Families of Gröbner 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 Gröbner fan of 𝐽 with 𝑚 rays. We construct a flat family over 𝔸ᵐ that assembles the Gröbner degenerations of 𝑉 associated with all faces of 𝐶. This is a multi-parameter generalization of th...

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Published in:	Symmetry, Integrability and Geometry: Methods and Applications
Date:	2021
Main Authors:	Bossinger, Lara, Mohammadi, Fatemeh, Nájera Chávez, Alfredo
Format:	Article
Language:	English
Published:	Інститут математики НАН України 2021
Online Access:	https://nasplib.isofts.kiev.ua/handle/123456789/211364
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Cite this:	Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras. Lara Bossinger, Fatemeh Mohammadi and Alfredo Nájera Chávez. SIGMA 17 (2021), 059, 46 pages

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author	Bossinger, Lara Mohammadi, Fatemeh Nájera Chávez, Alfredo
author_facet	Bossinger, Lara Mohammadi, Fatemeh Nájera Chávez, Alfredo
citation_txt	Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras. Lara Bossinger, Fatemeh Mohammadi and Alfredo Nájera Chávez. SIGMA 17 (2021), 059, 46 pages
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container_title	Symmetry, Integrability and Geometry: Methods and Applications
description	Let 𝑉 be the weighted projective variety defined by a weighted homogeneous ideal 𝐽 and 𝐶 a maximal cone in the Gröbner fan of 𝐽 with 𝑚 rays. We construct a flat family over 𝔸ᵐ that assembles the Gröbner degenerations of 𝑉 associated with all faces of 𝐶. This is a multi-parameter generalization of the classical one-parameter Gröbner degeneration associated with 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 𝑋C (the toric variety associated to 𝐶) along the universal torsor 𝔸ᵐ → 𝑋C. We apply this construction to the Grassmannians Gr(2, ℂⁿ) with their Plücker embeddings and the Grassmannian Gr(3, ℂ⁶) with its cluster embedding. In each case, there exists a unique maximal Gröbner cone whose associated initial ideal is the Stanley-Reisner ideal of the cluster complex. We show that the corresponding cluster algebra with universal coefficients arises as the algebra defining the flat family associated with this cone. Further, for Gr(2, ℂⁿ), we show how Escobar-Harada's mutation of Newton-Okounkov bodies can be recovered as a tropicalized cluster mutation.
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 17 (2021), 059, 46 pages Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras Lara BOSSINGER a, Fatemeh MOHAMMADI bc and Alfredo NÁJERA CHÁVEZ ad a) Instituto de Matemáticas UNAM Unidad Oaxaca, León 2, altos, Oaxaca de Juárez, Centro Histórico, 68000 Oaxaca, Mexico E-mail: lara@im.unam.mx, najera@matem.unam.mx URL: https://www.matem.unam.mx/~lara/, https://www.matem.unam.mx/~najera/ b) Department of Mathematics: Algebra and Geometry, Ghent University, 9000 Gent, Belgium E-mail: fatemeh.mohammadi@ugent.be URL: https://www.fatemehmohammadi.com c) Department of Mathematics and Statistics, UiT – The Arctic University of Norway, 9037 Tromsø, Norway d) Consejo Nacional de Ciencia y Tecnoloǵıa, Insurgentes Sur 1582, Alcald́ıa Benito Juárez, 03940 CDMX, Mexico Received October 21, 2020, in final form May 29, 2021; Published online June 10, 2021 https://doi.org/10.3842/SIGMA.2021.059 Abstract. Let V be the weighted projective variety defined by a weighted homogeneous ideal J and C a maximal cone in the Gröbner fan of J with m rays. We construct a flat family over Am that assembles the Gröbner degenerations of V associated with all faces of C. This is a multi-parameter generalization of the classical one-parameter Gröbner degeneration associated to a weight. We explain how our family can be constructed from Kaveh–Manon’s recent work on the classification of toric flat families over toric varieties: it is the pull-back of a toric family defined by a Rees algebra with baseXC (the toric variety associated to C) along the universal torsor Am → XC . We apply this construction to the Grassmannians Gr(2,Cn) with their Plücker embeddings and the Grassmannian Gr ( 3,C6 ) with its cluster embedding. In each case, there exists a unique maximal Gröbner cone whose associated initial ideal is the Stanley–Reisner ideal of the cluster complex. We show that the corresponding cluster algebra with universal coefficients arises as the algebra defining the flat family associated to this cone. Further, for Gr(2,Cn) we show how Escobar–Harada’s mutation of Newton– Okounkov bodies can be recovered as tropicalized cluster mutation. Key words: cluster algebras; Gröbner basis; Gröbner fan; Grassmannians; flat degenerations; Newton–Okounkov bodies 2020 Mathematics Subject Classification: 13F60; 14D06; 14M25; 14M15; 13P10 1 Introduction The theory of Gröbner fans, introduced by Mora and Robbiano [40], is a popular tool in commu- tative 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[x1, . . . , xn] a d-weighted homogeneous ideal for some d ∈ Zn>0. The Gröbner fan of J is a complete fan in Rn whose elements represent weight vectors on the variables x1, . . . , xn. 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 pro- jective space P(d). Every open cone in the Gröbner fan gives rise to a one-parameter flat family mailto:lara@im.unam.mx mailto:najera@matem.unam.mx https://www.matem.unam.mx/~lara/ https://www.matem.unam.mx/~najera/ mailto:fatemeh.mohammadi@ugent.be https://www.fatemehmohammadi.com https://doi.org/10.3842/SIGMA.2021.059 2 L. Bossinger, F. Mohammadi and A. Nájera Chávez 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 r1, . . . , rm and denote by r the (m × n)-matrix whose rows are r1, . . . , rm. For an element f = ∑ α∈Zn≥0 cαxα of J , we define µ(f) ∈ Zm as the vector whose i-th entry is mincα 6=0{ri · α}. Then the lift of f is f̃ := f ( tr·e1x1, . . . , t r·enxn ) t−µ(f), where ta for a ∈ Zm denotes the monomial ∏m i=1 t ai i . The lifted ideal J̃ ⊆ K[t1, . . . , tm][x1, . . . , xn] is the ideal generated by the lifts of all polynomials in J . We prove that J̃ 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 xi’s, see Proposition 3.3. Consequently, J̃ is independent of the choice of r and it defines a variety inside P(d)× Am. 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 algebra Ã := K[t1, . . . , tm][x1, . . . , xn]/J̃ is a free K[t1, . . . , tm]-algebra. It defines a flat family Proj ( Ã ) P(d)× Am Am π such that for every face τ of C there exists aτ ∈ Am with fiber π−1(aτ ) isomorphic to the variety defined by the initial ideal associated to τ . In particular, generic fibers are isomorphic to Proj(A), where A = K[x1, . . . , xn]/J , and there exist special fibers for every proper face τ ⊂ C. Next, we explain how the algebra Ã 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 XΣ 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 XΣ 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 XΣ 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[x1, . . . , xn] be a weighted homogeneous ideal and C a maximal cone in the Gröbner fan of J . Let RC(A) be the Rees algebra associated to the PL-quasivaluation on A = K[x1, . . . , xn]/J defined by C and let ψ : Spec(RC(A))→ XC be the corresponding toric family. The morphism π : Spec ( Ã ) → Am fits into a pull-back diagram as follows: Spec(RC(A)) Spec ( Ã ) XC Am. ψ π pC Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras 3 Here pC : Am → XC is the universal torsor of XC (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 Am contains the universal torsor TΣ of the toric variety XΣ and all fibers of π−1(TΣ)→ TΣ correspond to initial ideals of cones in Σ. In Corollary 3.20 we show how the family defined by π\|π−1(TΣ) also arises as a pull-back from a toric family defined by a sheaf of Rees algebras on XΣ. 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,Cn) and Gr ( 3,C6 ) whose coordinate rings are cluster algebras. For Gr(2,Cn) we choose its Plücker embedding and obtain the homogeneous coordinate ring A2,n as a quotient of the polynomial ring in Plücker variables C[pij : 1 ≤ i < j ≤ n] by the Plücker ideal I2,n. It was shown in [23] that A2,n is a cluster algebra in which the cluster variables are in one-to-one correspondence with Plücker coordinates p̄ij ∈ A2,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,Cn) obtained by adding principal coefficients to A2,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 construc- tion 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 prin- cipal 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 wT , see Definition 4.35. In Proposition 4.36 we prove that the initial ideal with respect to wT is binomial and prime, hence wT lies in the tropicalization of I2,n. Moreover, the central fiber of the Gröbner degen- eration induced by wT is isomorphic to the central fiber of the toric degeneration induced by endowing A2,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 Auniv 2,n associated to A2,n, see Defini- tion 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 I2,n as it encodes various toric degenerations of Gr(2,Cn) 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,Cn): Theorem 1.3. There exists a maximal cone C in the Gröbner fan of I2,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 A2,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 wT lies in the boundary of C and the intersection of C with the tropicalization of I2,n is the totally positive part of Trop(I2,n) (Proposition 4.44 and Theorem 4.46). 4 L. Bossinger, F. Mohammadi and A. Nájera Chávez (iii) Let Ĩ2,n be the lift of I2,n with respect to C and denote by Ã2,n the quotient C[tij : 2 ≤ i+1 < j ≤ n][pij : 1 ≤ i < j ≤ n]/Ĩ2,n. Then there exists a canonical isomorphism between the cluster algebra with universal coefficients Auniv 2,n and Ã2,n (Theorem 4.50). (iv) The monomial initial ideal of I2,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,C6 ) from the viewpoint of Gröbner theory. In this case, we fix its cluster embedding obtained as follows: consider the Plücker embedding and let A3,6 be the homogeneous coordinate ring. As a clus- ter algebra of type D4, A3,6 has 22 cluster variables, 20 of which are the Plücker coordinates. The additional two cluster variables are homogeneous binomials in Plücker coordinates of deg- ree 2. Hence, we can present A3,6 as the quotient of a polynomial ring in 22 variables by a weighted homogeneous ideal denoted by Iex. This yields an embedding of Gr ( 3,C6 ) in the weighted projective space P(1, . . . , 1, 2, 2), where the Plücker coordinates correspond to the coor- dinates 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 Iex such that (i) The algebra Ã3,6 is canonically isomorphic to the cluster algebra with universal coeffici- ents Auniv 3,6 , where rays of C are identified with mutable cluster variables of A3,6 (Theo- rem 4.53(ii)). (ii) For every seed of A3,6 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(Iex) is the totally positive part of Trop(Iex) (Theorem 4.53(iii)). (iii) The monomial initial ideal inC(Iex) 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 A3,6 and A2,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 A2,n each Newton–Okounkov body arising from Trop(I2,n) is unimod- ularly 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 σ1 and σ2 be two maximal prime cones in Trop(I2,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,Cn) 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 Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras 5 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,Cn) and Gr ( 3,C6 ) . 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,Cn) in Section 4.2. Then we apply the main construction to Gr(2,Cn) in Section 4.3 and afterwards to Gr ( 3,C6 ) 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,C6 ) . 2 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[x1, . . . , xn], we fix the notation xα with α = (a1, . . . , an) ∈ Zn≥0 denoting the monomial xa1 1 . . . xann . Throughout when we write f = ∑ α∈Zn≥0 cαxα for f ∈ K[x1, . . . , xn] we refer to the expression of f in the basis of monomials. 2.1 Weighted projective varieties Fix a vector d = (d1, . . . , dn) ∈ Rn. Let Kd[x1, . . . , xn] be the polynomial ring K[x1, . . . , xn] endowed with the R-grading determined by setting deg(xi) = di for i = 1, . . . , n. Definition 2.1. For c ∈ K∗ and α = (a1, . . . , an) ∈ Zn≥0 the weight of the monomial cxα ∈ Kd[x1, . . . , xn] is d · α = ∑n i=1 diai. We extend this definition and say f ∈ Kd[x1, . . . , xn] is d-homogeneous if f can be expressed as a sum of monomials of the same weight. Similarly, an ideal J ⊆ Kd[x1, . . . , xn] is d-homogeneous if it is generated by d-homogeneous elements. Assume d ∈ Zn>0. In this case Kd[x1, . . . , xn] is Z-graded and we can define the weighted projective space P(d1, . . . , dn) as the quotient of Kn under the equivalence relation (x1, . . . , xn) ∼ ( λd1x1, . . . , λ dnxn ) for λ ∈ K∗. For K = C it is well-known that P(d1, . . . , dn) is a projective toric variety [15]. Moreover, as a scheme P(d1, . . . , dn) is the homogeneous spectrum (or Proj) of Kd[x1, . . . , xn] [17, Nota- tion 1.1]. For simplicity, we denote P(d1, . . . , dn) by P(d) and let [x1 : · · · : xn] denote the class of (x1, . . . , xn) in P(d). Given a d-homogeneous ideal J ⊆ Kd[x1, . . . , xn] its set of zeros is V (J) = { [x1 : · · · : xn] ∈ P(d) : f(x1, . . . , xn) = 0 for all f ∈ J } . Conversely, given a subset X ⊆ P(d1, . . . , dn), its associated ideal I(X) is generated by polyno- mials:{ f ∈ Kd[x1, . . . , xn] : f is d-homogeneous and f(p) = 0 for all p ∈ X } . 6 L. Bossinger, F. Mohammadi and A. Nájera Chávez 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 the- ory 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 Pn−1 can be realized as Pn−1 = P(1), where 1 = (1, . . . , 1) ∈ Zn. 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 S(X) := Kd[x1, . . . , xn]/I(X). 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(d1, . . . , dn) 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 . 2.2 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. Definition 2.3. Let f = ∑ α∈Zn≥0 cαxα ∈ K[x1, . . . , xn]. Given a weight vector w ∈ Rn the initial form of f with respect to w is defined as inw(f) := ∑ α : w·α=a cαxα, where a = min{w · α : cα 6= 0}. For an ideal J ⊆ K[x1, . . . , xn] its initial ideal with respect to w is defined as inw(J) : = 〈inw(f) : f ∈ J〉. A finite set G = {g1, . . . , gr} ⊆ J is called a Gröbner basis for J with respect to w if inw(J) = 〈inw(g1), . . . , inw(gr)〉. Definition 2.4. A monomial term order on K[x1, . . . , xn] is a total order < on the set of monic monomials in K[x1, . . . , xn] such that for every α, β, γ in Zn≥0 we have that (i) 1 ≤ xα, and (ii) if xα < xβ, then xα+γ < xβ+γ . The initial monomial of an element f = ∑ α∈Zn≥0 cαxα ∈ K[x1, . . . , xn] with respect to < is in<(f) := cβx β, where xβ = max<{xα : cα 6= 0}. The initial ideal of an ideal J ⊆ K[x1, . . . , xn] with respect to < is defined as in<(J) := 〈in<(f) : f ∈ J〉. Recall that given an ideal J ⊆ K[x1, . . . , xn] and a monomial term order < there always exists a weight vector −w ∈ Zn≥0 such that inw(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 inw(J) is generated by monomials and there exists a monomial term order < such that inw(J) = in<(J), then we say that w is compatible with <. Definition 2.5. Let in<(J) be a monomial initial ideal of the ideal J for some monomial term order < on K[x1, . . . , xn]. Then the set B< := {x̄α : xα 6∈ in<(J)} is a vector space basis of K[x1, . . . , xn]/J (and K[x1, . . . , xn]/ in<(J)) called standard monomial basis. Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras 7 Theorem 2.6 (the division algorithm). Fix a monomial term order < on K[x1, . . . , xn] and let g1, . . . , gs be non-zero polynomials in K[x1, . . . , xn]. Given 0 6= f ∈ K[x1, . . . , xn] there exist f1, . . . , fs, f ′ ∈ K[x1, . . . , xn] with f = f1g1 + · · ·+ fsgs + f ′, such that (i) if f ′ 6= 0 then no monomial of f ′ is divisible by any of the monomials in<(g1), . . . , in<(gs); (ii) if fi 6= 0 then in<(f) ≥ in<(figi) for all i. We say that f reduces to f ′ with respect to {g1, . . . , gs}. Definition 2.7. Let < be a monomial term order on K[x1, . . . , xn] and G = {g1, . . . , gs} a finite generating set of an ideal J . Then the S-polynomial of gi and gj is defined as S(gi, gj) := lcm ( in<(gi), in<(gj) ) in<(gi) gi − lcm ( in<(gi), in<(gj) ) in<(gj) gj . We say that Buchberger’s criterion holds if for all 1 ≤ i < j ≤ s, the S-pairs reduce to zero with respect to {g1, . . . , gs}. 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 (i) in<(gi) is monic for all 1 ≤ i ≤ s, and (ii) for i 6= j no monomial in gi is divisible by in<(gj). 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]): Definition 2.8. Let J ⊆ K[x1, . . . , xn] be an ideal. The Gröbner region of J denoted by GR(J) is the set of w ∈ Rn such that there exists a monomial term order < with in<(inw(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 ∈ Rn lie in the relative interior of the same cone C ⊂ Rn (denoted by C◦) if and only if inu(J) = inw(J). We introduce the notation inC(J) := inw(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 ∈ Rn : inw(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 inu(J) = J . Integral weight vectors in GF(J) can be seen as points in a lattice N = Zn whose dual lattice M = (Zn)∗ contains exponent vectors of monomials in K[x1, . . . , xn]. 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 inw(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<(inw(g)) for every g ∈ G<(J). Lemma 2.10. Let d ∈ Zn>0 and J ⊆ Kd[x1, . . . , xn] be a d-homogeneous ideal. Then d ∈ L(J). More generally, L(J) contains all elements u ∈ Rn such that J is u-homogeneous. Proof. Observe that for every d-homogeneous polynomial f ∈ Kd[x1, . . . , xn] the equality ind(f) = f holds. Next, every f ∈ J can be uniquely written as a finite sum f = ∑d i=0 fi for some d ∈ Z≥0 and fi ∈ J of d-degree i (see [30, Lemma 3.0.7]). Then ind(f) = fj , where j = min{i : fi 6= 0}. In particular, for any f ∈ J we have that ind(f) ∈ J . Therefore, ind(J) = J . � 8 L. Bossinger, F. Mohammadi and A. Nájera Chávez 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 := {f1, . . . , ft} of J with respect to a fixed element ` ∈ L(J). Then J = in`(J) is generated by in(F ) := {in`(f1), . . . , in`(ft)}. 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 res- pect 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 ⊆ Kd[x1, . . . , xn] the support of its Gröbner fan, i.e., GR(J), is Rn. In other words, there exists a compatible monomial term order for every maximal cone C ∈ GF(J). Lemma 2.12. Let J ⊆ Kd[x1, . . . , xn] 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 ∈ Rn such that w := u+ v ∈ C◦. Then for every g ∈ G<(J), we have in<(g) = inw(g) = inu(inv(g)). Proof. Consider an element g in G<(J). Since w ∈ C◦ we have in<(g) = inw(g). On the other hand, since v ∈ C, by Remark 2.9 we have that in<(g) = in<(inv(g)). This implies that inw(g) is a refinement of inv(g). In other words, inv(g) contains the monomial inw(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. Lemma 2.13. For every cone C ∈ GF(J) ⊆ Rn, the cone C := C/L(J) ⊆ Rn/L(J) is strongly convex, that is C ∩ (−C) = {0}. Proof. As we have chosen J to be d-homogeneous every cone in GF(J) is a face of a full- dimensional cone. Hence, we may assume without loss of generality that C is a cone of dimen- sion n. Let < be the corresponding monomial term order and g ∈ G<(J). Let v and −v be in C \ C◦ and choose u, u′ ∈ Rn such that u + v, u′ − v ∈ C◦. By Lemma 2.12 we have that in<(g) = inu(inv(g)) = inu′(in−v(g)). However, in<(g) cannot simultaneously be a monomial in inv(g) and in−v(g) unless v ∈ L(J). � Definition 2.14. For a d-homogeneous ideal J ⊆ Kd[x1, . . . , xn] we define its tropicalization Trop(J) := { w ∈ Rn : inw(J) does not contain any monomial } . In fact, Trop(J) is a d-dimensional subfan of GF(J), where d is the Krull-dimension of Kd[x1, . . . , xn]/J . Definition 2.15. An ideal J ⊆ R[x1, . . . , xn] is called totally positive if it does not contain any non-zero element of R≥0[x1, . . . , xn]. For a d-homogeneous ideal J ⊆ R[x1, . . . , xn] the totally positive part of its tropicalization is defined as Trop+(J) := {w ∈ Trop(J) : inw(J) is totally positive}. Due to [50] Trop+(J) is a closed subfan of Trop(J) (that may be empty). Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras 9 3 Families of Gröbner degenerations In this section, we introduce the main construction of the paper. Let J ⊆ Kd[x1, . . . , xn] be a d-homogeneous ideal, C a maximal cone in GF(J) and A the quotient Kd[x1, . . . , xn]/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 Ĵw := 〈 ∑ α∈Zn≥0 cαxαtw·α−min{w·β : cβ 6=0} : ∑ α∈Zn≥0 cαxα ∈ J 〉 ⊆ K[t][x1, . . . , xn]. (3.1) Remark 3.1. The ideal Ĵw induces a flat family Spec ( K[t][x1, . . . , xn]/Ĵw ) → Spec(K[t]) whose fiber over the closed point (t) is isomorphic to Spec(K[x1, . . . , xn]/ inw(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 r1, . . . , rm ∈ C such that {r1, . . . , rm} 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 r1, . . . , rm are non-negative or positive vectors. We denote by r the (m × n)-matrix whose rows are r1, . . . , rm. Additionally, we write < for a monomial term order compatible with C and denote by G the associated reduced Gröbner basis. Definition 3.2. For f = ∑ α∈Zn≥0 cαxα ∈ J set µr(f) := ( mincα 6=0{ri · α} ) i=1,...,m ∈ Zm×1, that is we think of µr(f) as a column vector with m entries. We define the lift of f to be the polynomial f̃r ∈ K[t1, . . . , tm][x1, . . . , xn] given by the following formula f̃r := f̃r(t,x) := f(tr·e1x1, . . . , t r·enxn)t−µr(f) = ∑ α∈Zn≥0 cαxαtr·α−µr(f). Similarly, we define the lifted ideal as J̃r := 〈 f̃r : f ∈ J 〉 ⊆ K[t1, . . . , tm][x1, . . . , xn], and the lifted algebra as the quotient Ãr := K[t1, . . . , tm][x1, . . . , xn]/J̃r. (3.2) We proceed by showing that the lifted algebra Ãr is independent of the choice of vectors r1, . . . , rm ∈ C that represent the primitive ray generators r1, . . . , rm of C. For this, we identify generators of the ideal J̃r which is a crucial matter for applications. Proposition 3.3. Suppose r′1, . . . , r ′ m ∈ C are such that r′i = ri in Rn/L(J). Let r′ be the (m× n)-matrix whose rows are r′1, . . . , r ′ m. Then for g ∈ G we have that g̃r = g̃r′. Proof. We have that r′i = ri + li for some li ∈ L(J). Let L be the (m × n)-matrix whose rows are l1, . . . , lm. In particular, r′ = r + L. Write g = ∑ α∈Zn≥0 cαxα. Since g ∈ G we have by Lemma 2.11 that the value li ·α is the same for all α with cα 6= 0. Let ai be this common value. Observe that mincα 6=0{r′i · α} = mincα 6=0{ri · α} + ai. In particular, we have that the column vector (a1, . . . , am) is equal to L · α for all α with cα 6= 0 and therefore µr′(f) = µr(f) + L · α. Finally, we compute g̃r′ = ∑ α∈Zn≥0 cαxαtr ′·α−µr′ (g) = ∑ α∈Zn≥0 cαxαtr·α+L·α−(µr(g)+L·α) = g̃r. � 10 L. Bossinger, F. Mohammadi and A. Nájera Chávez In the following, when there is no risk of confusion, we write J̃ for J̃r, f̃ for f̃r and Ã for Ãr. Notation 3.4. For a = (a1, . . . , am) ∈ Km denote by (t − a) the maximal ideal 〈t1 − a1, . . . , tm − am〉 of K[t1, . . . , tm]. Then we denote by Ã\|t=a the tensor product Ã⊗K[t1,...,tm]K[t1, . . . , tm] /(t−a). Additionally, for f ∈ K[t1, . . . , tm][x1, . . . , xn] let f \|t=a ∈ K[x1, . . . , xn] be the evaluation of f at ti = ai for i ∈ [m]. Similarly, denote by J̃ \|t=a the ideal of K[x1, . . . , xn] generated by the set {f̃ \|t=a : f ∈ J}. For a set B ⊂ K[t1, . . . , tm][x1, . . . , xn] let B\|t=a := {b\|t=a : b ∈ B}. Recall the isomorphism K[t1, . . . , tm]/(t− a) → K that sends an element f̄ to f \|t=a. It extends to an isomorphism Ã\|t=a ∼= K[x1, . . . , xn]/J̃ \|t=a, which explains our choice of notation. Denote 1 := (1, . . . , 1) ∈ Zm. Lemma 3.5. Let w = (w1, . . . , wn) = r1 + · · ·+ rm, and w′ = (−1, . . . ,−1, w1, . . . , wn) ∈ Zm+n, where wi is the weight of xi and −1 is the weight of tj for i = 1, . . . , n and j = 1, . . . ,m. Then for every f ∈ J its lift f̃ is w′-homogeneous. In particular, the ideal J̃ is w′-homogeneous. Proof. Let f = ∑ α∈Zn≥0 cαxα. Then the lift of f is by definition f̃ = ∑ cαxαtr·α−µ(f). Notice that the w′-weight of a monomial cαxαtr·α−µ(f) in f̃ is w · α− 1 · (r · α− µ(f)) = m∑ i=1 ri · α− m∑ i=1 ri · α+ m∑ i=1 min cβ 6=0 (ri · β) = m∑ i=1 min cβ 6=0 (ri · β). This implies the first claim. The second claim is immediate as J̃ = 〈f̃ : f ∈ J〉. � Remark 3.6. In Lemma 3.5 more generally we can choose v = c1r1 + · · · + cmrm ∈ C◦ with ci ∈ R≥0 and v′ = (−c1, . . . ,−cm, v1, . . . , vn). Then J̃ is v′-homogeneous. Lemma 3.7. Let h ∈ J̃ be w′-homogeneous with w′ as in Lemma 3.5. Then h = tuf̃ for some f ∈ J and u ∈ Zm≥0. Proof. Let h = ∑s i=1 pif̃i with fi ∈ J and pi ∈ K[t1, . . . , tm][x1, . . . , xn] w′-homogeneous. Then h\|t=1 = ∑s i=1 pi\|t=1fi ∈ J . Furthermore, since h is w′-homogeneous, h = tuh̃\|t=1 for some u ∈ Zm≥0. Hence, we can take h\|t=1 as f . � Lemma 3.8. Consider f = ∑ α∈Zn≥0 cαxα ∈ J with a unique monomial cβx β ∈ in<(J). That is, for every α 6= β with cα 6= 0, there exists no g′ ∈ G such that xα is divisible by in<(g′). Then f̃ = ∑ α∈Zn≥0 cαxαtr·(α−β) = cβx β + ∑ α 6=β cαxαtr·(α−β), (3.3) contains a unique monomial with coefficient in K. In particular, this is the case for elements of the reduced Gröbner basis G. Proof. The assumption that only the monomial cβx β of f is contained in in<(J) ensures that cβx β is also a monomial appearing in inri(f) for all i. To see this, assume there is a ray ri such that cβx β is not a monomial in inri(f) and let w′ = ∑ j 6=i rj . By [38, Lemma 2.4.5] we can find an ε so that inεw′+ri(f) = inw′(inri(f)). As we assumed cβx β is not a monomial in inri(f), it is also not a monomial in inεw′+ri(f). But εw′+ri ∈ C◦ and so inεw′+ri(f) ∈ inεw′+ri(J) = in<(J). This however is a contradiction as we assumed that cβx β is the only monomial in f that is con- tained in in<(J). Therefore, ri · β = mincα 6=0{ri · α} for all i and r · β = µr(f). Assume on the contrary that there exists another monomial cγx γ in f with r · γ = r · β. Then cγx γ is also a monomial in inri(f) for all i. Moreover, for w = r1 + · · ·+rm ∈ C◦ the initial form inw(f) con- tains both monomials cβx β and cγx γ . But as inw(J) = in<(J) this implies that cγx γ ∈ in<(J), Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras 11 a contradiction. Hence, cβx β is the unique monomial in f with µr(f) = r ·β and we obtain (3.3) as r · (α− β) 6= 0 for all α 6= β. This completes the proof of the first claim. To prove the second part, assume that g ∈ G has a monomial term cγx γ ∈ in<(J) with cγx γ 6= in<(g). Then by the definition of the reduced Gröbner basis, cγx 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[x1, . . . , xn] to a monomial term order � on the polynomial ring K[t1, . . . , tm, x1, . . . , xn] in such a way that ti � xj for all 1 ≤ i ≤ m and 1 ≤ j ≤ n, and xαtλ � xβtµ if and only if (i) xα < xβ or (ii) xα = xβ and tλ <lex tµ. (3.4) Proposition 3.9. The set G̃ = {g̃ : g ∈ G} is a Gröbner basis for J̃ with respect to �. In par- ticular, G̃ is a generating set for J̃ . Proof. Since J̃ is w′-homogeneous by Lemma 3.5, it is enough to show that for every w′- homogeneous polynomial h ∈ J̃ there exists some g̃i ∈ G̃ whose initial term divides in�(h). Let h ∈ J̃ be w′-homogeneous. By Lemma 3.7 there exist f ∈ J and u ∈ Zm≥0 such that h = tuf̃ . We compute in�(h) = tu in�(f̃) Def. � = tu+v in<(f) ∃ g∈G = tu+vcαxα in<(g) Lemma 3.8 = tu+vcαxα in�(g̃), where g is a suitable element of the Gröbner basis G for J and v ∈ Zm≥0. As g̃ ∈ G̃, this shows that G̃ is a Gröbner basis for J̃ . Hence, by [29, Theorem 2.1.8] G̃ is a generating set for J̃ . � 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 r1, . . . , rm. Corollary 3.10. For r and r′ as in Proposition 3.3 we have that Ãr = Ãr′. The algebras Ã\|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. Proposition 3.11. Consider a = (a1, . . . , am) and b = (b1, . . . , bm) in Km with the property that ai = 0 if and only if bi = 0. Then the algebras Ã\|t=a and Ã\|t=b are isomorphic. Proof. Let ϕ be the K-algebra automorphism of K[t1, . . . , tm] defined by ϕ(ti) = ai bi ti if bi 6= 0 and ϕ(ti) = ti if bi = 0. Then ϕ(t− a) = (t− b), 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 = qjrj with qj ∈ Q for all j. Then we still have an isomorphism between Ãr\|t=a and Ãr′ \|t=a for all a ∈ Km. For the proof it is neces- sary to extend the polynomial ring K[t1, . . . , tm] to a ring K [ tQ1 , . . . , t Q m ] , where the ti’s are allowed to have rational exponents. The automorphism of K [ tQ1 , . . . , t Q m ] defined by h(ti) = tqii extends to an automorphism of K [ tQ1 , . . . , t Q m ] [x1, . . . , xn] with the property h ( f̃r ) = f̃r′ for all d-homogeneous polynomials f ∈ J . The rest follows from Proposition 3.11. Before presenting our main result we explain how the ideal J̃r is related to the ideal Ĵw in (3.1). 12 L. Bossinger, F. Mohammadi and A. Nájera Chávez Proposition 3.13. Consider a face τ of C spanned by a subset {ri1 , . . . , ris} of {r1, . . . , rm} and the lineality space L(J). We define tτ ∈ K[t]m by (tτ )k := { t if k ∈ {i1, . . . , is}, 1 otherwise, for k = 1, . . . ,m. Then for wτ = ri1 + · · ·+ ris we have J̃ \|t=tτ = Ĵwτ . Proof. Let < be the monomial term order compatible with C. Consider an element g =∑ α∈Zn≥0 cαxα in G with in<(g) = cβx β. Since the vectors wτ , w := wτ − ris and ris are all in C, by Remark 2.9: in<(inwτ (g)) = in<(inw(g)) = in<(inris (g)) = in<(g) = cβx β. This implies that the initial forms of inwτ (g), inw(g) and inris (g) contain cβx β. In other words, min cα 6=0 {wτ · α} = wτ · β = (w + ris) · β, min cα 6=0 {w · α} = w · β, min cα 6=0 {ris · α} = ris · β. Therefore, (w + ris) · β = min cα 6=0 {(w + ris) · α} = min cα 6=0 {w · α}+ min cα 6=0 {ris · α}. Using the same argument multiple times we obtain that min cα 6=0 {( s∑ j=1 rij ) · α } = s∑ j=1 min cα 6=0 {rij · α}. Now the claim follows by comparing the generating sets of Ĵwτ and J̃ \|t=tτ . � We are now prepared to present our main theorem: Theorem 3.14. Let J be a d-homogeneous ideal in Kd[x1, . . . , xn], A = Kd[x1, . . . , xn]/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 ⊂ Rn/L(J). Then: (i) The algebra Ãr is a free K[t1, . . . , tm]-module with basis B<, the standard monomial basis of A with respect to in<(J). In particular, we have a flat family Proj(Ã) P(d)× Am Am. π (ii) For every face τ of C there exists aτ ∈ Am such that π−1(aτ ) = Proj(Kd[x1, . . . , xn]/ inτ (J)). 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[t1, . . . , tm, x1, . . . , xn] as in (3.4). Let B� be the standard monomial basis for Ãr induced by in� ( J̃r ) . Then B� is a basis for Ãr as a K-vector space and a generating set for Ãr as a K[t1, . . . , tm]-module. By Proposition 3.9 and [29, Proposition 1.1.5] we have B� = { t̄βx̄α : in�(g̃) 6 \| tβxα for all g ∈ G } , B< = { x̄α : in<(g) 6 \| xα for all g ∈ G } . Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras 13 Observe that B� can be reduced to a K[t1, . . . , tm]-basis of Ãr by defining B′� := B� \ { t̄βx̄α : ∃ t̄γx̄α ∈ B� with γ < β } , where γ < β means that γi ≤ βi for all i and γj < βj for some j. Note that B< ⊆ B′�. Now assume there is a monomial t̄βx̄α ∈ B′� \ B<. By Proposition 3.9 we have that in� ( J̃ ) \|t=1 = in<(J). Hence, B′�\|t=1 = B�\|t=1 = B<. In particular, this implies that x̄α ∈ B<. But as B< ⊂ B� this is a contradiction to the definition of B′�. For the second claim, let K[t1, . . . , tm]d[x1, . . . , xn] denote the polynomial ring in x1, . . . , xn with coefficients in K[t1, . . . , tm] and grading induced by d. Then by Lemma 3.8 and Proposi- tion 3.9 J̃ ⊆ K[t1, . . . , tm]d[x1, . . . , xn] is d-homogeneous. This yields the embedding Proj ( Ã ) ↪→ P(d) × Am. The projection P(d) × Am � Am induces the flat morphism π : Proj(Ã) → Am as Ã has a K[t1, . . . , tm]-basis by the first claim. (ii) Every face τ of C induces a face τ of C with primitive ray generators ri1 , . . . , ris for some i1, . . . , is ∈ [m]. We define (aτ )k := { 0 if k ∈ {i1, . . . , is}, 1 otherwise, for k = 1, . . . ,m. Then by Proposition 3.13 we have J̃ \|t=aτ = Ĵwτ \|t=0 which is equal to inτ (J) by Remark 3.1. � 3.1 Torus equivariant families We explain how the above results are related to Kaveh–Manon’s recent work [33] on the classi- fication of torus equivariant families. Consider the lattice N = Zn, its dual lattice M = N∗ and a fan Σ ⊂ N ⊗Z R. We write XΣ for the toric variety associated to Σ. Furthermore, we define ON to be the semifield of piecewise linear functions on N and OΣ the semifield of piecewise linear functions on \|Σ\| ∩ N . For a, b ∈ OΣ we have a⊗ b := a+ b and a⊕ b := min{a, b}, where the minimum is taken pointwise. In particular, OΣ is partially ordered: a ≥ b if a⊕ b = b. A PL-quasivaluation on an algebra A is a map v : A→ OΣ that satisfies: (i) v(fg) ≥ v(f) + v(g) and (ii) v(f + g) ≥ min{v(f), v(g)}, where the minimum is taken pointwise in OΣ. If (i) is an equality, v is called a PL-valuation. If A is a graded algebra A = ⊕ n∈ZAn then v is called homogeneous if it is compatible with the grading, i.e., for f = ∑ n∈Z cngn with gn ∈ An and coefficients cn we have v(f) = v(gk) for the smallest k with ck 6= 0. Given a sheaf of algebras A on XΣ its relative spectrum denoted by Spec(A) is the scheme obtained from gluing affine schemes Spec(A(Ui)), where ⋃ i Ui is an open cover of XΣ and A(Ui) is the corresponding section of A. Theorem 3.15 ([33, Theorem 1.1]). Let A = ⊕ n≥0An be a positively graded algebra and a domain over an algebraically closed field of characteristic zero and Σ a fan in N ⊗Z R. Then the following are equivalent information: (i) a TN -equivariant flat sheaf A of positively graded algebras of finite type over XΣ such that Spec(A) has reduced and irreducible fibers and its generic fibers are isomorphic to Spec(A); (ii) a homogeneous PL-valuation v : A→ OΣ with finite Khovanskii basis. (Given v : A→ OΣ for every ray ρ ∈ Σ one constructs a quasivaluation vρ : A → Z ∪ {∞} with associated graded algebra grρ(A), see [33, Section 4.1]. A Khovanskii basis for v is a subset B ⊂ A such that for every ray ρ ∈ Σ the image of B in grρ(A) is a set of algebra generators.) 14 L. Bossinger, F. Mohammadi and A. Nájera Chávez Consider a presentation of A, i.e., a surjection pr : K[x1, . . . , xn]→ A whose kernel is a weig- hted homogeneous prime ideal J and so it induces an isomorphism of graded algebras A ∼= K[x1, . . . , xn]/J . Take a cone σ in GF(J). Then σ defines PL-quasivaluation vσ : A → Oσ 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 inσ(J) is prime, then vσ is a PL-valuation. Then for x̄α ∈ B< we have vσ(x̄α) := − · α : σ → Z, where − · − is the pairing between the lattices N and M , as above. For an arbitrary element f ∈ A write f = ∑ x̄α∈B< cαx̄α. Then vσ(f) := mincα 6=0{vσ(x̄α)}. Definition 3.16 ([33]). The PL-quasivaluation vσ : A → Oσ defines a filtration on A with filtered pieces Fm(vσ) := {f ∈ A : vσ(f) ≥ − · m} for m ∈ M . The Rees algebra of the PL- valuation vσ is Rσ(A) := ⊕ m∈M Fm(vσ)tm, where we think of tm as a character of the torus TN = N ⊗Z C∗. More generally, a PL-quasivaluation can be obtained from a collection of equidimensional cones σ1, . . . , σk 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 σ1, . . . , σk and vΣ : A → ON be the corresponding PL-valuation. By Kaveh–Manon’s classification of toric families this yields a flat family ψ : Spec(RΣ(A))→ XΣ. (3.5) Here RΣ(A) is the TN -equivariant flat sheaf of Rees algebras on XΣ. The scheme Spec(RΣ(A)) is glued from Spec(Rσ) for σ ∈ Σ. For p ∈ TN ⊂ XΣ we have ψ−1(p) ∼= Spec(A). Moreover, ψΣ has a special fiber over every torus fixed point of X. More precisely, let pσi be the torus fixed point corresponding to the (maximal) cone σi ∈ Σ. Then ψ−1(pσi) ∼= Spec ( K[x1, . . . , xn]/ inσi(J) ) . (3.6) 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 <. Definition 3.17 ([33]). The PL-quasivaluation wC : K[x1, . . . , xn]→ OC associated to C is defi- ned by wC(xα) = −·α : C → Z. For a polynomial f = ∑ cαxα we have wC(f) := mincα 6=0{−·α}. The associated Rees algebra isRC(K[x1, . . . , xn]) = ⊕ m∈M Fm(wC)tm, where Fm(wC) is defined analogous to Fm(vσ) in Definition 3.16. The PL-quasivaluation vC : A → OC is obtained as a pushforward of the PL-quasivaluation wC : K[x1, . . . , xn] → OC along the morphism pr: K[x1, . . . , xn] → A. An adapted basis for vC is a vector space basis B for A such that B ∩ Fm(vσ) is a vector space basis of Fm(vσ) for every m ∈ M . By [33] the standard monomial basis B< is an adapted basis for vC . Similarly, the monomial basis of K[x1, . . . , xn] is adapted to wC . Hence, RC(A) is a free K[SC ]-algebra with basis B< and RC(K[x1, . . . , xn]) is a free K[SC ]-algebra whose basis is the monomial basis, where SC := −C∨ ∩ Zn due to our min-convention. In particular: RC ( K[x1, . . . , xn] ) ∼= K[SC ][x1, . . . , xn]. (3.7) Manon explained the proof of the following proposition to us. It follows from results in [33]. Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras 15 Proposition 3.18. For any maximal cone C ∈ GF(J), the Rees algebra RC(A) has an explicit presentation of form K[SC ][x1, . . . , xn]/Ĵ . More precisely, the ideal Ĵ is generated by homoge- nizations of elements in the reduced Gröbner basis G: let g = xγ + ∑ cαxα ∈ G and xγ = inC(g), then the homogenization of g is ĝ = xγ + ∑ cαxαtβ with γ − α = β ∈ SC . (3.8) Proof. Consider f ∈ K[x1, . . . , xn]. It can be written as f = ∑ xα∈B< cαxα + ∑ xβ 6∈B< cβx β. In particular, if wC(f) ≥ −·m in OC , then vC(pr(f)) ≥ −·m. So the morphism pr: K[x1, . . . , xn] → A induces a map prm : Fm(wC)→ Fm(vC) for all m. Further, every element in B< ∩Fm(vC) is the image of a monomial in Fm(wC). Hence, prm is surjective and we obtain: p̂r := ⊕ m∈(Zn)∗ prm : RC(K[x1, . . . , xn])→ RC(A). By (3.7) this implies the first claim. Now let Ĵ be the kernel of p̂r. Then Ĵ = ⊕ m∈M Ĵm, where Ĵm = ker(prm) ⊂ Fm(wC). Now consider an element g ∈ G of form xγ + ∑ cαxα with xγ = inC(g). We want to show that the element ĝ defined in (3.8) lies in Ĵ 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 cα 6= 0 in g. Hence, w · (γ − α) ≤ 0 and so β = γ − α ∈ −C∨. In particular, this implies that ĝ ∈ Ĵγ ⊂ Fγ(wσ). Lastly, we need to show that Ĵ is generated by {ĝ : g ∈ G}. Consider h ∈ Ĵ . As Ĵ =⊕ m∈M Ĵm, we may assume that h ∈ Ĵm. Further, as Ĵm ⊂ Fm(wC) ⊂ K[x1, . . . , xn] we can think of h as an element of K[x1, . . . , xn] as well as an element of RC(K[x1, . . . , xn]). Then in<(h) = xµ in<(g) for some g ∈ G of form g = xγ + ∑ cαxα with xγ = in<(g). In particular, similar to the above the exponent of in<(xµg) = xµ+γ has the property that w·(µ+γ) ≤ w·(µ+α) for all w ∈ C and all α with cα 6= 0 in g. So, h − xµg ∈ Fm(wC). In particular, the division algorithm with respect to G takes place inside Fm(wC) 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 r1, . . . , rm are representatives of the primitive ray generators of C ⊂ Rn/L(J). The rays of Σ, denoted by Σ(1), form a subset of {r1, . . . , rm}. Recall the definition of the lifted algebra Ãr from (3.2). We can apply the Cox construction in two ways, to XC and to XΣ: First, let XC be the affine toric variety associated to C. Recall that a quasitorus is a product of a torus with a finite abelian group. Then XC is isomorphic to the almost geometric quotient Am//G (see, e.g., [15, Theorem 5.1.11]), where the group G by [15, Lemma 5.1.1(b)] is the quasitorus G = {(t1, . . . , tm) ∈ (C∗)m : tr1·ei1 · · · trm·eim = 1∀ i ∈ [n]}. (3.9) One can easily check that if p, q ∈ Am lie in the same G-orbit, then π−1(p) ∼= π−1(q) by Propo- sition 3.11. Hence, the flat family π : Spec(Ãr)→ Am induces the commutative diagram Spec(Ãr) XC Am. π pC◦π pC (3.10) If p, q ∈ XC lie in the same torus orbit, then (pC ◦ π)−1(p) ∼= (pC ◦ π)−1(q) by Proposition 3.11. Note that pC : Am → XC is indeed a morphism. Namely, it is the universal torsor for XC . 16 L. Bossinger, F. Mohammadi and A. Nájera Chávez Similarly, we can apply the Cox construction to the toric variety XΣ. For simplicity we assume that Σ(1) = {r1, . . . , rm}. In this case the construction has two steps: first we remove the locus of Am that does not correspond to torus orbits in XΣ, denoted by Z(Σ). Recall that C ⊂ {r1, . . . , rm} is a primitive collection for Σ, if (i) C 6⊂ 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] Z(Σ) = ⋃ C primitive collection V (ti : ri ∈ C) ⊂ Am. (3.11) Now XΣ is isomorphic to the almost geometric quotient (Am−Z(Σ))//G, where G is as in (3.9) (see, e.g., [15, Theorem 5.1.11]). In contrast to the morphism pC : Am → XC from (3.10), we obtain a rational map pΣ : Am 99K XΣ as pΣ is only defined on TΣ := Am−Z(Σ). Moreover, pΣ : TΣ → XΣ is the universal torsor of XΣ. Similarly to the first case, this gives the diagram: π−1(TΣ) Spec(Ãr) XΣ TΣ Am π\|π−1(TΣ) pΣ◦π π pΣ (3.12) which is commutative. The fibers of the family defined by pΣ ◦π\|π−1(TΣ) coincide with the fibers of ψ defined in (3.5): (pΣ ◦ π)−1(x) ∼= ψ−1(x) for every x ∈ XΣ, see (3.6). 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. Theorem 3.19. Let J ⊆ Kd[x1, . . . , xn] 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 ( Ãr ) → Am fits into a pull-back diagram Spec(RC) Spec ( Ãr ) XC Am. ψ π pC Here pC : Am → XC is the universal torsor of XC obtained from the Cox construction as in (3.10). Proof. We prove equivalently that the corresponding diagram between the algebras is a push- out. Hence, it suffices to show that RC ⊗K[SC ] K[t1, . . . , tm] ∼= Ã. Note that the map pC : Am → XC corresponds to p# C : K[SC ]↪→K[t1, . . . , tm]. In particular, for a character t−r ∗ i ∈ K[SC ] with r∗i ∈ C∨ ⊂ M the dual of ri ∈ C ⊂ N we have p# C (t−r ∗ i ) = ti ∈ K[t1, . . . , tm]. Then under the extension of scalars (i.e., applying the functor − ⊗K[SC ] K[t1, . . . , tm]) the homogenization ĝ of an element g ∈ G is sent to the lift g̃. This can be seen as follows: by Lemma 3.8 we have g̃ = xγ + ∑ cαxαtr·(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 {`1, . . . , `k} be a basis for L(J) ∩N . Then {r1, . . . , rm, `1, . . . , `k} spans N and {r∗1, . . . , r∗m} spans (L(J) ∩N)⊥ ⊂M . So every α− γ has an expression of form α− γ = ∑ cir ∗ i . We compute t−(α−γ) = t−c1r ∗ 1 · · · t−cmr∗m p# C7−→ tc11 · · · t cm m = tr·(α−γ). (3.13) Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras 17 Note that γ has the property that ri · γ = mincα 6=0{ri · α}. So, ri · (α − γ) ≥ 0 and therefore α− γ ∈ C∨. Hence, we have RC ⊗K[SC ] K[t1, . . . , tm] Proposition 3.18∼= ( K[SC ][x1, . . . , xn]/Ĵ ) ⊗K[SC ] K[t1, . . . , tm] (3.13)∼= K[t1, . . . , tm][x1, . . . , xn]/J̃ = Ã. � Corollary 3.20. Let J ⊆ K[x1, . . . , xn] 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 denote Ãr by Ã. Let Σ be the intersection of C with Trop(J). Then the restriction of morphism π : Spec ( Ã ) → Am to π−1(TΣ) = Spec ( Ã ) − π−1(Z(Σ)) fits into a pull-back diagram Spec(RΣ) π−1(TΣ) XΣ TΣ. ψ π\|π−1(TΣ) pΣ Here pΣ : TΣ → XΣ is the universal torsor of XΣ obtained from the Cox construction as in (3.12). Proof. The scheme XΣ and the sheaf RΣ are defined locally for affine pieces Uσ = Spec(K[Sσ]) ⊂ XΣ for σ ∈ Σ. Algebraically, for every σ ∈ Σ the pull-back of ψ and pΣ corresponds to the following push-out diagram: Rσ Rσ ⊗K[Sσ ] K[t1, . . . , tm]〈ti : ri 6∈σ〉 K[Sσ] K[t1, . . . , tm]〈ti : ri 6∈σ〉. Observe that by (3.11) the localization K[t1, . . . , tm]〈ti : ri 6∈σ〉 corresponds to an affine piece in Am that does not intersect Z(Σ). So it is in fact an affine piece of TΣ. We have to show that Rσ ⊗K[Sσ ] K[t1, . . . , tm]〈ti : ri 6∈σ〉 ∼= Ã〈ti : ri 6∈σ〉. As σ is a face of the maximal cone C by [33, Proposition 3.13] we have Rσ ∼= RC ⊗K[SC ] K[Sσ]. Using Theorem 3.19 we compute Rσ ⊗K[Sσ ] K[t1, . . . , tm]〈ti : ri 6∈σ〉 ∼= ( RC ⊗K[SC ] K[Sσ] ) ⊗K[Sσ ] K[t1, . . . , tm]〈ti : ri 6∈σ〉 ∼= RC ⊗K[SC ] K[t1, . . . , tm]〈ti : ri 6∈σ〉 ∼= Ã〈ti : ri 6∈σ〉. � 4 Grassmannians and cluster algebras We now apply the machinery developed in Section 3 to the Grassmannians Gr(2,Cn) and Gr ( 3,C6 ) . 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,Cn) in Section 4.1 and then in Sec- tion 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,C6 ) . We first fix our notation for Gr(2,Cn). Denote by S the polynomial ring in Plücker variables C[pij : 1 ≤ i < j ≤ n]. Here, Plücker variables are associated with arcs in an n-gon whose 18 L. Bossinger, F. Mohammadi and A. Nájera Chávez 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 pij := −pji. In S we define for every i < j < k < l the Plücker relation: Rijkl := pijpkl − pikpjl + pilpjk ∈ S. The ideal I2,n ⊂ S generated by all Plücker relations Rijkl is called the Plücker ideal. The quo- tient A2,n := S/I2,n is the Plücker algebra which is the homogeneous coordinate ring of the Grassmannian Gr(2,Cn) with respect to its Plücker embedding into P(n2)−1. The cosets of Plü- cker variables in the quotient are denoted by p̄ij ∈ A2,n and are called Plücker coordinates. Denote by d := 2(n− 2) the dimension of Gr(2,Cn), so d+ 1 is the Krull-dimension of A2,n. In S, we fix the notation pm with m = (mij)ij ∈ Z(n2) ≥0 denoting the monomial ∏ 1≤i<j≤n p mij ij . The maximal cones in the Gröbner fan GF(I2,n) are those of full dimension in the ambient space R(n2) with associated monomial initial ideals. The lineality space L2,n := L(I2,n) ⊂ GF(I2,n) is n-dimensional and spanned by the elements `s for s ∈ [n] which are defined by (`s)ij := { 1 if s = i or s = j, 0 otherwise. Note that 1 := (1, . . . , 1) ∈ R(n2) lies in L2,n. The Gröbner fan of the Plücker ideal has a subfan that is particularly interesting when studying toric degenerations of Gr(2,Cn). Definition 4.1. The tropical Grassmannian is a simplicial (d+1)-dimensional subfan of GF(I2,n) defined as Trop(I2,n). Maximal cones in Trop(I2,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(I2,n) by wΥ(pij) := −#{edges on the unique path i→ j in Υ}. (4.1) To simplify notation we denote inΥ(I) := inwΥ(I). Theorem 4.2 ([49]). For every trivalent tree Υ with n leaves the ideal inΥ(I) is toric, i.e., bino- mial and prime. Moreover, it is generated by inΥ(Rijkl) for all 1 ≤ i < j < k < l ≤ n. Definition 4.3. The totally positive tropical Grassmannian is Trop+(I2,n). In [50, Section 5], it was shown that maximal cones in Trop+(I2,n) are in bijection with planar trivalent trees, i.e., trivalent trees whose vertices are labeled 1, . . . , n in a clockwise manner. 4.1 Preliminaries on cluster algebras The Plücker algebra A2,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(u1, . . . , um+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 ( x̃, B̃ ) , Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras 19 where x̃ = {x1, . . . , xm+f} is a free generating set of F and B̃ = (bij) is an (m+f)×m rectangular matrix with the following property: the top square submatrix of B̃ (that is the m×m submatrix of B̃ formed by its first m rows) is skew-symmetric. We call B̃ an extended exchange matrix and refer to its top square submatrix B as an exchange matrix. We call x = {x1, . . . , xm} a cluster, {xm+1, . . . , xm+f} the set of frozen variables and x̃ an extended cluster. Given k ∈ {1, . . . ,m}, the mutation in direction k of a seed ( x̃, B̃ ) is the new seed µk ( x̃, B̃ ) =( x̃′, B̃′ ) , defined as follows: � The extended cluster x̃′ = x̃ \ {xk} ∪ {x′k}, where xkx ′ k = ∏ i : bik>0 xbiki + ∏ i : bik<0 x−biki . (4.2) We call an expression of the form (4.2) an exchange relation and the monomial xkx ′ k an exchange monomial. � The extended exchange matrix B̃′ = (b′ij) is defined by the following formula b′ij := { −bij if i = k or j = k, bij + sgn(bik) max(bikbkj , 0) else, where sgn(bik) :=  −1 if bik < 0, 0 if bik = 0, 1 if bik > 0. Definition 4.4. The cluster algebra A ( x̃, B̃ ) associated to a seed ( x̃, B̃ ) is the C-subalgebra of F generated by the elements of all the extended clusters that can be obtained from the initial seed ( x̃, B̃ ) by a finite sequence of mutations. The elements of the clusters constructed in this way are called the cluster variables. If f 6= 0 we say that the cluster algebra A ( x̃, B̃ ) has frozen variables, namely, xm+1, . . . , xm+f . Remark 4.5. It can be easily verified that, up to a canonical isomorphism, A ( x̃, B̃ ) is inde- pendent of x̃. Therefore, we write A ( B̃ ) instead of A ( x̃, B̃ ) . 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 con- struction. Theorem 4.7 ([25, Laurent phenomenon]). The cluster algebra A ( B̃ ) is a C[xm+1, . . . , xm+f ]- subalgebra of C [ x±1 1 , . . . , x±1 m , xm+1, . . . , xm+f ] . Definition 4.8. The cluster algebra with principal coefficients associated to an m ×m skew- symmetric matrix B is the cluster algebra A(Bprin), where Bprin is the 2m ×m matrix whose top m × m square submatrix is B and whose lower m × m square submatrix is the identity matrix 1m. The notion of cluster algebras with principal coefficients is central in cluster theory. Its rele- vance in the framework of toric degenerations of cluster varieties was highlighted in [28]. In par- ticular, in loc. cit. the authors work implicitly with cluster algebras with frozen variables and principal coefficients. This is precisely the notion we need here. 20 L. Bossinger, F. Mohammadi and A. Nájera Chávez Definition 4.9. Let B̃ be an extended exchange matrix. Let B̂ be the (m + f) × (m + f) skew-symmetric square matrix whose left (m + f) × m submatrix is B̃ and whose bottom right f × f square submatrix is the zero matrix (this information fully determines B̂ since it is skew-symmetric). Denote by t1, . . . , tm+f the variables of A ( B̂prin ) associated to the last m+f directions, which in this case are the frozen directions. Then the cluster algebra with prin- cipal coefficients and frozen variables A ( B̃prin ) associated to B̃ is the C[t1, . . . , tm+f ]-subalgebra of A ( B̂prin ) spanned by the cluster variables that can be obtained from the initial extended cluster by iterated mutation in the first m mutable directions. Theorem 4.10 ([16, Theorem 1.7]). Let X be a cluster variable associated to A ( B̃prin ) . By Theorem 4.7 we know that X ∈ C[t1, . . . , tm+f ] [ x±1 1 , . . . , x±1 m , xm+1, . . . , xm+f ] . Let X\|t=0 be the Laurent polynomial in C [ x±1 1 , . . . , x±1 m , xm+1, . . . , xm+f ] obtained after the evaluation of X at t1 = 0, . . . , tm+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) = ( gX1 , . . . , g X m+f ) ∈ Zm+f we have that X\|t=0 = m+f∏ i=1 x gXi i . Moreover, if X and X ′ are different cluster variables then g(X) 6= g(X ′). Definition 4.11. Let X be a cluster variable associated to A ( B̃prin ) . The vector g(X) ∈ Zm+f introduced in Theorem 4.10 is the g-vector associated to X. We refer to the vector( gX1 , . . . , g X m ) ∈ Zm as the truncated g-vector of X. The set of (truncated) g-vectors associated to B̃ is the set of (truncated) g-vectors of all the cluster variables in A ( B̃prin ) . Remark 4.12. Let X\|t=1 be the Laurent polynomial in C [ x±1 1 , . . . , x±1 m , xm+1, . . . , xm+f ] obtai- ned after the evaluation of X at t1 = 1, . . . , tm+f = 1. Then X\|t=1 is a cluster variable of A ( B̃ ) . 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 intro- duced 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. Definition 4.13. An ice quiver is a pair (Q,F ), where Q is a finite quiver with the vertex set V (Q) and F ⊂ V (Q) is a set of frozen vertices. The vertices in V (Q)\F are called mutable. If (Q,F ) is an ice quiver then the corresponding \|V (Q)\| × \|V (Q) \ F \| matrix is B(Q,F ) := (bij), where bij := #{arrows i→ j in Q} −#{arrows j → i in Q} (4.3) for i ∈ V (Q) and j ∈ V (Q) \ F . Note that (4.3) can be similarly used to recover (Q,F ) from B(Q,F ). The quiver mutation is the operation induced by matrix mutation at the level of quivers. If F = ∅ we set BQ := B(Q,∅). Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras 21 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 ( Buniv Q ) , where Buniv Q = ( BQ UQ ) and UQ is the rectangular matrix whose rows are the g-vectors of the opposite quiver Qop. Observe that we have described the rows of UQ but did not specify in which order they appear. Any order we choose provides a realization of A ( Buniv Q ) since a rearrangement of the rows of UQ 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 ( Buniv Q ) is canonically isomorphic to A ( Buniv Q′ ) . The canonical isomorphism is completely determined by sending the elements of the initial cluster of Buniv Q to those of Buniv Q′ . Remark 4.15. Cluster algebras with universal coefficients as treated by Fomin and Zelevinsky are defined by a universal property [25, Definition 12.1]. We have chosen Reading’s equivalent definition since we will not use the universal property here. If Q is a bipartite orientation of a Dynkin diagram ∆ then the matrix Buniv Q has a particularly nice and explicit description. More precisely, let Φ∨ be the root system dual to the root system associated to ∆. Fix Φ∨+ ⊂ Φ∨ a subset of positive coroots with {α1, . . . , αn} ⊂ Φ∨+ the set of sim- ple coroots. The almost positive coroots are the coroots in the set Φ∨≥−1 := Φ∨+∪{−α1, . . . ,−αn}. The cardinality of Φ∨≥−1 coincides with the number of cluster variables in A ( Buniv Q ) which is the number of coefficients for A ( Buniv Q ) . Hence, we may consider { yα : α ∈ Φ∨≥−1 } as the set of coeffi- cients for A ( Buniv Q ) . Next, for a coroot β = ∑n i=1 ciαi we define [β : αi] := ci. With this notation we have the following description of Buniv Q which is a reformulation of [25, Theorem 12.4]. Proposition 4.16. The entry in Buniv Q in the column i and the row corresponding to yβ is [β : αi] if i is a source and −[β : αi] if i is a sink. Definition 4.17. Let (Q,F ) be an ice quiver whose mutable part is a quiver Qmut of finite cluster type. The associated cluster algebra with universal coefficients is A ( Buniv (Q,F ) ) , where Buniv (Q,F ) = ( B(Q,F ) UQmut ) and UQmut is the matrix whose rows are the truncated g-vectors of the opposite quiver (Qmut)op. Remark 4.18. The algebra A ( Buniv (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 A2,n. We now turn to the cluster algebra that is of most interest to us, and use the combinatorics governing the cluster structure of A2,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 coordinate p̄ij ∈ A2,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 xT := { p̄ij : ij is an arc in T } which is the associated cluster. The elements of xT are the cluster variables. The monomials in Plücker coordinates which are all part of one cluster are the cluster monomials. 22 L. Bossinger, F. Mohammadi and A. Nájera Chávez Note that every triangulation T contains the boundary edges 12, 23, . . . , n1. The correspon- ding cluster variables p̄12, p̄23, . . . , p̄1n are frozen cluster variables. All other variables p̄ij with i 6= 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 ( QT , FT ) : the vertices V (QT ) are in correspondence with arcs and boundary edges ij ∈ T ; Two vertices vij and vkl are connected by an arrow, if they correspond to arcs in one triangle. Inside every triangle, we orient the arrows clockwise. The set FT 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. 1 2 3 4 56 7 8 1 1′ 2 2′ 3 3′ 4 4′ 5 5′ 6 6′ 7 7′ 8 8′ 1 2 3 4 56 7 8 Figure 1. On the right a triangulation T of the 8-gon and its corresponding quiver QT . On the left its corresponding extended tree D̂T 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. Theorem 4.21 ([23]). For all n ≥ 3 the ring A2,n is a cluster algebra with frozen variables. More precisely, if ( QT , FT ) is the ice quiver associated to a triangulation T of the n-gon then A2,n ∼= A ( B(QT ,FT ) ) as C-algebras. Moreover, the cluster variables associated to B(QT ,FT ) are precisely the Plücker coordinates and the exchange relations are precisely the Plücker relations. Theorem 4.22 ([21]). The set of all cluster monomials is a C-vector space basis for the alge- bra A2,n called the basis of cluster monomials. We endow A2,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 ( QT , FT ) be the quiver associated to a triangulation of the n-gon. We de- note by Qmut T the full subquiver of QT supported in the mutable vertices V (QT ) \ FT . In par- ticular, Qmut 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 Auniv 2,n is the cluster algebra defined by the extended exchange matrix Buniv (QT ,FT ) = ( B(QT ,FT ) UQmut T ) . Up to canonical isomorphism, Auniv 2,n = A ( Buniv (QT ,FT ) ) is independent of T . The quivers Qmut T and ( Qmut T )op define canonically isomorphic cluster algebras whose cluster variables (denoted by p̄ij) are both in bijection with arcs ij of the n-gon for which 2 ≤ i+ 1 < j ≤ n. Hence, the rows of UQmut T are also in bijection with these arcs. We write yij for the coefficient variable of Auniv 2,n corresponding to the arc ij. Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras 23 1 2 34 5 Buniv (QT ,FT ) =  0 1 −1 0 1 0 0 −1 0 1 1 −1 −1 0 1 0 0 −1 0 1 1 −1 −1 0  = ( B(QT ,FT ) UQmut T ) Figure 2. A triangulation of the 5-gon and the associated matrix Buniv (QT ,FT ). Example 4.25. Let ( QT , FT ) 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 uni- versal coefficients coincide with the rows corresponding to the frozen part of ( QT , FT ) . However, for polygons with more than 5 sides this will not be the case. To get a better understanding of the cluster algebra Auniv 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 DT and its extended dual graph D̂T . 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 DT 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 D̂T is obtained from DT 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 D̂T the extended tree dual to T . Note that the tree DT is by definition planar. See Figure 1 for an example. Moreover, the extended tree D̂T can alternatively be defined as the dual graph of a triangulation T̂ of the 2n-gon with vertices 1′, 1, 2′, 2, . . . , n′, n in the clockwise order. Here T̂ is obtained from T by replacing every boundary edge i− 1, i by a triangle with a new vertex labeled by i′. Then D̂T = D T̂ . 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 D̂T . First, observe that the interior edges of D̂T 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 D̂T with the end points in {1, . . . , n} (not in {1′, . . . , n′}). Say the path goes from i to j. For simplicity, we fix an ori- entation of the path and denote it by i j (what follows is independent of the choice of orientation). As D̂T is trivalent, at every interior vertex the path can either turn right or left. Denote by eab the edge in D̂T corresponding to an arc ab ∈ T and let vabc and vabd be the vertices in D̂T adjacent to eab. Given the path i j we associate signs σabi j ∈ {−1, 0,+1} to every ab ∈ T as follows: 1. If the path i j either does not pass through eab, or it passes through eab by turning right at vabc and turning right at vabd, or it passes through eab by turning left at vabc and turning left at vabd: then σabi j = 0. 24 L. Bossinger, F. Mohammadi and A. Nájera Chávez 2. If the path i j passes through the eab by turning left at vabc and turning right at vabd: then σabi j = +1. 3. If the path i j passes through the eab by turning right at vabc and turning left at vabd: then σabi j = −1. Cases 2 and 3 are depicted in Figure 3. We leave it to the reader to verify that σabi j = σabj i. Later we replace the end points i and j by the interior vertices of D̂T and use the same symbol. − + Figure 3. Sign conventions to compute g-vectors. Definition 4.27. Let {fab}ab∈T be the standard basis of Z2n−3. For the Plücker coordinate p̄ij consider the path from i to j in D̂T . The g-vector of p̄ij with respect to T is defined as gT̂ij := ∑ ab∈T σabi jfab. Similarly, we define the g-vector of a cluster monomial as the sum of the g-vectors of its factors. Note that for ij ∈ T we have gij = fij . When there is no ambiguity, we write gij instead of gT̂ij . 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 gT̂ij of a Plücker coordinate pij with respect to T introduced in Definition 4.27 coincides with the g-vector g(p̄ij) (from Defini- tion 4.11) of the cluster variable p̄ij of A ( B(QT ,FT ) ) . Proof. We use some of the terminology of cluster algebras associated to surfaces developed in [22]. The algebra A2,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 of p̄ij 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 gT̂ij . 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 B̂(QT ,FT ) is the submatrix of BQ T̂ ,F T̂ associated to the internal arcs of the triangulation T̂ of the 2n-gon. As a consequ- ence g(p̄ij) = gT̂ij . � Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras 25 i vi l vl a1 b1 c1 v′ v j vj k vk a2 b2 c2 i′ vi′ d Figure 5. The full subtree of D̂T 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 vi, vj , vk, vl, v, v′ to the corresponding elements of Z2n−3. We follow the convention in Figure 3. Example 4.29. Consider the triangulation T depicted in Figure 1 and its corresponding tree D̂T . Then gij = fij for ij ∈ T . We compute the g-vectors of the remaining Plücker coordinates: g15 = f17 − f47 + f45, g24 = f12 − f13 + f34, g25 = f12 − f14 + f45, g46 = f47 − f57 + f56, g58 = f78 − f47 + f45, g16 = f17 − f57 + f56, g27 = f12 − f14 + f47, g28 = f12 − f17 + f78, g48 = f14 − f17 + f78, g68 = f56 − f57 + f78, g37 = f47 − f14, g35 = f45 − f14, g38 = f78 − f17, g26 = f12 − f14 + f47 − f57 + f56, g36 = −f14 + f47 − f57 + f56. 4.2 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 I2,n arising this way are toric. Throughout this section we fix a triangulation T . Definition 4.30. Consider a quadrilateral with vertices i, j, k, l inside the n-gon. The arcs ik and jl are called crossing arcs, if when drawn inside the n-gon the corresponding arcs cross each other. Let M2,n be the set of crossing monomials pikpjl, where in the quadrilateral with vertices i, j, k, l the arcs ik and jl are crossing. LetM2,n ⊂ S be the monomial ideal generated by M2,n. Lemma 4.31. Given a Plücker relation pijpkl−pikpjl+pilpjk 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 pikpjl. Proof. Consider the extended tree D̂T 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 a1, b1, c1, respectively a2, b2, c2. Denote the vertex of D̂T corresponding to the triangle a1, b1, c1 (respectively a2, b2, c2) 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 vi := ∑ pq∈T σpqi ∆1 fpq, vj := ∑ pq∈T σpqj ∆2 fpq, vl := ∑ pq∈T σpql ∆1 fpq, vk := ∑ pq∈T σpqk ∆2 fpq. 26 L. Bossinger, F. Mohammadi and A. Nájera Chávez So vi 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 fa1c1 , etc. Further, let v := ∑ pq∈T σ pq ∆1 ∆2 fpq. Then gij = vi + σa1c1 i j fa1c1 + σa1b1 i j fa1b1 + v + σa2c2 i j fa2c2 + σa2b2 i j fab + vj , gik = vi + σa1c1 i j fa1c1 + σa1b1 i j fa1b1 + v + ( 1 + σa2c2 i j ) fa2c2 + σb2c2i kfb2c2 + vk, gjl = vl + σb1c1j l fb1c1 − ( 1− σa1b1 i j ) fa1b1 + v + σa2c2 i j fa2c2 + σa2b2 i j fa2b2 + vj , gkl = vl + σb1c1j l fb1c1 − ( 1− σa1b1 i j ) fa1b1 + v + ( 1 + σa2c2 i j ) fa2c2 + σb2c2i kfb2c2 + vk, gil = vi + ( 1 + σa1c1 i j ) fa1c1 − ( 1− σb1c1j l ) fb1c1 + vl, gjk = vj − ( 1 + σa2b2 i j ) fa2b2 + ( 1 + σb2c2i k ) fb2c2 + vk. In particular, we have gij + gkl = gik + gjl 6= gil + gjk. � Recall from (4.1) how to obtain a weight vector from a trivalent tree with n leaves in Trop(I2,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, gij + gkl = gik + gjl 6= gil + gjk. Then inDT (pijpkl − pikpjl + pilpjk) = pijpkl − pikpjl. Definition 4.33. A quadrilateral weight map for a triangulation T is a linear map wT : R2n−3 → R, such that for every quadrilateral abcd with diagonal ac in T we have: wT (fad + fbc) > wT (fab + fcd). (4.4) Note that (4.4) is equivalent to saying that for every mutable vertex ac of the quiver QT we require wT (∑ pq→ac gpq ) > wT (∑ ac→pq gpq ) . 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(I2,n) from g-vectors. Proposition 4.34. Consider any i < j < k < l with gij + gkl = gik + gjl 6= gil + gjk and any quadrilateral linear map wT : R2n−3 → R. Then wT (gij + gkl) = wT (gik + gjl) < wT (gil + gjk). Proof. Without loss of generality assume that the full subtree of D̂T with leaves i, j, k, l is of form as in Figure 5. For s = 1, 2 let ∆s be the vertex of D̂T corresponding to the tri- angle as, bs, cs 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 a1 = d = a2 and b1 = c2 in Figure 5. Note that every quadrilateral linear map wT satisfies wT (fa2c1 + fb2c2) > wT (fa2b2 + fc2c1). Using Lemma 4.31 we compute wT (gil + gjk) = wT (gij + gkl + fa2c1 + fb2c2 − fa2b2 − fc2c1) > wT (gij + gkl). p ≥ 1: Assume that the result holds for p. When the number of edges along the path from va2b2c2 to va1b1c1 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 Rijkl 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 Rii′jl we have wT (gij + gi′l) < wT (gil + gi′j). Further, by assumption on wT we have wT (fa2d + fb2c2) > wT (fa2b2 + fc2d). One verifies by direct computation that wT (gil + gjk) = wT (gil + gi′j) + wT (gjk − gi′j) > wT (gij + gkl) + wT (gjk − gi′j + gi′l − gkl) = wT (gij + gkl) + wT (fa2d + fb2c2 − fa2b2 − fc2d) > wT (gij + gkl). � Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras 27 Definition 4.35. Fix a quadrilateral linear map wT : R2n−3 → R. We define wT ∈ R(n2) the weight vector associated to T and wT as (wT )ij := wT (gij). Proposition 4.36. For each quadrilateral weight map wT : R2n−3 → R, the initial ideal inwT (I2,n) is toric. Moreover, every binomial in the minimal generating set of inwT (I2,n) cor- responds 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 inDT (Rijkl) = inwT (Rijkl) for every Plücker relation Rijkl. 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 inDT (I2,n) is generated by inDT (Rijkl) for all 1 ≤ i < j < k < l ≤ n. Moreover, by [49, Corollary 4.4] the ideal inDT (I2,n) is binomial and prime. � Remark 4.37. As shown in Proposition 4.36, each triangulation of the n-gon gives rise to a Gröbner toric degeneration of Gr(2,Cn). For general Grassmannians, there are other examples of combinatorial objects leading to toric degenerations such as plabic graphs [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 xT , which is the output of Algorithm 1. We use it to define a linear map vT : R2n−3 → R and show in Lemma 4.39 that it is quadrilateral for T . Input: An initial cluster xT with its corresponding triangulation T and its quiver QT . Output: A partitioning of the initial seed indices. Initialization: i← 0; T ← the corresponding triangulation of xT ; Q← QT ; while T is not empty do Vi := {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 Vi then T ← T \ {t} (Remove the triangle t from T ); Q← QT ; end end i← i+ 1; end Vi ← xT \ (V1 ∪ · · · ∪ Vi−1); return (V0, . . . , Vi); Algorithm 1: Partitioning the cluster xT 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 xT . We repeatedly shrink T to obtain a quiver associated to smaller triangulations. More precisely, in step i we identify the frozen vertices of QT that are sinks, i.e., they have no outgoing arrows. Then we add the corresponding arcs to the set Vi and remove the corresponding triangles from T . Note that in this process, we might remove edges whose corresponding vertices are not in Vi, but they are part of a triangle with an edge in Vi. 28 L. Bossinger, F. Mohammadi and A. Nájera Chávez Let V0, . . . , Vi be the partition of the initial seed indices obtained by applying Algorithm 1 to the triangulation T associated to the initial seed xT . Refining the output of the algorithm yields an ordering on the variables of our initial seed, or equivalently the arcs of our initial triangulation. Example 4.38. We continue with our running example and compute a partition of the initial seed indices as V0 = {56, 12, 78}, V1 = {13, 45}, V2 = {17, 14, 47} and V3 = {57, 34, 23, 67, 18}. Lemma 4.39. Given a triangulation T consider the partition of xT = V0 ∪ · · · ∪ Vs obtained from Algorithm 1. Let vT : R2n−3 → R be the linear map defined by sending basis elements fij with ij ∈ T to q, where ij ∈ Vq. Then vT 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 ∈ Vp, lk ∈ Vq and jk, il ∈ Vs 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 vT (fij+fkl) = p+ q < s+ s = vT (fjk + fil). � 4.3 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(I2,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 A2,n. Definition 4.40. Let u ∈ Q(n2) be the weight vector such that the weight of pij is: u(pij) := − ( j − i− n 2 )2 for 1 ≤ i < j ≤ n. 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: u(pij) =  0, if min{\|j − i\|, \|i− j\|} = 4, −1, if min{\|j − i\|, \|i− j\|} = 3, −4, if min{\|j − i\|, \|i− j\|} = 2. Lemma 4.42. With u as in Definition 4.40 we have inu(I2,n) =M2,n from Definition 4.30. Proof. Consider the quadrilateral with vertices i < j < k < l and its corresponding Plücker relation Rijkl = pikpjl − pijpkl − pilpjk. The arcs ik and jl are crossing, hence we have to show that (a) inu(Rijkl) = pikpjl and (b) the crossing monomials generate inu(I2,n). To prove (a) we verify that u(pikpjl) = u(pilpjk) + 2(j − i)(l − k) > u(pilpjk), u(pikpjl) = u(pijpkl) + 2(k − j)(n+ i− l) > u(pijpkl). To establish (b), we apply Buchberger’s criterion and show that all the S-pairs S(Rijkl, Rabcd) reduce to 0 modulo {Rijkl}1≤i<j<k<l≤n. If inu(Rijkl) and inu(Rabcd) have no common factor, then S(Rijkl, Rabcd) reduces to zero (see, for example, [29, Lemma 2.3.1]). Assuming that inu(Rijkl) Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras 29 and inu(Rabcd) have a common factor we distinguish two cases, in each we underline the crossing monomials. Consider Rijls and Riklr, where without loss of generality i < j < k < l < r < s: S(Rijls, Riklr) = pkrplspij + pkrpjlpis − pjspikplr − pjspirpkl = −pijRklrs − pisRjklr + plrRijks + pklRijrs. The second case for Rijkl and Rjklm with i < j < k < l < m is similar. � Let C ∈ GF(I2,n) be the maximal cone corresponding to the monomial initial idealM2,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. Proposition 4.43. The basis of non-crossing monomials of A2,n coincides with the basis of cluster monomials (see Theorem 4.22). Proof. For every standard monomial p̄m with m ∈ Z(n2) ≥0 we draw all arcs ij in the n-gon, for which mij 6= 0. By definition there is no pair of crossing arcs. Hence, the set of arcs can be completed to a triangulation and p̄m is a cluster monomial for the corresponding seed. On the other hand, every cluster monomial is contained in B< which completes the proof. � Proposition 4.44. The monomial ideal M2,n is the unique common monomial degeneration of the toric ideals inT (I2,n) associated to triangulations T constructed in Proposition 4.36. Proof. For every weight vector wT associated with a triangulation T and a quadrilateral lin- ear map wT we have inu ( inwT (I2,n) ) = inwT ( inu(I2,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 inwT (Rijkl) = pijpkl − pikpjl by Proposition 4.34 and Corollary 4.32. Now consider T ′ obtained from T by flipping the arc ik, so T ′ contains jl. Then inwT ′ (Rijkl) = −pikpjl+pilpjk. So −pikpjl is the only monomial that simultaneously is an initial of inwT (Rijkl) and inwT ′ (Rijkl) for all i < j < k < l. � As mentioned in Remark 4.37 there are various ways to associate a toric degeneration to a tri- angulation 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,Cn) in Proposition 4.36. Corollary 4.45. The quotient S/ inT (I2,n) is isomorphic to the algebra of the semigroup gene- rated by {gij : 1 ≤ i < j ≤ n} ⊂ Z2n−3. In particular, the central fiber of the toric Gröbner degeneration induced by T is isomorphic to that of the degeneration induced by principal coeffi- cients at T . Proof. The quotient S/ inT (I2,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/ inT (I2,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 {gij : 1 ≤ i < j ≤ n} ⊂ Z2n−3. 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 {gij : 1 ≤ i < j ≤ n} ⊂ Z2n−3. � 30 L. Bossinger, F. Mohammadi and A. Nájera Chávez Theorem 4.46. In R(n2) we have C ∩ Trop(I2,n) = Trop+(I2,n). In particular, every (d + 1)- dimensional face of C corresponds to a planar trivalent tree and Trop+(I2,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 wT gives a toric initial ideal. Hence, wT lies in the relative interior of a maximal (i.e., (d+ 1)-dimensional) cone σT in Trop+(I2,n). Furthermore, σT 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 inΥ(Rijkl) contains the monomial pikpjl 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 inΥ(Rijkl) = pijpkl + pilpjk, a contra- diction. � Corollary 4.47. The cone C is defined by the lineality space L2,n and additional ( n 2 ) − n rays rij ∈ R(n2) with 2 ≤ i+ 1 < j ≤ n defined by (rij)kl = { −1 if k ∈ [i+ 1, j] 63 l or k 6∈ [i+ 1, j] 3 l, 0 otherwise. 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 = dimL2,n) are also rays of Trop+(I2,n). By the combinatorial description of Trop(I2,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 2rij 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 denote f̃r by f̃ for all f ∈ S. Proposition 4.48. For all i < j < k < l we have that R̃ijkl = Rijkl(t), where Rijkl(t) := −pikpjl + pilpjk ∏ a∈[i,j−1], b∈[k,l−1] tab + pijpkl ∏ a∈[j,k−1], b∈[l,i−1] tab. Proof. We show that every variable tab occurs with the same exponent in R̃ijkl and Rijkl(t). For simplicity, we adopt the notation Rijkl = −peik+ejl + peil+ejk + peij+ekl . To compute the exponents of a variable tab in R̃ijkl 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] 63 j, k, l. Note that due to symmetries this is equivalent to assuming i 6∈ [a+ 1, b] 3 j, k, l. Then 1 2rab · (eik + ejl) = 1 2rab · (eil + ejk) = 1 2rab · (eij + ekl) = −1 2 . Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras 31 Hence, in R̃ijkl the variable tab does not appear. One can see that tab also does not appear in Rijkl(t), since neither b ∈ [l, i− 1] nor b ∈ [k, l − 1]. Case 2. Assume that i, j ∈ [a + 1, b] 63 k, l. Note that this is equivalent to assuming i, j 6∈ [a + 1, b] 3 k, l. Then 1 2rab · (eik + ejl) = 1 2rab · (eil + ejk) = −1 and 1 2rab · (eij + ekl) = 0. Hence, in R̃ijkl the variable tab does only appear with exponent 1 in the coefficients of the monomial pijpkl. As we assumed i, j ∈ [a + 1, b] 63 k, l it follows that a ∈ [l, i − 1] and b ∈ [j, k − 1]. So in Rijkl(t) the variable tab appears also with exponent 1 in the coefficient of the monomial pijpkl. � Corollary 4.49. The algebra Ã2,n = S[tij : 2 ≤ i+ 1 < j ≤ n]/Ĩ2,n is a free C[tij : 2 ≤ i+ 1 < j ≤ n]-module with basis given by the cluster monomials. Moreover, fibers of the flat family π : Proj(Ã2,n)→ A n(n−3) 2 are in correspondence with collections of arcs in the n-gon. Proof. The first part is a direct corollary of Proposition 4.48, Theorem 3.14(i) and Proposi- tion 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,Cn). Theorem 4.50. The Plücker algebra with universal coefficients Auniv 2,n is canonically isomorphic to the quotient Ã2,n = S[tij : 2 ≤ i+ 1 < j ≤ n]/Ĩ2,n. Proof. Recall that Auniv 2,n has the same set of (frozen and mutable) cluster variables as A2,n, namely {p̄ij : 1 ≤ i < j ≤ n}, and additionally coefficients {yij : 2 ≤ i+ 1 < j ≤ n}. We define Ψ: S[tij : 2 ≤ i+ 1 < j ≤ n]→ Auniv 2,n , pij 7→ p̄ij for 1 ≤ i < j ≤ n, tij 7→ yij for 2 ≤ i+ 1 < j ≤ n. This morphism of C-algebras induces the desired isomorphism between Auniv 2,n and Ã2,n. By Pro- position 3.9 the lifts of Plücker relations R̃ijkl generate the lifted ideal Ĩ2,n. We proceed by showing that Ψ ( R̃ijkl ) is the corresponding exchange relations in Auniv 2,n . Since the mutable parts of Buniv (QT ,FT ) and B(QT ,FT ) coincide for every triangulation T there is a natural bijection between cluster monomials of A2,n and Auniv 2,n . It is the only bijection that sends the initial cluster variables of B(QT ,FT ) to those of Buniv (QT ,FT ) and commutes with mutation. Further, it induces a bijection between the exchange relations associated to B(QT ,FT ) and Buniv (QT ,FT ), which yields bijections between the sets: R̃ijkl ∈ Ĩ2,n with i, j, k, l ∈ [n], i < j < k < l ←→ { quadrilaterals with vertices i < j < k < l in the n-gon } ←→ { exchange relations of Auniv 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 Auniv 2,n associated with the quadrilateral i < j < k < l is of form: Euniv ijkl := −p̄ikp̄jl + p̄ij p̄kl ∏ (g∨ab)ik=−1 yab + p̄ilp̄jk ∏ (g∨ab)ik=+1 yab, where g∨ab is the abth row of UQmut T and ( g∨ab ) ik is its entry in the column of Buniv (QT ,FT ) corresponding to the mutable variable p̄ik. Hence, we need to compute those rows of UQmut T with non-zero entries 32 L. Bossinger, F. Mohammadi and A. Nájera Chávez ... . . . ... . . . i j kl H ... . . . ... . . . j′ i′ l′k′ (j − 1)′ i′ (i− 1)′ l′ (l − 1)′k′ (k − 1)′ j′ . . .. . . ...... . . .. . . ... ... j′ i′ l′k′ 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 ( Qmut T )op (i.e., columns of UQmut T ) can be read off. in the ikth column. To do so, we embed T into R2 and consider a hyperplane H ⊂ R2 which does not intersect T . Let T∨ be the image of T under the reflection sH with respect to H and denote m′ := sH(m) for all m ∈ [n]. Naturally, we have QT∨ = (QT )op, so Qmut T∨ = ( Qmut T )op . Using the right side of Figure 6 we compute the ikth entry of truncated g-vectors with respect to Qmut T∨ : (g∨ab)ik =  +1 if a ∈ [i, j − 1] and b ∈ [k, l − 1] (or vice versa), −1 if a ∈ [j, k − 1] and b ∈ [l, i− 1] (or vice versa), 0 otherwise. (4.5) We compute: Ψ ( R̃ijkl ) = −p̄ikp̄jl + p̄ilp̄jk ∏ a∈[i,j−1], b∈[k,l−1] yab + p̄ij p̄kl ∏ a∈[j,k−1], b∈[l,i−1] yab (4.5) = Euniv ijkl . In particular, Ψ induces a surjective map Ψ̄: Ã2,n → Auniv 2,n . By Corollary 4.49, Ã2,n is a free C[tij ]-module whose basis are the cluster monomials. Similarly, after identifying tij , 2 ≤ i+ 1 < j ≤ n, with Ψ(tij) = yij , A univ 2,n is a free C[tij ]-module with basis given by the cluster monomials by [28, Theorem 0.3 and p. 502]. Hence, Ψ̄ is also injective and the claim follows. � As Auniv 2,n is by definition a domain the following is now a direct consequence. Corollary 4.51. The ideal Ĩ2,n ⊂ S[t] is prime. Example 4.52. We list the lifted Plücker relations, respectively the exchange relations of Auniv 2,5 , associated to our running example. These relations also constitute a Gröbner basis for Ĩ2,5, 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: R̃1234 = p13 p24 − p12 p34 t24 t25 − p14 p23 t13, R̃1235 = p13 p25 − p15 p23 t13 t14 − p12 p35 t25, R̃1245 = p14 p25 − p12 p45 t25 t35 − p15 p24 t14, R̃1345 = p14 p35 − p15 p34 t14t24 − p13 p45 t35, R̃2345 = p24 p35 − p23 p45 t13 t35 − p25 p34 t24. 4.4 The Grassmannian Gr ( 3,C6 ) We now turn to the case of Gr ( 3,C6 ) and prove an analogue of Theorem 4.50. To highlight var- ious important differences between this case and the case of Gr(2,Cn), we focus more on explicit computations. We believe that the explicit computations help to understand the difficulties that Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras 33 may arise when studying other compactifications of finite type cluster varieties such as Gr(3,C7), Gr(3,C8) or (skew-)Schubert varieties inside Grassmannians as in [48]. Denote by A3,6 the homogeneous coordinate ring of the Grassmannian Gr ( 3,C6 ) with res- pect to its Plücker embedding. We use the cluster structure on A3,6 to consider Gr ( 3,C6 ) as a weighted projective variety as follows. The cluster algebra A3,6 has 22 cluster variables, out of which 20 are the Plücker coordinates and the 2 additional elements are X̄ = p̄145p̄236 − p̄123p̄456 and Ȳ = p̄125p̄346 − p̄126p̄345, see [47, Theorem 6]. We systematically write the frozen Plücker coordinates in blue. We list the 22 cluster variables of A3,6 in the following order and fix this order for later use p̄123, p̄124, p̄125, p̄126, p̄134, p̄135, p̄136, p̄145, p̄146, p̄156, p̄234, p̄235, p̄236, p̄245, p̄246, p̄256, p̄345, p̄346, p̄356, p̄456, X̄, Ȳ . (4.6) Every seed s of A3,6 consists of the six frozen variables p̄123, p̄234, p̄345, p̄456, p̄156, p̄126 and four additional mutable cluster variables. Consider the following bijection that sends the Plücker variable pijk to the Plücker coordinate p̄ijk ∈ A3,6 and X to X̄ ∈ A3,6 as well as Y to Ȳ ∈ A3,6: {p123, . . . , p456, X, Y } ←→ {cluster variables of A3,6}. (4.7) Denote by d ∈ Z22 the vector (1, . . . , 1, 2, 2). That is, the first 20 entries of d are 1 and the last two are 2. The bijection (4.7) induces a surjective map ψ : Cd[p123, . . . , p456, X, Y ]� A3,6. (4.8) Let Iex := ker(ψ), so we obtain a ring isomorphism A := Cd[p123, . . . , p456, X, Y ]/Iex ∼= A3,6. Moreover, the weighted projective variety V (Iex) ⊂ P(d) is isomorphic to Gr ( 3,C6 ) as a weig- hted projective variety. Indeed, d was chosen so that every exchange relation in the cluster structure of A3,6 is identified with a d-homogeneous element of Cd[p123, . . . , p456, X, Y ] and Iex is prime (hence, radical) since A3,6 is a domain. Therefore, I(V (Iex)) = Iex and ψ induces an iso- morphism of Z-graded rings S(V (Iex))→ A3,6. One can verify (e.g., using Macaulay2 [27]) that the ideal Iex is generated by all three-term Plücker relations and the following seven additional relations: p145p236 − p123p456 −X, p136p245 − p126p345 −X, p146p235 − p156p234 −X, p124p356 − p123p456 − Y, p125p346 − p126p345 − Y, p134p256 − p156p234 − Y, p135p246 − p134p256 − p126p345 − p145p236. (4.9) Note that with exception of the last relation all of them correspond to exchange relations in A3,6. We study the Gröbner fan of Iex. It contains the lineality space L(Iex) generated by `1 = (1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1), `2 = (1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1), `3 = (1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1), `4 = (0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1), `5 = (0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1), `6 = (0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1). The order of the entries corresponds to the order on cluster variables of A3,6 in (4.6). Note that d = 1 3(`1 + · · · + `6). In the Gröbner fan GF(Iex) we identify a maximal cone C and consider its image C ⊂ GF(Iex)/L(Iex). The cone C is strongly convex (by Lemma 2.13) and simplicial. We choose the following representatives of the 16 minimal ray generators of C: r1 = (0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), 34 L. Bossinger, F. Mohammadi and A. Nájera Chávez r2 = (1, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 2, 2), r3 = (0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1), r4 = (1, 0, 1, 2, 0, 1, 2, 0, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 1, 3, 2), r5 = (1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 2, 1), r6 = (1, 1, 1, 0, 1, 1, 0, 2, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 2, 1), r7 = (0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), r8 = (1, 2, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 2), r9 = (2, 3, 2, 1, 3, 2, 1, 3, 2, 1, 2, 1, 0, 2, 1, 0, 2, 1, 0, 2, 3, 3), r10 = (2, 3, 2, 1, 3, 2, 1, 3, 2, 2, 2, 1, 0, 2, 1, 1, 2, 1, 1, 2, 3, 4), r11 = (2, 3, 3, 2, 3, 2, 1, 3, 2, 2, 2, 1, 0, 2, 1, 1, 2, 1, 1, 2, 3, 4), r12 = (1, 1, 1, 2, 0, 0, 1, 0, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 1, 1, 2, 2), r13 = (1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 2, 1), r14 = (1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1), r15 = (1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1), r16 = (1, 2, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 1, 1, 2). (4.10) From now on, we denote by r the (16×22)-matrix with rows r1, . . . , r16. Recall that a monomial is called squarefree if the exponent of each variable in it is either 0 or 1. Theorem 4.53. There exists a unique maximal cone C ∈ GF(Iex) with the following properties: (i) the associated initial ideal inC(Iex) is generated by squarefree monomials of degree 2 and it contains all exchange monomials; (ii) the free C[t1, . . . , t16]-algebra associated to C and r defined in Definition 3.2 has the pro- perty Ãr := C[t1, . . . , t16][p123, . . . , p456, X, Y ]/Ĩex r ∼= Auniv 3,6 ; (iii) for every seed s in the cluster structure of A3,6 the cone C has a 10-dimensional face τs whose associated initial ideal inτs(I ex) is a totally positive binomial prime ideal (hence τs ∈ Trop+(Iex)). Moreover, C ∩ Trop(Iex) = Trop+(Iex). Before we prove Theorem 4.53 we explain the conventions we use to describe the exchange relations of Auniv 3,6 . The algebra Auniv 3,6 is of cluster type D4. In particular, as explained in Sec- tion 4.1, we label the coefficients of Auniv 3,6 with the set of almost positive coroots Φ∨≥−1 in the root system dual to a root system of type D4. For this we fix an initial seed for Auniv 3,6 such that the mutable part of its quiver is a bipartite orientation of D4. We choose the seed that contains the mutable variables p246, p346, p124 and p256 together with the frozen variables. The part of the quiver that involves only the vertices corresponding to these variables is the following: p246 p256 p346 p124 p126 p456 p234 p156 p345 p123 Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras 35 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 p246 correspond to 1, p346 to 2, p124 to 3 and p256 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 monomi- als for Auniv 3,6 . By [28, Theorem 0.3 and p. 502] cluster monomials form a C[yα : α ∈ Φ∨≥−1]- basis for Auniv 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, Theo- rem 7.12(b)] this gives us the following characterization of cluster monomials. Write x̄a ∈ Auniv 3,6 , a ∈ Z22 ≥0, to denote a monomial in the (mutable and frozen) cluster variables p̄123, . . . , p̄456, X̄, Ȳ . Then x̄a ∈ Auniv 3,6 is a cluster monomial if and only if m 6 \| x̄a for all m ∈M3,6, (4.11) where M3,6 = {exchange monomials}∪ { X̄Ȳ , p̄135p̄246 } . We write M3,6 to denote the C[yα : α ∈ Φ∨≥−1]-basis of cluster monomials for Auniv 3,6 . Proof of Theorem 4.53. We prove the statements of the theorem in order. (i) In Macaulay2 [27] we compute the initial ideal of Iex with respect to the cone C. For the computation, we fix the weight vector w = r1 + · · ·+ r16 in the relative interior of C: w = (16, 19, 16, 16, 16, 11, 10, 19, 16, 16, 16, 10, 7, 16, 11, 10, 16, 10, 7, 16, 27, 27). We compute a minimal generating set of inC(Iex): it has 54 elements, 52 of those are exchange monomials, and the other two are p135p246 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 ideal Ĩex r , then we define a surjective map Ãr → Auniv 3,6 , and lastly, we show that the map is also injective. Step 1: By Proposition 3.9 the lifted ideal Ĩex r is generated by the lifts of elements of a Gröbner basis for Iex 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 p135p246) and the following element (whose initial form is XY ): S(p134p256 − p156p234 − Y, p134X − p136p145p234 − p123p146p345), 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 (mini- mal) Gröbner basis for Iex with respect to C. The reduced Gröbner basis GC(Iex) consists of the 52 exchange relations and the additional two elements f := p135p246 − p156p234 − Y − p123p456 −X − p126p345, g := XY − p123(p156p246p345 + p156p234p456 + p126p345p456) − p126(p135p234p456 + p156p234p345). (4.12) The first monomial in f (and g) is its leading monomial. We now compute the lifts of the elements in GC(Iex) with respect to the matrix r in Definition 3.2, which are given explicitly in Appendix A. 36 L. Bossinger, F. Mohammadi and A. Nájera Chávez Step 2: Auniv 3,6 has 22 cluster variables (in one-to-one correspondence with those of A3,6) and 16 coefficients labeled by almost positive roots of type D4: yα with α ∈ Φ∨≥−1. We extend the morphism ψ in (4.8) to Ψ: C[t1, . . . , t16][p123, . . . , p456, x, y] → Auniv 3,6 by sending ti’s to yα’s according to the following identification: y−α1 ↔ t16, yα1 ↔ t14, yα1+α2 ↔ t6, yα1+α2+α4 ↔ t9, y−α2 ↔ t12, yα2 ↔ t10, yα1+α3 ↔ t4, yα1+α2+α3 ↔ t2, y−α3 ↔ t15, yα3 ↔ t3, yα1+α4 ↔ t13, y2α1+α2+α3+α4 ↔ t5, y−α4 ↔ t7, yα4 ↔ t8, yα1+α3+α4 ↔ t1, yα1+α2+α3+α4 ↔ t11. (4.13) We now verify that Ĩex r ⊆ ker(Ψ): the lifts of those elements in Iex that correspond to exchange relations in A3,6 are sent to exchange relations in Auniv 3,6 by Ψ, so they lie in ker(Ψ). For the elements f, g ∈ GC(Iex) in (4.12) note that, for example, p245f has an expression in terms of ex- change relations with coefficients that are Plücker variables (see (A.1) for the precise expression). Hence, p245t µ(f)f̃ = p245t 2 2t 2 4t5t6t8t 3 9t 3 10t 3 11t 2 12t13t14t15t16f̃ has an expression in terms of the lifts of those exchange relations with monomial coefficients in t’s and Plücker variables. So, 0 = Ψ ( f̃p245t µ(f) ) = Ψ ( f̃ ) p̄245y µ(f), where yµ(f) = Ψ ( tµ(f) ) . As Auniv 3,6 is a domain, this implies f̃ ∈ ker(Ψ). A similar argument implies that g̃ ∈ ker(Ψ) (see (A.2)) and so we obtain the induced morphism Ψ̄: Ãr → Auniv 3,6 . Note that the image of Ψ̄ contains all cluster variables and all coefficients of Auniv 3,6 , so Ψ̄ is in fact surjective. Step 3: Lastly, we show that Ψ̄ is injective. By Theorem 3.14(i) the standard monomial basis B< (for < a monomial term order compatible with C) is a C[t1, . . . , t16]-basis for Ãr. Similarly, Auniv 3,6 has the C[yα : α ∈ Φ∨≥−1]-basis of cluster monomials M3,6. The test for membership in M3,6 is given byM3,6 in (4.11), which is in one-to-one correspondence with the set {in<(g) : g ∈ G<(Iex)}. Hence, there is a bijection between the standard monomial basis B< for Ãr (see Theorem 3.14(i)) and the cluster monomial basis M3,6 of Auniv 3,6 induced by Ψ̄. In particular, Ψ̄ is injective and Ãr ∼= Auniv 3,6 . (iii) To prove this part, we identify the rays r1, . . . , r16 with mutable cluster variables. As we have already identified yα’s with ti’s in (4.4) (and by definition ti’s correspond to ri’s) it is enough to identify the yα’s with the mutable cluster variables of A3,6. 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 yα (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 7→ yα defines a bijection between mutable cluster variables and coefficients. Combining with the identification of yα’s with ti’s in (4.4) we obtain: r1 ↔ p̄125, r2 ↔ p̄134, r3 ↔ p̄124, r4 ↔ p̄145 r5 ↔ p̄135, r6 ↔ p̄136, r7 ↔ p̄146, r8 ↔ p̄256, r9 ↔ p̄356, r10 ↔ p̄346, r11 ↔ Y, r12 ↔ p̄245, r13 ↔ p̄235, r14 ↔ X, r15 ↔ p̄236, r16 ↔ p̄246. Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras 37 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 = {p̄124, p̄125, p̄245, p̄256} is ws = r1 + r3 + r8 + r12. Using Macaulay2 [27] we verify that inws(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(Iex) = Trop+(Iex) observe that for every element h ∈ GC(Iex) its initial monomial inC(h) is the unique monomial with positive coefficient (the complete list of GC(Iex) can be found in Section A). Hence, a weight vector w lies in C ∩ Trop(Iex) if and only if it lies in Trop+(Iex). � Remark 4.54. There are various methods in cluster theory to compute the exchange relations for Auniv 3,6 and M3,6, e.g., one can use the categorification of finite type cluster algebras with universal coefficients introduced in [42]. To compute M3,6 one can use the compatibility degree of cluster variables from [24]. In fact, the elements ofM3,6 are exactly those pairs of cluster variables whose compatibility degrees are positive. However, we have presented the case of Gr ( 3,C6 ) with as few machinery from the cluster theory as possible to make it more digestible to a broader audience. Example 4.55. Here we demonstrate the need of Lemma 2.12 and that in fact the Lemma is not true for arbitrary elements of the ideal J . To see this, let J = Iex and take v := 1 4 16∑ i=1 i 6=2,8 ri + 5 4 r8 and w := r2 + v. 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), w = 1 4 (23, 30, 26, 23, 23, 17, 13, 29, 23, 22, 23, 13, 7, 23, 15, 13, 23, 14, 10, 23, 37, 41). Now take the S-pair of two exchange relations h := S(p134p256 − p156p234 − Y, p134x− p136p145p234 − p123p146p345) = −XY − p156p234x+ p136p145p234p256 − p123p146p256p345 ∈ Iex. The weights of the non-zero monomials in h with respect to v are (in order) 31 2 , 31 2 , 29 2 , 33 2 and with respect to w are 39 2 , 41 2 , 39 2 , 41 2 . We compute inv(h) = inr2(inv(h)) = p136p145p234p256 6= −XY + p136p145p234p256 = inw(h). 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 , p136p145p234p256 and p123p146p256p345 all lie in inC(Iex). 4.5 Stanley–Reisner ideals and the cluster complex Definition 4.56. Let (Q,F ) be an ice-quiver and V the set of mutable cluster variables of A(Q,F ). We call x and x′ in V compatible if there exists a cluster containing both of them. Similarly, a subset of V is compatible if it consists of pairwise compatible cluster variables. The cluster complex ∆(A(Q,F )) is the simplicial complex on V whose simplices are the compatible subsets. 38 L. Bossinger, F. Mohammadi and A. Nájera Chávez 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. Definition 4.58. Let ∆ be a simplicial complex with vertex set V = {x1, . . . , xn}. The Stanley– Reisner ideal of ∆ is generated by monomials associated to the minimal non-faces of ∆ as: I∆ := 〈xi1 · · ·xis : {xi1 , . . . , xis} 6∈ ∆〉 ⊆ K[x1, . . . , xn]. 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 A2,n or A3,6 and ∆(A) the associated cluster complex. Similarly, let I be the ideal I2,n or Iex 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 inC(I). Proof. The initial ideal inC(I) is squarefree and generated by the monomials in the set M , which is respectively, M2,n or M3,6. Observe that M defines the minimal non-faces of ∆(A). Hence, ∆(A) is the Stanley–Reisner complex of inC(I). � 4.6 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[x1, . . . , xn] assume there exists a maximal cone σ in Trop(J) whose associated ideal is toric. Then [34, Section 4] provides a recipe to construct a full rank valuation vσ : A \ {0} → Zd, where A = K[x1, . . . , xn]/J and d is the dimension of the affine variety V (J). Furthermore, [34, Corol- lary 4.7] shows how to compute the corresponding Newton–Okounkov body ∆(σ). Without loss of generality we may assume that the first entry of vσ(f) equals the degree of f (with respect to the standard grading). Then the Newton–Okounkov body of vσ and K[x1, . . . , xn]/J is ∆(σ) := Cone(vσ(f) : f ∈ A \ {0}) ∩ {1} × Rd−1. 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,Cn). We fix a triangulation T of the n-gon. The output of Algorithm 1 can be used to define a total order on Z2n−3 as follows. Definition 4.60. Let V0, . . . , Vi be the output of Algorithm 1 for some triangulation T . To each Vj we associate a sequence V j with the same elements as Vj . Let V be the sequence( V 0, . . . , V i ) and ≺ the associated lexicographic order on Z2n−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 gT̂ : B< → Z2n−3 given by p̄a 7→ ∑ ij∈T aijg T̂ ij . (4.14) extends to a full rank valuation on A2,n \ {0} as follows. Fix a total order ≺ compatible with T . Every 0 6= f ∈ A2,n is a unique linear combination of elements in B<, that is f = ∑ p̄a∈B< cap̄ a. Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras 39 We define the valuation of f as gT̂ (f) := min≺ { gT̂ (p̄a) : ca 6= 0 } . Denote the associated graded algebra by grT (A2,n) and the corresponding Newton–Okounkov body by ∆(T ). By [5, Corollary 2] and Proposition 4.34 we have grT (A2,n) ∼= S/ inT (I2,n) and ∆(T ) = conv ( gT̂ij ) 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 A2,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 theo- retic applications of cluster algebras. The following example shows that truncated g-vectors do not have the desired properties for applications. Example 4.62. Consider for Gr(2, 6) the triangulation T consisting of arcs 24, 25, 26. Order the cluster variables corresponding to T in a compatible way using the output of Algorithm 1, e.g., p16, p23, p24, p56, p12, p25, p26, p34, p45. Now compute the g-vectors with respect to T and consider the following elements in A2,n endowed with their g-vectors: (1, 0, 0, 0, 0, 1, 0, 0, 0) = (1, 0, 0, 0, 0, 1, 0, 0, 0) ≺ (0, 0, 0, 1, 1, 0, 0, 0, 0), p̄15p̄26 = p̄16p̄25 + p̄12p̄56, (0, 1, 0, 0, 0, 0, 0, 0, 1) = (0, 1, 0, 0, 0, 0, 0, 0, 1) ≺ (0, 0, 0, 0, 0, 1, 0, 1, 0), p̄24p̄35 = p̄23p̄45 + p̄25p̄34. 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 tri- angulation. 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 T1 and T2 be two triangulations such that there exist a < b < c < d in [n] with ac ∈ T1, bd ∈ T2 and T1 \ {ac} = T2 \ {bd}. In other words, T2 is obtained from T1 by flipping the arc ac. We denote by RT1 the vector space R2n−3 with standard basis {fij : ij ∈ T1} and similarly, RT2 for R2n−3 with basis {f ′ij : ij ∈ T2}. The theory of cluster varieties gives us two distinguished maps from R2n−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]): ζac : RT1 → RT1 , m = ∑ ij∈T1 mijfij 7→ { m if mac ≤ 0, m+macvac if mac > 0, where vac = −fab − fcd + fad + fbc. The second map is a GL2n−3(Z)-base change from {fij : ij ∈ T1} to {f ′ij : ij ∈ T2} corresponding to a seed mutation (see [20, equation (8)]) denoted by µT1 T2 : RT1 → RT2 . It is given by f ′ij = fij for ij ∈ T1 ∩ T2 and f ′bd = −fac + fab + fcd. 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. 40 L. Bossinger, F. Mohammadi and A. Nájera Chávez Lemma 4.63. Let T1 and T2 be two triangulations related by a flip as above. Then ∆(T2) = µT1 T2 ◦ ζac(∆(T1)). More precisely, for every p̄a ∈ B< we have gT̂2(p̄a) = µT1 T2 ◦ ζac ◦ gT̂1(p̄a). Proof. By definition of gT̂p for p = 1, 2 from (4.14) it is enough to show the second claim for the Plücker coordinates. As ∆(Tp) is the convex hull of g T̂p ij for 1 ≤ i < j ≤ n, this implies the first claim. Given 1 ≤ i < j ≤ n, by Definition 4.27 the coefficient of fac in gT̂1(p̄ij) = gT̂1 ij can be 1, −1 or 0. We go through the details of the case when it is 1, the other two cases are similar. Recall the sign conventions for the g-vectors from Figure 3. Take gT̂1 ij = fac + ∑ kl∈T1\{ac} σ kl i jfkl. Recall that T1 \ {ac} = T2 \ { bd } and fkl = f ′kl for kl ∈ T1 \ {ac}. Now we compute µT1 T2 ◦ ζac ◦ gT̂1(pij) step-by-step as follows: gT̂1(pij) = fac + ∑ kl∈T1\{ac} σkli jfkl ζac7−−→ fac + vac + ∑ kl∈T1\{ac} σkli jfkl µ T1 T27−−→ −f ′bd + f ′ad + f ′bc + ∑ kl∈T2\{bd} σkli jf ′ kl = gT̂2(pij). Observe that the sign σkli j depends on the triangulation, but the above equation takes into account the change of signs that might happen when we pass from T1 to T2. � 1 3 5 2 4Υ1 1 3 2 5 4Υ 1 3 4 2 5Υ2 ←→ 1 3 4 5 2 Υ′2 Figure 7. Trivalent trees for Gr(2,C5) corresponding to maximal cones in Trop(I2,n) that intersect in a facet (Υ1 and Υ2) and the tree corresponding to the intersection is Υ, see Example 4.65. Note that although in Section 4.3 we are only considering a fixed maximal Gröbner cone C, all other maximal cones can be obtained from C through a symmetric group action (see [38, Second Proof of ⊇ Theorem 4.3.5]). Moreover, [19, Lemma 5.13] shows that any two maximal cones in Trop(I2,n) that intersect in a facet are faces of one maximal Gröbner cone (up to symmetry). Corollary 4.64. Let σ1, σ2 ∈ Trop(I2,n) be two maximal prime cones that intersect in a facet. Then ∆(σ1) and ∆(σ2) are related by the shear map ζac for appropriate a, c ∈ [n]. Proof. The cones σ1 and σ2 are in correspondence with trivalent trees Υ1, Υ2 by Theorem 4.2. By [19, Lemma 5.13] we can assume that σ1 and σ2 are faces of the same maximal Gröbner cone C. Hence, we may assume that the leaves of Υ1 and Υ2 are labeled in the same order. The symmetric group action on Trop(I2,n) translates to permuting the leaf labeling of trivalent trees. So we can simultaneously move the cones σ1 and σ2 to faces of C. Say these faces correspond to triangulation T1 and T2 of the n-gon, respectively. One consequence of the symmetric group action is that V (inσi(I)) ∼= V (inTi(I)) for i = 1, 2. In particular, these isomorphisms induce unimodular equivalences ∆(σi) ∼= ∆(Ti) for i = 1, 2. The assumption that σ1 and σ2 intersect in a facet, directly implies that T1 and T2 differ by flipping one arc. So there exist a < b < c < d in [n] with ac ∈ T1, bd ∈ T2 and T1 \ {ac} = T2 \ {bd}. Now applying Lemma 4.63 completes the proof. � Example 4.65. We illustrate the proof of Corollary 4.64 for Gr(2,C5). Let Υ1 and Υ2 be the trees in Figure 7. They correspond to cones σ1 and σ2 sharing a facet corresponding to the tree Υ Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras 41 from Figure 7. First, we apply [19, Lemma 5.13] and replace Υ2 by a tree Υ′2 that gives the same initial ideal as Υ2, and its leaves are labeled in the same order as Υ1. This implies that σ1 and σ2 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 Υ1 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): 1 2 3 4 5 1 5 2 3 4 s(Υ1) 1 5 4 2 3 s(Υ′2) 4 3 2 1 5 The symmetric group element s induces an automorphism s : S→S given by s(pij)= ps−1(i)s−1(j). It is straight-forward to verify that s(ins(Υ)(I)) = inΥ(I) for Υ ∈ {Υ1,Υ ′ 2}. Consider σ1 and σ2 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 semi- groups im(vσ1) and im(vσ2) (see [19, Section 4.2], where this is called an algebraic wall-crossing). As seen in Lemma 4.63 the map µT1 T2 ◦ζac extends to a map between the value semigroups im ( gT̂1 ) and im ( gT̂2 ) . In [19, Theorem 5.15] the authors show that their piecewise linear flip map in- duces the algebraic wall-crossing for Gr(2,Cn). Hence, Corollary 4.64 implies that the flip map for Gr(2,Cn) 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 alge- braic 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]. A Computational data Here, we present data on the ideal Iex. Recall that the weighted homogeneous coordinate ring of Gr ( 3,C6 ) with respect to this embedding is A3,6 ∼= C[p123, . . . , p456, X, Y ]/Iex. We frequently identify the variables p123, . . . , p456, X, Y with their cosets p̄123, . . . , p̄456, X̄, Ȳ in A3,6, as well as elements of the ideal Iex with the corresponding relations in A3,6; that is, we identify, for example p145p236 − p123p456 −X ∈ Iex and p̄145p̄236 = p̄123p̄456 + X̄ in A3,6. A minimal generating set for Iex. We now list a minimal generating set for Iex consisting of elements of the reduced Gröbner basis GC(Iex). Note that except the last polynomial f , these are the exchange relations of A3,6 with the first monomial being the exchange monomial. p145p236 − p123p456 −X, p124p356 − p123p456 − Y, p136p245 − p126p345 −X, p125p346 − p126p345 − Y, p146p235 − p156p234 −X, p134p256 − p156p234 − Y, p246p356 − p346p256 − p236p456, p245p356 − p345p256 − p235p456, 42 L. Bossinger, F. Mohammadi and A. Nájera Chávez p146p356 − p346p156 − p136p456, p145p356 − p345p156 − p135p456, p245p346 − p345p246 − p234p456, p235p346 − p345p236 − p234p356, p145p346 − p345p146 − p134p456, p135p346 − p345p136 − p134p356, p146p256 − p246p156 − p126p456, p145p256 − p245p156 − p125p456, p136p256 − p236p156 − p126p356, p135p256 − p235p156 − p125p356, p235p246 − p245p236 − p234p256, p145p246 − p245p146 − p124p456, p136p246 − p236p146 − p126p346, p134p246 − p234p146 − p124p346, p125p246 − p245p126 − p124p256, p134p245 − p234p145 − p124p345, p135p245 − p235p145 − p125p345, p135p236 − p235p136 − p123p356, p134p236 − p234p136 − p123p346, p125p236 − p235p126 − p123p256, p124p236 − p234p126 − p123p246, p134p235 − p234p135 − p123p345, p124p235 − p234p125 − p123p245, p135p146 − p145p136 − p134p156, p125p146 − p145p126 − p124p156, p125p136 − p135p126 − p123p156, p124p136 − p134p126 − p123p146, p124p135 − p134p125 − p123p145, f = p135p246 − p156p234 − Y − p123p456 −X − p126p345. The reduced Gröbner basis GC(Iex). Let C be the maximal cone in the Gröbner fan of Iex whose rays are given in (4.10). Recall from Theorem 4.53(i) that the associated monomial initial ideal of C is generated by the exchange monomials, together with the monomials p135p246 and XY . To obtain the reduced Gröbner basis GC(Iex) we need to add the missing exchange relations: p235Y − p125p234p356 − p123p256p345, p134X − p136p145p234 − p123p146p345, p146Y − p124p156p346 − p126p134p456, p256X − p156p236p245 − p126p235p456, p136Y − p123p156p346 − p126p134p356, p346X − p136p234p456 − p146p236p345, p245Y − p125p234p456 − p124p256p345, p125X − p123p156p245 − p126p145p235, p145Y − p125p134p456 − p124p156p345, p124X − p126p145p234 − p123p146p245, p236Y − p126p234p356 − p123p256p346, p356X − p136p235p456 − p156p236p345, p135Y − p125p134p356 − p123p156p345, p135X − p136p145p235 − p123p156p345, p246Y − p124p256p346 − p126p234p456, p246X − p146p236p245 − p126p234p456. Further, we need to add the following additional element to the generating list above (the first monomial is its leading monomial in inC(Iex)): g = XY − p123p156p246p345 − p126p135p234p456 − p126p156p234p345 − p123p156p234p456 − p123p126p345p456. Here, we list the identities used in the proof of Theorem 4.53: p245f = p246(p135p245 − p235p145 − p125p345) + p145(p235p246 − p245p236 − p234p256) + p245(p145p236 − p123p456 −X) + p234(p145p256 − p245p156 − p125p456) − p125(p245p346 − p345p246 − p234p456) + p245(p125p346 − p126p345 − Y ). (A.1) g = p256(p134X − p136p145p234 − p123p146p345)−X(p134p256 − p156p234 − Y ) − p145p234(p136p256 − p236p156 − p126p356)− p156p234(p145p236 − p123p456 −X) − p126p234(p145p356 − p345p156 − p135p456) + p123p345(p146p256 − p246p156 − p126p456). (A.2) Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras 43 The exchange relations of Auniv 3,6 (or equivalently the lifts of the elements of GC(Iex)) are: p245p356 − t3t12t16p345p256 − t5t6t9p235p456, p146p356 − t3t7t16p346p156 − t5t9t13p136p456, p145p256 − t4t7t14p245p156 − t8t9t11p125p456, p136p256 − t6t7t14p236p156 − t1t8t11p126p356, p134p245 − t2t5t6p234p145 − t8t12t16p124p345, p134p236 − t2t4t5p234p136 − t8t15t16p123p346, p125p146 − t1t5t13p145p126 − t7t10t16p124p156, p124p136 − t1t3t11p134p126 − t6t14t15p123p146, p125p236 − t1t4t5p235p126 − t10t15t16p123p256, p124p235 − t2t3t11p234p125 − t13t14t15p123p245, p235p346 − t12t13t14p345p236 − t2t10t11p234p356, p145p346 − t4t12t14p345p146 − t9t10t11p134p456, p146p256 − t7p246p156 − t1t5t8t9t11t13p126p456, p124p236 − t15p123p246 − t1t2t3t4t5t11p234p126, p125p136 − t1p135p126 − t6t7t10t14t15t16p123p156, p145p236 − t4X − t8t9t10t11t15t16p123p456, p136p245 − t6X − t1t3t8t11t12t16p126p345, p146p235 − t13X − t2t3t7t10t11t16p156p234, p135p256 − t8t11p125p356 − t4t5t6t7t14p235p156, p134p246 − t8t16p124p346 − t2t4t5t6t14p234p146, p246p356 − t3t16p346p256 − t5t6t9t13t14p236p456, p135p245 − t5t6p235p145 − t3t8t11t12t16p125p345, p136p246 − t6t14p236p146 − t1t3t8t11t16p126p346, p145p246 − t4t14p245p146 − t8t9t10t11t16p124p456, p125p246 − t10t16p124p256 − t1t4t5t13t14p245p126, p135p236 − t4t5p235p136 − t8t10t11t15t16p123p356, p135p146 − t5t13p145p136 − t3t7t10t11t16p134p156, p124p135 − t3t11p134p125 − t5t6t13t14t15p123p145, p135p346 − t10t11p134p356 − t4t5t12t13t14p345p136, p235p246 − t13t14p245p236 − t2t3t10t11t16p234p256, p245p346 − t12p345p246 − t2t5t6t9t10t11p234p456, p145p356 − t9p135p456 − t3t4t7t12t14t16p345p156, p134p235 − t2p234p135 − t8t12t13t14t15t16p123p345, p124p356 − t3Y − t5t6t9t13t14t15p123p456, p134p256 − t8Y − t2t4t5t6t7t14p156p234, p346p125 − t10Y − t1t4t5t12t13t14p126p345, p125X − t1t5p126p145p235 − t7t10t14t15t16p123p156p245, p145Y − t9t11p125p134p456 − t4t7t12t14t16p124p156p345, 44 L. Bossinger, F. Mohammadi and A. Nájera Chávez p124X − t14t15p123p146p245 − t1t2t3t5t11p126p145p234, p236Y − t15t16p123p256p346 − t1t2t4t5t11p126p234p356, p134X − t2t5p136p145p234 − t8t12t14t15t16p123p146p345, p235Y − t2t11p125p234p356 − t12t13t14t15t16p123p256p345, p246X − t14p146p236p245 − t1t2t3t5t8t9t10t 2 11t16p126p234p456, p246Y − t16p124p256p346 − t1t2t4t25t6t9t11t13t14p126p234p456, p256X − t7t14p156p236p245 − t1t5t8t9t11p126p235p456, p146Y − t7t16p124p156p346 − t1t5t9t11t13p126p134p456, p346X − t12t14p146p236p345 − t2t5t9t10t11p136p234p456, p136Y − t1t11p126p134p356 − t6t7t14t15t16p123p156p346, p356X − t5t9p136p235p456 − t3t7t12t14t16p156p236p345, p245Y − t12t16p124p256p345 − t2t5t6t9t11p125p234p456, p135X − t5p136p145p235 − t3t7t8t10t11t12t14t15t 2 16p123p156p345, p135Y − t11p125p134p356 − t4t5t6t7t12t13t 2 14t15t16p123p156p345. The lifts of f , g (the elements of GC(Iex) that do not correspond to exchange relations) are: f̃ = p135p246 − t2t3t4t5t6t7t10t11t14t16p156p234 − t3t8t10t11t16Y − t5t6t8t9t10t11t13t14t15t16p123p456 − t4t5t6t13t14X − t1t3t4t5t8t11t12t13t14t16p126p345, g̃ = XY − t7t12t14t15t16p123p156p246p345 − t1t2t5t9t11p126p135p234p456 − t1t2t3t4t5t7t11t12t14t16p126p156p234p345 − t2t5t6t7t9t10t11t14t15t16p123p156p234p456 − t1t5t8t9t11t12t13t14t15t16p123p126p345p456. Acknowledgements This work was partially supported by CONACyT grant CB2016 no. 284621. L.B. was sup- ported by “Programa de Becas Posdoctorales en la UNAM 2018” Instituto de Matemáticas, Universidad Nacional Autónoma de México. F.M. thanks the Instituto de Matemáticas, UNAM Unidad Oaxaca for their hospitality during this project and also acknowledges partial supports by the EPSRC Early Career Fellowship EP/R023379/1, the Starting Grant of Ghent University BOF/STA/201909/038, and the FWO grants (G023721N and G0F5921N). 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institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
issn	1815-0659
language	English
last_indexed	2026-03-12T14:01:37Z
publishDate	2021
publisher	Інститут математики НАН України
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spelling	Bossinger, Lara Mohammadi, Fatemeh Nájera Chávez, Alfredo 2025-12-30T15:56:47Z 2021 Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras. Lara Bossinger, Fatemeh Mohammadi and Alfredo Nájera Chávez. SIGMA 17 (2021), 059, 46 pages 1815-0659 2020 Mathematics Subject Classification: 13F60; 14D06; 14M25; 14M15; 13P10 arXiv:2007.14972 https://nasplib.isofts.kiev.ua/handle/123456789/211364 https://doi.org/10.3842/SIGMA.2021.059 Let 𝑉 be the weighted projective variety defined by a weighted homogeneous ideal 𝐽 and 𝐶 a maximal cone in the Gröbner fan of 𝐽 with 𝑚 rays. We construct a flat family over 𝔸ᵐ that assembles the Gröbner degenerations of 𝑉 associated with all faces of 𝐶. This is a multi-parameter generalization of the classical one-parameter Gröbner degeneration associated with 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 𝑋C (the toric variety associated to 𝐶) along the universal torsor 𝔸ᵐ → 𝑋C. We apply this construction to the Grassmannians Gr(2, ℂⁿ) with their Plücker embeddings and the Grassmannian Gr(3, ℂ⁶) with its cluster embedding. In each case, there exists a unique maximal Gröbner cone whose associated initial ideal is the Stanley-Reisner ideal of the cluster complex. We show that the corresponding cluster algebra with universal coefficients arises as the algebra defining the flat family associated with this cone. Further, for Gr(2, ℂⁿ), we show how Escobar-Harada's mutation of Newton-Okounkov bodies can be recovered as a tropicalized cluster mutation. This work was partially supported by CONACyT grant CB2016 no. 284621. L.B. was supported by Programa de Becas Posdoctorales en la UNAM 2018, Instituto de Matemáticas, Universidad Nacional Autonoma de México. F.M. thanks the Instituto de Matemáticas, UNAM Unidad Oaxaca, for their hospitality during this project and also acknowledges partial support by the EPSRC Early Career Fellowship EP/R023379/1, the Starting Grant of Ghent University BOF/STA/201909/038, and the FWO grants (G023721N and G0F5921N). L.B. would like to thank Kiumars Kaveh for providing an opportunity to present this work, and further Nathan Ilten for his insightful comments. A special thanks goes to Christopher Manon, who patiently explained his joint work with Kaveh to us. The connection between cluster mutation and Escobar-Harada's flip map was first observed in a discussion with Megumi Harada during the MFO-Workshop on Toric geometry in 2019, see [10]. We would like to thank the organizers of the meeting and, in particular, Megumi Harada for explaining to us the results of [19]. We are grateful to Brenda Policarpo Sibaja and Nathan Ilten for pointing out misprints in an earlier version of this paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras Article published earlier
spellingShingle	Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras Bossinger, Lara Mohammadi, Fatemeh Nájera Chávez, Alfredo
title	Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras
title_full	Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras
title_fullStr	Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras
title_full_unstemmed	Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras
title_short	Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras
title_sort	families of gröbner degenerations, grassmannians and universal cluster algebras
url	https://nasplib.isofts.kiev.ua/handle/123456789/211364
work_keys_str_mv	AT bossingerlara familiesofgrobnerdegenerationsgrassmanniansanduniversalclusteralgebras AT mohammadifatemeh familiesofgrobnerdegenerationsgrassmanniansanduniversalclusteralgebras AT najerachavezalfredo familiesofgrobnerdegenerationsgrassmanniansanduniversalclusteralgebras

Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras

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