torsor of X C obtained from the Cox construction as in (3.10).
Proof . We prove equivalently that the corresponding diagram between the algebras is a pushout. Hence, it suffices to show that Then under the extension of scalars (i.e., applying the functor − ⊗ K[S C ] K[t 1 , . . . , t m ]) the homogenizationĝ of an element g ∈ G is sent to the liftg. This can be seen as follows: by Lemma 3.8 we haveg = x γ + c α x α t r·(α−γ) . By Lemma 2.11 for every ∈ L(J) we have in (g) = g. In particular, · α = · γ and so α −γ ∈ (L(J) ∩ N ) ⊥ . Let Note that γ has the property that r i · γ = min cα =0 {r i · α}. So, r i · (α − γ) ≥ 0 and therefore α − γ ∈ C ∨ . Hence, we have . , x n ] be a weighted homogeneous ideal as in Definition 2.1 and C a maximal cone in GF(J). Let r be the matrix whose rows are representatives of primitive ray generators for C and denoteÃ r byÃ. Let Σ be the intersection of C with Trop(J). Then the restriction of morphism π : Here p Σ : T Σ → X Σ is the universal torsor of X Σ obtained from the Cox construction as in (3.12).
Proof . The scheme X Σ and the sheaf R Σ are defined locally for affine pieces Algebraically, for every σ ∈ Σ the pull-back of ψ and p Σ corresponds to the following push-out diagram: Observe that by (3.11) the localization K[t 1 , . . . , t m ] t i : r i ∈σ corresponds to an affine piece in A m that does not intersect Z(Σ). So it is in fact an affine piece of T Σ . We have to show that As σ is a face of the maximal cone C by [33, Proposition 3.13] we have Using Theorem 3.19 we compute

Grassmannians and cluster algebras
We now apply the machinery developed in Section 3 to the Grassmannians Gr(2, C n ) and Gr 3, C 6 . Our aim is to make this paper as self-contained as possible and accessible to readers of broad mathematical background. Therefore, we recall background on tropical Grassmannians below, on cluster algebras and the cluster structure of Gr(2, C n ) in Section 4.1 and then in Section 4.2 we translate between toric degenerations obtained from the tropicalizations and from the cluster structure. In Section 4.3, we turn to the application of the flat families introduced in Section 3 and show how to recover the cluster algebra with universal coefficients in this case. Similarly, in Section 4.4 we treat the case of Gr 3, C 6 . We first fix our notation for Gr(2, C n ). Denote by S the polynomial ring in Plücker variables C[p ij : 1 ≤ i < j ≤ n]. Here, Plücker variables are associated with arcs in an n-gon whose vertices are labeled by [n] in clockwise order. Hence, we think of [n] := {1, . . . , n} as a cyclically ordered set. In particular, i < j < k < l indicates that in clockwise order starting at the vertex i the other vertices appear in order j, k, l. Note that in general we might not have 1 ≤ i < j < k < l ≤ n. We set the convention p ij := −p ji . In S we define for every i < j < k < l the Plücker relation: The ideal I 2,n ⊂ S generated by all Plücker relations R ijkl is called the Plücker ideal. The quotient A 2,n := S/I 2,n is the Plücker algebra which is the homogeneous coordinate ring of the Grassmannian Gr(2, C n ) with respect to its Plücker embedding into P ( n 2 )−1 . The cosets of Plücker variables in the quotient are denoted byp ij ∈ A 2,n and are called Plücker coordinates. Denote by d := 2(n − 2) the dimension of Gr(2, C n ), so d + 1 is the Krull-dimension of A 2,n .
In S, we fix the notation p m with m = (m ij ) ij ∈ Z ( n 2 ) ≥0 denoting the monomial 1≤i<j≤n p  Maximal cones in Trop(I 2,n ) are called tropical maximal cones. They are in correspondence with trivalent trees with n leaves as shown by Speyer and Sturmfels in [49]. To a trivalent tree Υ one associates a weight vector in the relative interior of the corresponding cone of Trop(I 2,n ) by w Υ (p ij ) := −#{edges on the unique path i → j in Υ}. (4.1) To simplify notation we denote in Υ (I) := in w Υ (I).
Theorem 4.2 ( [49]). For every trivalent tree Υ with n leaves the ideal in Υ (I) is toric, i.e., binomial and prime. Moreover, it is generated by in Υ (R ijkl ) for all 1 ≤ i < j < k < l ≤ n. In [50,Section 5], it was shown that maximal cones in Trop + (I 2,n ) are in bijection with planar trivalent trees, i.e., trivalent trees whose vertices are labeled 1, . . . , n in a clockwise manner.

Preliminaries on cluster algebras
The Plücker algebra A 2,n is in fact a cluster algebra (see Definition 4.4) whose cluster structure is closely related to the combinatorics governing the tropical Grassmannian. To exhibit this connection, we recall some facts about skew-symmetric cluster algebras following Fomin and Zelevinsky [23,25]. Afterwards, we define finite type cluster algebras with frozen variables and universal coefficients.
Let m, f ∈ N be positive integers and F = C(u 1 , . . . , u m+f ) be the field of rational functions on m + f variables. Here m stands for mutable and f for frozen. A seed in F is a pair x, B , where x = {x 1 , . . . , x m+f } is a free generating set of F and B = (b ij ) is an (m+f )×m rectangular matrix with the following property: the top square submatrix of B (that is the m × m submatrix of B formed by its first m rows) is skew-symmetric. We call B an extended exchange matrix and refer to its top square submatrix B as an exchange matrix. We call x = {x 1 , . . . , x m } a cluster, {x m+1 , . . . , x m+f } the set of frozen variables and x an extended cluster.
Given k ∈ {1, . . . , m}, the mutation in direction k of a seed x, B is the new seed µ k x, B = x , B , defined as follows: We call an expression of the form (4.2) an exchange relation and the monomial x k x k an exchange monomial.
• The extended exchange matrix B = (b ij ) is defined by the following formula Definition 4.4. The cluster algebra A x, B associated to a seed x, B is the C-subalgebra of F generated by the elements of all the extended clusters that can be obtained from the initial seed x, B by a finite sequence of mutations. The elements of the clusters constructed in this way are called the cluster variables. If f = 0 we say that the cluster algebra A x, B has frozen variables, namely, x m+1 , . . . , x m+f .
Remark 4.5. It can be easily verified that, up to a canonical isomorphism, A x, B is independent of x. Therefore, we write A B instead of A x, B .
Remark 4.6. Fomin and Zelevinsky usually define cluster algebras over Q. We recover the construction described above applying tensor product with C to Fomin and Zelevinsky's construction.

Theorem 4.7 ([25, Laurent phenomenon]). The cluster algebra
Definition 4.8. The cluster algebra with principal coefficients associated to an m × m skewsymmetric matrix B is the cluster algebra A(B prin ), where B prin is the 2m × m matrix whose top m × m square submatrix is B and whose lower m × m square submatrix is the identity The notion of cluster algebras with principal coefficients is central in cluster theory. Its relevance in the framework of toric degenerations of cluster varieties was highlighted in [28]. In particular, in loc. cit. the authors work implicitly with cluster algebras with frozen variables and principal coefficients. This is precisely the notion we need here. Definition 4.9. Let B be an extended exchange matrix. Let B be the (m + f ) × (m + f ) skew-symmetric square matrix whose left (m + f ) × m submatrix is B and whose bottom right f × f square submatrix is the zero matrix (this information fully determines B since it is skew-symmetric). Denote by t 1 , . . . , t m+f the variables of A B prin associated to the last m + f directions, which in this case are the frozen directions. Then the cluster algebra with principal coefficients and frozen variables A B prin associated to B is the C[t 1 , . . . , t m+f ]-subalgebra of A B prin spanned by the cluster variables that can be obtained from the initial extended cluster by iterated mutation in the first m mutable directions. . Let X be a cluster variable associated to A B prin . By Theorem 4.7 we know that X ∈ C[t 1 , . . . , . , x m+f obtained after the evaluation of X at t 1 = 0, . . . , t m+f = 0. Then X| t=0 is a non-constant Laurent monomial with coefficient 1. In other words, for a non-zero vector g(X) = g X 1 , . . . , g X m+f ∈ Z m+f we have that Moreover, if X and X are different cluster variables then g(X) = g(X ).
Definition 4.11. Let X be a cluster variable associated to A B prin . The vector g(X) ∈ Z m+f introduced in Theorem 4.10 is the g-vector associated to X. We refer to the vector g X 1 , . . . , g X m ∈ Z m as the truncated g-vector of X. The set of (truncated) g-vectors associated to B is the set of (truncated) g-vectors of all the cluster variables in A B prin .
Remark 4.12. Let X| t=1 be the Laurent polynomial in C x ±1 1 , . . . , x ±1 m , x m+1 , . . . , x m+f obtained after the evaluation of X at t 1 = 1, . . . , t m+f = 1. Then X| t=1 is a cluster variable of A B . By a slight abuse of terminology we say g(X) is the g-vector of the cluster variable X| t=1 .
Universal coefficients. We now turn to the definition of finite type cluster algebras with frozen variables and universal coefficients. This notion is a natural (and slight) generalization of the usual notion of universal coefficients for cluster algebras without frozen variables introduced by Fomin and Zelevinsky in [25,Section 12]. The difference between these notions arises from the distinction we make between coefficients and frozen variables, notions that are usually identified. This distinction was first suggested in [9] and is of particular importance in the study of toric degenerations of cluster varieties. It was shown in [25] that finite type cluster algebras can be endowed with universal coefficients. This construction was categorified in [42] and the categorification was used to perform various computations in the following sections. To further clarify the preceding discussion let us first recall the notion of finite type cluster algebras with universal coefficients. Theorem-Definition 4.14 ([43, Corollary 8.15]). Let Q be a quiver of finite cluster type, that is Q is mutation equivalent to an orientation of a Dynkin diagram. The cluster algebra with universal coefficients associated to Q is A B univ Q , where and U Q is the rectangular matrix whose rows are the g-vectors of the opposite quiver Q op .
Observe that we have described the rows of U Q but did not specify in which order they appear. Any order we choose provides a realization of A B univ Q since a rearrangement of the rows of U Q amounts to reindexing the corresponding coefficients. So any choice gives rise to an isomorphic algebra. It can be verified that the cluster algebra with universal coefficients associated to Q is an invariant of the mutation class of Q. In other words, if Q is mutation equivalent to Q then A B univ Q is canonically isomorphic to A B univ Q . The canonical isomorphism is completely determined by sending the elements of the initial cluster of B univ Q to those of B univ Q .  Definition 4.17. Let (Q, F ) be an ice quiver whose mutable part is a quiver Q mut of finite cluster type. The associated cluster algebra with universal coefficients is A B univ (Q,F ) , where and U Q mut is the matrix whose rows are the truncated g-vectors of the opposite quiver (Q mut ) op .
Remark 4.18. The algebra A B univ (Q,F ) satisfies a universal property which is a straight forward generalization of the universal property of a cluster algebra without frozen directions and with universal coefficients, see Remark 4.15.
The Grassmannian cluster algebra A 2,n . We now turn to the cluster algebra that is of most interest to us, and use the combinatorics governing the cluster structure of A 2,n : triangulations of an n-gon. The vertices of the n-gon are labeled by [n] in the clockwise order. An arc connecting two vertices i and j is denoted by ij and we associate to it the Plücker coordinatep ij ∈ A 2,n .
Definition 4.19. A triangulation of the n-gon is a maximal collection of non-crossing arcs dividing the n-gon into n − 2 triangles. To a triangulation T we associate a collection of Plücker coordinates x T := p ij : ij is an arc in T which is the associated cluster. The elements of x T are the cluster variables. The monomials in Plücker coordinates which are all part of one cluster are the cluster monomials.
Note that every triangulation T contains the boundary edges 12, 23, . . . , n1. The corresponding cluster variablesp 12 ,p 23 , . . . ,p 1n are frozen cluster variables. All other variablesp ij with i = j ± 1 are mutable cluster variables. To every triangulation T we associated an ice quiver. Definition 4.20. For a triangulation T , we define its associated ice quiver Q T , F T : the vertices V (Q T ) are in correspondence with arcs and boundary edges ij ∈ T ; Two vertices v ij and v kl are connected by an arrow, if they correspond to arcs in one triangle. Inside every triangle, we orient the arrows clockwise. The set F T consists of vertices corresponding to the boundary edges of the n-gon, hence F does not depend on T . We neglect arrows between vertices in F , see  Figure 1. On the right a triangulation T of the 8-gon and its corresponding quiver Q T . On the left its corresponding extended tree D T dual to T which is a trivalent tree with 16 leaves.
We first recall some classic results due to Fomin-Zelevinsky and Fomin-Shapiro-Thurston.  ). The set of all cluster monomials is a C-vector space basis for the algebra A 2,n called the basis of cluster monomials.
We endow A 2,n with universal coefficients associated to its mutable part. This is an example of a cluster algebra with frozen directions and coefficients in the sense of [9]. Notation 4.23. Let Q T , F T be the quiver associated to a triangulation of the n-gon. We denote by Q mut T the full subquiver of Q T supported in the mutable vertices V (Q T ) \ F T . In particular, Q mut T is mutation equivalent to an orientation of a type A Dynkin diagram. Definition 4.24. Let T be a triangulation of the n-gon. The Plücker algebra with universal coefficients A univ 2,n is the cluster algebra defined by the extended exchange matrix Up to canonical isomorphism, A univ 2,n = A B univ (Q T ,F T ) is independent of T .
The quivers Q mut T and Q mut T op define canonically isomorphic cluster algebras whose cluster variables (denoted byp ij ) are both in bijection with arcs ij of the n-gon for which 2 ≤ i + 1 < j ≤ n. Hence, the rows of U Q mut T are also in bijection with these arcs. We write y ij for the coefficient variable of A univ 2,n corresponding to the arc ij.  Figure 2. A triangulation of the 5-gon and the associated matrix B univ (Q T ,F T ) .
Example 4.25. Let Q T , F T be the ice quiver associated to the triangulation of the pentagon depicted in Figure 2. The corresponding rectangular matrix for universal coefficients can be found on the right side of Figure 2. Incidentally, in this case the rows corresponding to the universal coefficients coincide with the rows corresponding to the frozen part of Q T , F T . However, for polygons with more than 5 sides this will not be the case.
To get a better understanding of the cluster algebra A univ 2,n we now turn to the combinatorial gadgets that govern its definition: g-vectors of Plücker coordinates. Their description involves two more combinatorial objects associated to a triangulation T , namely its dual graph D T and its extended dual graph D T . Recall that non-interior vertices of a tree graph are called leaves. Moreover, two leaves that are connected to the same interior vertex form a so-called cherry.
Definition 4.26. Fix a triangulation T of the n-gon. The trivalent tree D T is the dual graph or ribbon graph of T . It has n leaves and its interior vertices correspond to triangles in T . Two interior vertices are connected by an edge if their corresponding triangles share an arc. Vertices corresponding to triangles involving a boundary edge i − 1, i are connected to the leaf i. The trivalent tree D T is obtained from D T by replacing every leaf i by a cherry with leaves i and i , where (in the clockwise order) i labels the first leaf. We call D T the extended tree dual to T .
Note that the tree D T is by definition planar. See Figure 1 for an example. Moreover, the extended tree D T can alternatively be defined as the dual graph of a triangulation T of the 2n-gon with vertices 1 , 1, 2 , 2, . . . , n , n in the clockwise order. Here T is obtained from T by replacing every boundary edge i − 1, i by a triangle with a new vertex labeled by i . Then We now focus on giving a combinatorial definition of the g-vector associated to a Plücker coordinate with respect to a triangulation T . The reader can find an equivalent definition above in Definition 4.11. The g-vectors can be read from the extended tree D T . First, observe that the interior edges of D T correspond to the arcs and the boundary edges in T and, therefore to the cluster variables of the associated seed.
Sign conventions for g-vectors. Consider a path in the tree D T with the end points in {1, . . . , n} (not in {1 , . . . , n }). Say the path goes from i to j. For simplicity, we fix an orientation of the path and denote it by i j (what follows is independent of the choice of orientation). As D T is trivalent, at every interior vertex the path can either turn right or left. Denote by e ab the edge in D T corresponding to an arc ab ∈ T and let v abc and v abd be the vertices in D T adjacent to e ab . Given the path i j we associate signs σ ab i j ∈ {−1, 0, +1} to every ab ∈ T as follows: 1. If the path i j either does not pass through e ab , or it passes through e ab by turning right at v abc and turning right at v abd , or it passes through e ab by turning left at v abc and turning left at v abd : then σ ab i j = 0.
2. If the path i j passes through the e ab by turning left at v abc and turning right at v abd : then σ ab i j = +1. 3. If the path i j passes through the e ab by turning right at v abc and turning left at v abd : then σ ab i j = −1. Figure 3. We leave it to the reader to verify that σ ab i j = σ ab j i . Later we replace the end points i and j by the interior vertices of D T and use the same symbol. Similarly, we define the g-vector of a cluster monomial as the sum of the g-vectors of its factors.

Cases 2 and 3 are depicted in
Note that for ij ∈ T we have g ij = f ij . When there is no ambiguity, we write g ij instead of g T ij . Figure 4. Combinatorial translation between laminations on triangulations (as in [22]) and paths on trees (as in Definition 4.27) to compute truncated g-vectors. The red lines corresponds to the simple lamination associated to ij.
Lemma 4.28. Let T be a triangulation of the n-gon. The g-vector g T ij of a Plücker coordinate p ij with respect to T introduced in Definition 4.27 coincides with the g-vector g(p ij ) (from Definition 4.11) of the cluster variablep ij of A B (Q T ,F T ) .
Proof . We use some of the terminology of cluster algebras associated to surfaces developed in [22]. The algebra A 2,n is a cluster algebra arising from an orientable surface with marked points, in this case a disc with n marked points in its boundary. As such, the truncated g-vectors of its cluster variables can be computed using laminations in the surface as explained in [22,Proposition 17.3]. To be more precise, the truncated g-vector ofp ij is the vector containing the shear coordinates of the internal arcs of T with respect to the simple lamination associated to the arc ij. The key observation is that the combinatorial rule defining shear coordinates translates to our combinatorial rule to compute g T ij . In Figure 4 we visualize the translation from our combinatorial rule to the one defining shear coordinates from [22, Definition 12.2, Figure 34]. Shear coordinates are only associated to internal arcs and, therefore, can only compute truncated g-vectors. However, the extended exchange matrix B (Q T ,F T ) is the submatrix of B Q T ,F T associated to the internal arcs of the triangulation T of the 2n-gon. As a consequ- Figure 5. The full subtree of D T with leaves i, i , j, k, l and labeling as in the proof of Lemma 4.31 (ignoring i , d, v ) and Proposition 4.34. Blue marks refer to certain paths in the tree and the labels v i , v j , v k , v l , v, v to the corresponding elements of Z 2n−3 . We follow the convention in Figure 3.
Example 4.29. Consider the triangulation T depicted in Figure 1 and its corresponding tree D T . Then g ij = f ij for ij ∈ T . We compute the g-vectors of the remaining Plücker coordinates:

Toric degenerations
In this section, we use the g-vectors from Definition 4.27 to obtain a weight vector for every triangulation of the n-gon. The initial ideals of I 2,n arising this way are toric. Throughout this section we fix a triangulation T . Lemma 4.31. Given a Plücker relation p ij p kl −p ik p jl +p il p jk with i < j < k < l the multidegree induced by the g-vectors for the triangulation T on the monomials agrees for exactly two terms. Moreover, one of those terms is the crossing monomial p ik p jl .
Proof . Consider the extended tree D T and restrict it to the full subtree with leaves i, j, k, l. By definition, the subtree has four leaves and edges of valency three or less. Without loss of generality we may assume the leaves are arranged as in Figure 5. Two vertices of the subtree are trivalent, namely the one where the paths i j and l j meet, respectively where the paths j i and k i meet. Assume these vertices come from triangles in T labeled by a 1 , b 1 , c 1 , respectively a 2 , b 2 , c 2 . Denote the vertex of D T corresponding to the triangle a 1 , b 1 , c 1 (respectively a 2 , b 2 , c 2 ) by ∆ 1 (respectively ∆ 2 ). We compute the g-vectors for all Plücker coordinates whose indices are a subset of {i, j, k, l}. To simplify the notation in the computation we use the following abbreviations So v i is the weighted sum (with respect to the sign convention as in Figure 3) of the edges on the path from i to the vertex ∆ 1 excluding f a 1 c 1 , etc. Further, let v := pq∈T σ pq ∆ 1 ∆ 2 f pq . Then In particular, we have g ij + g kl = g ik + g jl = g il + g jk .
Recall from (4.1) how to obtain a weight vector from a trivalent tree with n leaves in Trop(I 2,n ). The following corollary is a direct consequence of the proof of Lemma 4.31.
Corollary 4.32. Without loss of generality by Lemma 4.31 assume that for i < j < k < l, g ij + g kl = g ik + g jl = g il + g jk . Then in D T (p ij p kl − p ik p jl + p il p jk ) = p ij p kl − p ik p jl . Definition 4.33. A quadrilateral weight map for a triangulation T is a linear map w T : R 2n−3 → R, such that for every quadrilateral abcd with diagonal ac in T we have: Note that (4.4) is equivalent to saying that for every mutable vertex ac of the quiver Q T we require w T pq→ac g pq > w T ac→pq g pq . Before we show that quadrilateral weight maps do in fact exist (in Lemma 4.39, using Algorithm 1) we proceed by showing how they can be used to construct weight vectors in Trop(I 2,n ) from g-vectors.
Proposition 4.34. Consider any i < j < k < l with g ij + g kl = g ik + g jl = g il + g jk and any quadrilateral linear map w T : R 2n−3 → R. Then w T (g ij + g kl ) = w T (g ik + g jl ) < w T (g il + g jk ).
Proof . Without loss of generality assume that the full subtree of D T with leaves i, j, k, l is of form as in Figure 5. For s = 1, 2 let ∆ s be the vertex of D T corresponding to the triangle a s , b s , c s in T . We proceed by induction on the number of edges along the unique path from ∆ 2 to ∆ 1 , denote it by p. p = 1: In this case we have a 1 = d = a 2 and b 1 = c 2 in Figure 5. Note that every quadrilateral p ≥ 1: Assume that the result holds for p. When the number of edges along the path from v a 2 b 2 c 2 to v a 1 b 1 c 1 is p + 1 we know there exists at least one leaf of form either i with i < i < j or k with k < k < l. We treat the first case, where i exists depicted in Figure 5. The proof of the second case is similar. All expressions of g-vectors for Plücker variables in the relation R ijkl appear in the proof of Lemma 4.31. The g-vectors involving i can be computed similarly. By induction we know that for the relation R ii jl we have w T (g ij + g i l ) < w T (g il + g i j ). Further, by assumption on w T we have w T (f a 2 d + f b 2 c 2 ) > w T (f a 2 b 2 + f c 2 d ). One verifies by direct computation that Definition 4.35. Fix a quadrilateral linear map w T : R 2n−3 → R. We define w T ∈ R ( n 2 ) the weight vector associated to T and w T as (w T ) ij := w T (g ij ).
Proposition 4.36. For each quadrilateral weight map w T : R 2n−3 → R, the initial ideal in w T (I 2,n ) is toric. Moreover, every binomial in the minimal generating set of in w T (I 2,n ) corresponds to a quadrilateral and it contains the monomial associated to the crossing in that quadrilateral.
Proof . By Proposition 4.34 and Corollary 4.32 we have in D T (R ijkl ) = in w T (R ijkl ) for every Plücker relation R ijkl . By Lemma 4.31 these initial forms are binomials and one monomial corresponds to the crossing in the quadrilateral ijkl. By [49, Proof of Theorem 3.4] we know that in D T (I 2,n ) is generated by in D T (R ijkl ) for all 1 ≤ i < j < k < l ≤ n. Moreover, by [49,Corollary 4.4] the ideal in D T (I 2,n ) is binomial and prime.  [7,46] and matching fields [12,13,39]. All such degenerations can be realized as Gröbner degenerations, nevertheless, this is not true in general; see, e.g., [32] for a family of toric degenerations arising from graphs that cannot be identified as Gröbner degenerations.
Existence of quadrilateral weight maps: For every triangulation T we give a weight map obtained from a partition of the cluster x T , which is the output of Algorithm 1. We use it to define a linear map v T : R 2n−3 → R and show in Lemma 4.39 that it is quadrilateral for T .
Input: An initial cluster x T with its corresponding triangulation T and its quiver Q T .
Output: A partitioning of the initial seed indices.
while T is not empty do V i := {v : v is a frozen vertex in Q and a sink}; foreach triangle t in T do if t has an edge whose corresponding vertex in Q is in V i then T ← T \ {t} (Remove the triangle t from T ); Algorithm 1: Partitioning the cluster x T based on its triangulation T (union of triangles).
Description of Algorithm 1. The input of the algorithm is a triangulation T of the n-gon. The output is an ordered partition of x T . We repeatedly shrink T to obtain a quiver associated to smaller triangulations. More precisely, in step i we identify the frozen vertices of Q T that are sinks, i.e., they have no outgoing arrows. Then we add the corresponding arcs to the set V i and remove the corresponding triangles from T . Note that in this process, we might remove edges whose corresponding vertices are not in V i , but they are part of a triangle with an edge in V i . Let V 0 , . . . , V i be the partition of the initial seed indices obtained by applying Algorithm 1 to the triangulation T associated to the initial seed x T . Refining the output of the algorithm yields an ordering on the variables of our initial seed, or equivalently the arcs of our initial triangulation. Lemma 4.39. Given a triangulation T consider the partition of x T = V 0 ∪ · · · ∪ V s obtained from Algorithm 1. Let v T : R 2n−3 → R be the linear map defined by sending basis elements f ij with ij ∈ T to q, where ij ∈ V q . Then v T is a quadrilateral linear map for T .
Proof . Consider a quadrilateral i < j < k < l in T with ij, jk, kl, li and ik ∈ T . As the algorithm ends when there is only one triangle or one arc left, there are two possibilities: either one of the triangles ijk, kli is cut off first, or both triangles are cut off at the same time. In the first case we may assume the triangle ijk is removed first. Then ij ∈ V p , lk ∈ V q and jk, il ∈ V s for some p ≤ q ≤ s with p < s. In the first case p and q are different, while in the second they are equal. So in either case, we have v T (f ij +f kl ) = p + q < s + s = v T (f jk + f il ).

A distinguished maximal Gröbner cone
In order to apply the construction from Section 3 in this section we identify a particular maximal cone C in GF(I 2,n ). We analyze the cone C and apply Theorem 3.14. In the following section we will show how this construction is related to adding universal coefficients to the cluster algebra A 2,n . Definition 4.40. Let u ∈ Q ( n 2 ) be the weight vector such that the weight of p ij is: Example 4.41. For n = 8, the Plücker variables associated to boundary arcs receive the lowest weight, which is −9. Mutable Plücker variables have the following weights: Proof . Consider the quadrilateral with vertices i < j < k < l and its corresponding Plücker relation R ijkl = p ik p jl − p ij p kl − p il p jk . The arcs ik and jl are crossing, hence we have to show that (a) in u (R ijkl ) = p ik p jl and (b) the crossing monomials generate in u (I 2,n ). To prove (a) we verify that u(p ik p jl ) = u(p il p jk ) + 2(j − i)(l − k) > u(p il p jk ), u(p ik p jl ) = u(p ij p kl ) + 2(k − j)(n + i − l) > u(p ij p kl ).
To establish (b), we apply Buchberger's criterion and show that all the S-pairs S(R ijkl , R abcd ) reduce to 0 modulo {R ijkl } 1≤i<j<k<l≤n . If in u (R ijkl ) and in u (R abcd ) have no common factor, then S(R ijkl , R abcd ) reduces to zero (see, for example, [29, Lemma 2.3.1]). Assuming that in u (R ijkl ) and in u (R abcd ) have a common factor we distinguish two cases, in each we underline the crossing monomials. Consider R ijls and R iklr , where without loss of generality i < j < k < l < r < s: S(R ijls , R iklr ) = p kr p ls p ij + p kr p jl p is − p js p ik p lr − p js p ir p kl = −p ij R klrs − p is R jklr + p lr R ijks + p kl R ijrs .
The second case for R ijkl and R jklm with i < j < k < l < m is similar.
Let C ∈ GF(I 2,n ) be the maximal cone corresponding to the monomial initial ideal M 2,n and let < denote the compatible monomial term order on S. By definition the standard monomial basis B < (see Definition 2.5) is the set of monomials that are not contained in the ideal generated by crossing monomials. So we call it the basis of non-crossing monomials. Proof . For every standard monomialp m with m ∈ Z ( n 2 ) ≥0 we draw all arcs ij in the n-gon, for which m ij = 0. By definition there is no pair of crossing arcs. Hence, the set of arcs can be completed to a triangulation andp m is a cluster monomial for the corresponding seed. On the other hand, every cluster monomial is contained in B < which completes the proof. Proof . For every weight vector w T associated with a triangulation T and a quadrilateral linear map w T we have in u in w T (I 2,n ) = in w T in u (I 2,n ) for u as in Definition 4.40. Given i < j < k < l, we choose a triangulation T containing the arcs ij, jk, kl, li and ik. Then in w T (R ijkl ) = p ij p kl − p ik p jl by Proposition 4.34 and Corollary 4.32. Now consider T obtained from T by flipping the arc ik, so T contains jl. Then in w T (R ijkl ) = −p ik p jl + p il p jk . So −p ik p jl is the only monomial that simultaneously is an initial of in w T (R ijkl ) and in w T (R ijkl ) for all i < j < k < l.
As mentioned in Remark 4.37 there are various ways to associate a toric degeneration to a triangulation or more generally a seed. The most general construction is the principal coefficient construction introduced by Gross-Hacking-Keel-Kontsevich in [28,Section 8]. We can now relate their construction to the Gröbner toric degenerations for Gr(2, C n ) in Proposition 4.36.
Corollary 4.45. The quotient S/ in T (I 2,n ) is isomorphic to the algebra of the semigroup generated by {g ij : 1 ≤ i < j ≤ n} ⊂ Z 2n−3 . In particular, the central fiber of the toric Gröbner degeneration induced by T is isomorphic to that of the degeneration induced by principal coefficients at T .
Proof . The quotient S/ in T (I 2,n ) has a vector space basis consisting of standard monomials with respect to C. By Proposition 4.43 this basis is in bijection with cluster monomials which are further in bijection with their g-vectors. In particular, S/ in T (I 2,n ) is a direct sum of cluster monomials with multiplication induced by the addition of their g-vectors by Proposition 4.34. Hence, it is isomorphic to the algebra of the semigroup generated by {g ij : Moreover, by [28,Theorem 8.30] the central fiber of the toric degeneration induced by adding principal coefficients to the seed corresponding to T is defined by the algebra of the semigroup generated by Theorem 4.46. In R ( n 2 ) we have C ∩ Trop(I 2,n ) = Trop + (I 2,n ). In particular, every (d + 1)dimensional face of C corresponds to a planar trivalent tree and Trop + (I 2,n ) is the closed subfan whose support is the set of (d + 1)-dimensional faces of the cone C.
Proof . Every planar trivalent tree with n leaves arises as the dual graph of a triangulation T of the n-gon. By Proposition 4.36, every weight vector w T gives a toric initial ideal. Hence, w T lies in the relative interior of a maximal (i.e., (d + 1)-dimensional) cone σ T in Trop + (I 2,n ). Furthermore, σ T is a face of C by Proposition 4.44. The opposite direction follows from the following claim.
Claim: Let Υ be a trivalent tree with n leaves. If in Υ (R ijkl ) contains the monomial p ik p jl for all i < j < k < l then Υ is planar.
Let Υ be a non-planar trivalent tree. Without loss of generality we assume that 1 ≤ i < j < k < l ≤ n are labels of Υ appearing either in clockwise order: i, k, j, l or in anticlockwise order: i, k, j, l. We prove the claim for the clockwise case, the anticlockwise case is analogous. We consider the full subtree with leaves i, j, k, l (similar to the picture in Figure 5, but ignoring the doubled leaves and exchanging k and j). Then in Υ (R ijkl ) = p ij p kl + p il p jk , a contradiction.
Corollary 4.47. The cone C is defined by the lineality space L 2,n and additional n 2 − n rays Moreover, C is a rational simplicial cone whose faces correspond to collections of arcs in the n-gon.
Proof . As we have identified a fan consisting of all (d + 1)-dimensional (closed) faces of C, all the rays of C (i.e., (n + 1)-dimensional faces, where n = dim L 2,n ) are also rays of Trop + (I 2,n ). By the combinatorial description of Trop(I 2,n ) from [49], we know that the rays correspond to trees with only one interior edge (corresponding to a partition of [n] into two sets). The rays of the cones corresponding to planar trivalent trees are therefore in correspondence with partitions of [n] into two cyclic intervals. To a (non-trivalent) planar tree we associate a weight vector as in (4.1).
Let r denote the matrix whose rows are 1 2 r ij for all 2 ≤ i + 1 < j ≤ n and recall the lifting of elements from Definition 3.2 (and Remark 3.12). In the following we denotef r byf for all f ∈ S.
Proof . We show that every variable t ab occurs with the same exponent inR ijkl and R ijkl (t). For simplicity, we adopt the notation R ijkl = −p e ik +e jl + p e il +e jk + p e ij +e kl . To compute the exponents of a variable t ab inR ijkl we have to distinguish several cases. We give the proof for only two of them as all others are similar. Case 1. Assume that i ∈ [a + 1, b] j, k, l. Note that due to symmetries this is equivalent to assuming i ∈ [a + 1, b] j, k, l. Then 1 2 r ab · (e ik + e jl ) = 1 2 r ab · (e il + e jk ) = 1 2 r ab · (e ij + e kl ) = − 1 2 .
Hence, inR ijkl the variable t ab does not appear. One can see that t ab also does not appear in Case 2. Assume that i, j ∈ [a + 1, b] k, l. Note that this is equivalent to assuming i, j ∈ [a + 1, b] k, l. Then 1 2 r ab · (e ik + e jl ) = 1 2 r ab · (e il + e jk ) = −1 and 1 2 r ab · (e ij + e kl ) = 0. Hence, inR ijkl the variable t ab does only appear with exponent 1 in the coefficients of the monomial p ij p kl . As we assumed i, j ∈ [a + 1, b] k, l it follows that a ∈ [l, i − 1] and b ∈ [j, k − 1]. So in R ijkl (t) the variable t ab appears also with exponent 1 in the coefficient of the monomial p ij p kl . Proof . The first part is a direct corollary of Proposition 4.48, Theorem 3.14(i) and Proposition 4.43. The second part follows from Theorem 3.14(ii) and Corollary 4.47.
We are now prepared to state and prove our main result for Gr(2, C n ).
Proof . Recall that A univ 2,n has the same set of (frozen and mutable) cluster variables as A 2,n , namely {p ij : 1 ≤ i < j ≤ n}, and additionally coefficients {y ij : 2 ≤ i + 1 < j ≤ n}. We define This morphism of C-algebras induces the desired isomorphism between A univ 2,n andÃ 2,n . By Proposition 3.9 the lifts of Plücker relationsR ijkl generate the lifted idealĨ 2,n . We proceed by showing that Ψ R ijkl is the corresponding exchange relations in A univ 2,n . Since the mutable parts of B univ (Q T ,F T ) and B (Q T ,F T ) coincide for every triangulation T there is a natural bijection between cluster monomials of A 2,n and A univ 2,n . It is the only bijection that sends the initial cluster variables of B (Q T ,F T ) to those of B univ (Q T ,F T ) and commutes with mutation. Further, it induces a bijection between the exchange relations associated to B (Q T ,F T ) and B univ (Q T ,F T ) , which yields bijections between the sets: quadrilaterals with vertices i < j < k < l in the n-gon ←→ exchange relations of A univ 2,n .
Consider a quadrilateral with vertices i < j < k < l in the n-gon and fix a triangulation T in which this quadrilateral occurs. So without loss of generality we have ij, ik, il, jk, kl ∈ T (see, e.g., left side of Figure 6). The exchange relation of A univ 2,n associated with the quadrilateral i < j < k < l is of form: where g ∨ ab is the ab th row of U Q mut T and g ∨ ab ik is its entry in the column of B univ (Q T ,F T ) corresponding to the mutable variablep ik . Hence, we need to compute those rows of U Q mut  Figure 6. A quadrilateral i < j < k < l in a triangulation T and the reflected triangulation T ∨ from which truncated g-vectors with respect to Q mut Using the right side of Figure 6 we compute the ik th entry of truncated g-vectors with respect to Q mut T ∨ : We compute: = E univ ijkl .
In particular, Ψ induces a surjective mapΨ :Ã 2,n → A univ 2,n . By Corollary 4.49,Ã 2,n is a free C[t ij ]-module whose basis are the cluster monomials. Similarly, after identifying t ij , 2 ≤ i + 1 < j ≤ n, with Ψ(t ij ) = y ij , A univ 2,n is a free C[t ij ]-module with basis given by the cluster monomials by [28,Theorem 0.3 and p. 502]. Hence,Ψ is also injective and the claim follows.
As A univ 2,n is by definition a domain the following is now a direct consequence.
Example 4.52. We list the lifted Plücker relations, respectively the exchange relations of A univ 2,5 , associated to our running example. These relations also constitute a Gröbner basis forĨ 2,5 , the crossing monomial of each relation is the first one. As Lemma 3.8 predicts, this is the only term with coefficient in C. Plücker variables of frozen cluster variables are marked in blue:

The Grassmannian Gr 3, C 6
We now turn to the case of Gr 3, C 6 and prove an analogue of Theorem 4.50. To highlight various important differences between this case and the case of Gr(2, C n ), we focus more on explicit computations. We believe that the explicit computations help to understand the difficulties that may arise when studying other compactifications of finite type cluster varieties such as Gr(3, C 7 ), Gr(3, C 8 ) or (skew-)Schubert varieties inside Grassmannians as in [48].
Theorem 4.53. There exists a unique maximal cone C ∈ GF(I ex ) with the following properties: (i) the associated initial ideal in C (I ex ) is generated by squarefree monomials of degree 2 and it contains all exchange monomials; (ii) the free C[t 1 , . . . , t 16 ]-algebra associated to C and r defined in Definition 3.2 has the pro-pertyÃ (iii) for every seed s in the cluster structure of A 3,6 the cone C has a 10-dimensional face τ s whose associated initial ideal in τs (I ex ) is a totally positive binomial prime ideal (hence τ s ∈ Trop + (I ex )). Moreover, C ∩ Trop(I ex ) = Trop + (I ex ).
Before we prove Theorem 4.53 we explain the conventions we use to describe the exchange relations of A univ 3,6 . The algebra A univ 3,6 is of cluster type D 4 . In particular, as explained in Section 4.1, we label the coefficients of A univ 3,6 with the set of almost positive coroots Φ ∨ ≥−1 in the root system dual to a root system of type D 4 . For this we fix an initial seed for A univ 3,6 such that the mutable part of its quiver is a bipartite orientation of D 4 . We choose the seed that contains the mutable variables p 246 , p 346 , p 124 and p 256 together with the frozen variables. The part of the quiver that involves only the vertices corresponding to these variables is the following: The coefficients are labeled by Φ ∨ ≥−1 and can be realized as frozen vertices of the quiver. In order to compute the arrows between the coefficient vertices and the vertices corresponding to cluster variables we identify the mutable vertices of this quiver with {1, 2, 3, 4}: let p 246 correspond to 1, p 346 to 2, p 124 to 3 and p 256 to 4. Now Proposition 4.16 contains all the information necessary to compute the arrows. As the resulting quiver is rather complicated, we refrain from visualizing it here. It is available for download on the homepage [4] in a format that can directly be opened in quiver mutation app [35]. Finally, we use the quiver mutation app to compute all exchange relations. They can be found explicitly in the Appendix A.
One more ingredient we need for the proof of Theorem 4.53 is the basis of cluster monomials for A univ 3,6 . By [28, Theorem 0.3 and p. 502] cluster monomials form a C[y α : α ∈ Φ ∨ ≥−1 ]basis for A univ 3,6 . If x and x are cluster variables that do not occur together in any seed, then any monomial divisible by xx cannot be a cluster monomial. In fact, using [36, Theorem 7.12(b)] this gives us the following characterization of cluster monomials. Writex a ∈ A univ 3,6 , a ∈ Z 22 ≥0 , to denote a monomial in the (mutable and frozen) cluster variablesp 123 , . . . ,p 456 ,X,Ȳ . Thenx a ∈ A univ 3,6 is a cluster monomial if and only if m |x a for all m ∈ M 3,6 , (4.11) where M 3,6 = {exchange monomials} ∪ XȲ ,p 135p246 . We write M 3,6 to denote the C[y α : α ∈ Φ ∨ ≥−1 ]-basis of cluster monomials for A univ 3,6 .
Proof of Theorem 4.53. We prove the statements of the theorem in order.
We compute a minimal generating set of in C (I ex ): it has 54 elements, 52 of those are exchange monomials, and the other two are p 135 p 246 and XY . This implies the first claim of the Theorem.
(ii) We prove this part in three steps: first, we compute the generators of the idealĨ ex r , then we define a surjective mapÃ r → A univ 3,6 , and lastly, we show that the map is also injective.
Step 1: By Proposition 3.9 the lifted idealĨ ex r is generated by the lifts of elements of a Gröbner basis for I ex and C. As a Gröbner basis we choose the exchange relations (whose initial forms are the exchange monomials), the four-term relation in (4.9) (whose initial form is p 135 p 246 ) and the following element (whose initial form is XY ): which is computed explicitly in Example 4.55 below. By the proof of (i) above, the set of exchange relations together with the four-term relation (4.9) and the above S-pair form a (minimal) Gröbner basis for I ex with respect to C. The reduced Gröbner basis G C (I ex ) consists of the 52 exchange relations and the additional two elements (4.12) The first monomial in f (and g) is its leading monomial. We now compute the lifts of the elements in G C (I ex ) with respect to the matrix r in Definition 3.2, which are given explicitly in Appendix A.
Step 2: A univ 3,6 has 22 cluster variables (in one-to-one correspondence with those of A 3,6 ) and 16 coefficients labeled by almost positive roots of type D 4 : y α with α ∈ Φ ∨ ≥−1 . We extend the morphism ψ in (4.8) to Ψ : C[t 1 , . . . , t 16 ][p 123 , . . . , p 456 , x, y] → A univ 3,6 by sending t i 's to y α 's according to the following identification: (4.13) We now verify thatĨ ex r ⊆ ker(Ψ): the lifts of those elements in I ex that correspond to exchange relations in A 3,6 are sent to exchange relations in A univ 3,6 by Ψ, so they lie in ker(Ψ). For the elements f, g ∈ G C (I ex ) in (4.12) note that, for example, p 245 f has an expression in terms of exchange relations with coefficients that are Plücker variables (see (A.1) for the precise expression). Hence, has an expression in terms of the lifts of those exchange relations with monomial coefficients in t's and Plücker variables.
As A univ 3,6 is a domain, this impliesf ∈ ker(Ψ). A similar argument implies thatg ∈ ker(Ψ) (see (A.2)) and so we obtain the induced morphismΨ :Ã r → A univ 3,6 . Note that the image ofΨ contains all cluster variables and all coefficients of A univ 3,6 , soΨ is in fact surjective.
Step 3: Lastly, we show thatΨ is injective. By Theorem 3.14(i) the standard monomial basis B < (for < a monomial term order compatible with C) is a C[t 1 , . . . , t 16 ]-basis forÃ r . Similarly, A univ 3,6 has the C[y α : α ∈ Φ ∨ ≥−1 ]-basis of cluster monomials M 3,6 . The test for membership in M 3,6 is given by M 3,6 in (4.11), which is in one-to-one correspondence with the set {in < (g) : g ∈ G < (I ex )}. Hence, there is a bijection between the standard monomial basis B < forÃ r (see Theorem 3.14(i)) and the cluster monomial basis M 3,6 of A univ 3,6 induced byΨ. In particular,Ψ is injective and A r ∼ = A univ 3,6 . (iii) To prove this part, we identify the rays r 1 , . . . , r 16 with mutable cluster variables. As we have already identified y α 's with t i 's in (4.4) (and by definition t i 's correspond to r i 's) it is enough to identify the y α 's with the mutable cluster variables of A 3,6 . This can be done by considering the exchange relations obtained by repeatedly mutating our bipartite initial seed at a sink. More precisely, we only consider the mutable part of the initial quiver and mutate at all the vertices with only incoming arrows from mutable vertices, which (by slight abuse of language) we refer to as sinks. The order of the individual mutations in this mutation sequence is irrelevant as they pairwise commute. Every exchange relation produced by mutation at a sink corresponding to a cluster variable x has the property that one of the cluster monomials involves exactly one coefficient y α (see [25,Lemma 12.7]). When iterating the process of mutating at sinks, every mutable cluster variable appears as a sink at some point. Moreover, [25,Lemma 12.8] implies that the assignment x → y α defines a bijection between mutable cluster variables and coefficients. Combining with the identification of y α 's with t i 's in (4.4) we obtain: Next, to every seed we associate a weight vector that is the sum of the rays corresponding to its mutable cluster variables. For example, the weight vector associated to s = {p 124 ,p 125 ,p 245 ,p 256 } is w s = r 1 + r 3 + r 8 + r 12 . Using Macaulay2 [27] we verify that in ws (I ex ) is a totally positive binomial prime ideal for every seed listed above. The initial ideals can be found on [4].
To see that C ∩ Trop(I ex ) = Trop + (I ex ) observe that for every element h ∈ G C (I ex ) its initial monomial in C (h) is the unique monomial with positive coefficient (the complete list of G C (I ex ) can be found in Section A). Hence, a weight vector w lies in C ∩ Trop(I ex ) if and only if it lies in Trop + (I ex ).
Remark 4.54. There are various methods in cluster theory to compute the exchange relations for A univ 3,6 and M 3,6 , e.g., one can use the categorification of finite type cluster algebras with universal coefficients introduced in [42]. To compute M 3,6 one can use the compatibility degree of cluster variables from [24]. In fact, the elements of M 3,6 are exactly those pairs of cluster variables whose compatibility degrees are positive. However, we have presented the case of Gr 3, C 6 with as few machinery from the cluster theory as possible to make it more digestible to a broader audience. Note that w ∈ C • , hence the assumptions of Lemma 2.12 hold. One can explicitly compute v = 1 4 (19,26,18,19,19,9,9,21,19,14,19,9,7,19,15,9,19,14,6,19,29,33), In particular, the statement of Lemma 2.12 does not hold for h, hence it is false in general for arbitrary elements of J. More importantly, the initial form of an arbitrary element h ∈ J need not be the same with respect to different weight vectors in the relative interior of a maximal Gröbner cone of J. This may occur when h contains more than one monomial in the monomial initial ideal. Here, the monomials XY , p 136 p 145 p 234 p 256 and p 123 p 146 p 256 p 345 all lie in in C (I ex ). Remark 4.57. Note that by definition the cluster complex can be realized by the g-fan with simplices corresponding to simplicial cones. It was shown in [28] that the g-fan is a simplicial fan in complete generality. This occurred several years after the cluster complex had been defined.

Stanley-Reisner ideals and the cluster complex
Definition 4.58. Let ∆ be a simplicial complex with vertex set V = {x 1 , . . . , x n }. The Stanley-Reisner ideal of ∆ is generated by monomials associated to the minimal non-faces of ∆ as: Reversely, to every squarefree monomial ideal one can associate its Stanley-Reisner complex, whose non-faces are defined by the monomials in the ideal.
Corollary 4.59. Let A be A 2,n or A 3,6 and ∆(A) the associated cluster complex. Similarly, let I be the ideal I 2,n or I ex and C the maximal cone in GF(I) whose initial ideal contains all the exchange monomials. Then the Stanley-Reisner ideal I ∆(A) coincides with the initial ideal in C (I).
Proof . The initial ideal in C (I) is squarefree and generated by the monomials in the set M , which is respectively, M 2,n or M 3,6 . Observe that M defines the minimal non-faces of ∆(A). Hence, ∆(A) is the Stanley-Reisner complex of in C (I).

Newton-Okounkov bodies and mutations
In this section, we explain how our results relate to Escobar-Harada's work on wall-crossing phenomena for Newton-Okounkov bodies in [19]. Given a homogeneous ideal J ⊆ K[x 1 , . . . , x n ] assume there exists a maximal cone σ in Trop(J) whose associated ideal is toric. Then Escobar and Harada study the relation between the Newton-Okounkov bodies of two maximal prime cones intersecting in a facet. They give two piecewise linear maps, called flip and shift which send one Newton-Okounkov body to another.
Here, we focus on the case of Gr(2, C n ). We fix a triangulation T of the n-gon. The output of Algorithm 1 can be used to define a total order on Z 2n−3 as follows.
Definition 4.60. Let V 0 , . . . , V i be the output of Algorithm 1 for some triangulation T . To each V j we associate a sequence V j with the same elements as V j . Let V be the sequence V 0 , . . . , V i and ≺ the associated lexicographic order on Z 2n−3 . Call ≺ a total order compatible with T .
Recall from Section 4.3 the standard monomial basis B < consisting of non-crossing monomials (i.e., every monomial in B < corresponds to a collection of non-crossing (boundary and internal) arcs, where arcs may appear with multiplicity greater than 1). Then the map g T : B < → Z 2n−3 given byp a → ij∈T a ij g T ij . (4.14) extends to a full rank valuation on A 2,n \ {0} as follows. Fix a total order ≺ compatible with T .
Every 0 = f ∈ A 2,n is a unique linear combination of elements in B < , that is f = p a ∈B< c ap a .
We define the valuation of f as g T (f ) := min ≺ g T (p a ) : c a = 0 . Denote the associated graded algebra by gr T (A 2,n ) and the corresponding Newton-Okounkov body by ∆(T ). By [5,Corollary 2] and Proposition 4.34 we have gr T (A 2,n ) ∼ = S/ in T (I 2,n ) and ∆(T ) = conv g T ij ij .
Remark 4.61. The g-vectors and the corresponding valuation make sense in greater generality. For example, the case of arbitrary Grassmannians is treated in [6], where the authors (among other things) explain how the associated Newton-Okounkov bodies arise in the context of [28]. Even more generally, the algebra A 2,n can be replaced by the middle cluster algebra in the sense of [28], the standard monomial basis by the corresponding theta basis, and the total order ≺ by a lexicographic refinement of the dominance order. Similarly, Fujita and Oya in [26, Section 3] define a valuation on the ambient field of a cluster algebra (whose exchange matrix has full rank). Their valuation recovers Fomin-Zelevinsky's g-vectors for cluster monomials.
Interpreting g-vectors as a valuation reveals the necessity for working with extended g-vectors as opposed to their truncated counterparts that are popular in algebraic or representation theoretic applications of cluster algebras. The following example shows that truncated g-vectors do not have the desired properties for applications.
The total order ≺ is compatible with T . The truncated g-vectors are the underlined parts of the g-vectors above. So if we decided to only consider those we would need to find an order that satisfies (0, 0, 0) (0, 1, 0) and (0, 1, 0) (0, 0, 0), a contradiction.
Next we describe how the Newton-Okounkov body ∆(T ) behaves under changes of the triangulation. As all triangulations can be obtained from one another by a sequence of flips of arcs, we focus on performing a single such flip. Let T 1 and T 2 be two triangulations such that there exist a < b < c < d in [n] with ac ∈ T 1 , bd ∈ T 2 and T 1 \ {ac} = T 2 \ {bd}. In other words, T 2 is obtained from T 1 by flipping the arc ac. We denote by R T 1 the vector space R 2n−3 with standard basis {f ij : ij ∈ T 1 } and similarly, R T 2 for R 2n−3 with basis {f ij : ij ∈ T 2 }.
The theory of cluster varieties gives us two distinguished maps from R 2n−3 to itself. The first map is a piecewise linear shear that can be obtained by the Fock-Goncharov tropicalization of a cluster mutation (see [28,Definition 1.22]): where v ac = −f ab − f cd + f ad + f bc . The second map is a GL 2n−3 (Z)-base change from {f ij : ij ∈ T 1 } to {f ij : ij ∈ T 2 } corresponding to a seed mutation (see [20, equation (8)]) denoted by µ T 1 T 2 : R T 1 → R T 2 . It is given by f ij = f ij for ij ∈ T 1 ∩ T 2 and f bd = −f ac + f ab + f cd . The following Lemma follows by combining the results in [28,Sections 1.3 and 5]. For the convenience of the reader, we give a self-contained elementary proof below. from Figure 7. First, we apply [19,Lemma 5.13] and replace Υ 2 by a tree Υ 2 that gives the same initial ideal as Υ 2 , and its leaves are labeled in the same order as Υ 1 . This implies that σ 1 and σ 2 are indeed faces of the same maximal Gröbner cone (corresponding to the clockwise labeling of the leaves: 15243). Then we apply the symmetric group element s = (235) to the leaves of Υ 1 and Υ 2 to obtain the following trees which are both dual to triangulations of the 5-gon. Moreover, their triangulations are related by exchanging the diagonal 24 for 13 (this is also called a flip): The symmetric group element s induces an automorphism s : S → S given by s(p ij ) = p s −1 (i)s −1 (j) . It is straight-forward to verify that s(in s(Υ) (I)) = in Υ (I) for Υ ∈ {Υ 1 , Υ 2 }.
Consider σ 1 and σ 2 as in Corollary 4.64 and assume they lie in the same maximal Gröbner cone C. Then the standard monomial basis of C induces a bijective map between the value semigroups im(v σ 1 ) and im(v σ 2 ) (see [19,Section 4.2], where this is called an algebraic wall-crossing). As seen in Lemma 4.63 the map µ T 1 T 2 •ζ ac extends to a map between the value semigroups im g T 1 and im g T 2 . In [19,Theorem 5.15] the authors show that their piecewise linear flip map induces the algebraic wall-crossing for Gr(2, C n ). Hence, Corollary 4.64 implies that the flip map for Gr(2, C n ) is of cluster nature in the sense that (up to unimodular equivalence) it is given by the Fock-Goncharov tropicalization of a cluster mutation.
Remark 4.66. Cluster mutations are a very special class of automorphisms of a complex algebraic torus preserving its canonical volume form. Automorphisms preserving this form have various names in the literature such as Laurent polynomial mutations [31], elements of the special Cremona group [52] or combinatorial mutations [2]. Ilten in [19,Appendix] related the wall-crossing formulas to the theory of polyhedral divisors for complexity-one T -varieties [3] and outlined how this relates to combinatorial mutations in the sense of [2].