Toric Flat Families, Valuations, and Tropical Geometry Over the Semifield of Piecewise Linear Functions

Home , Map of lattices

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Acknowledgement: We would like to thank Laura Escobar and Megumi Harada for pa- tiently reading the paper and giving valuable comments.

1.1. Classification of toric flat families. By a flat toric family we mean a flat family π : X → Y (Σ) equipped with an action of the torus TN lifting its action on the toric variety Y (Σ). Roughly speaking, we view the coordinate sheaf A of a flat toric family π : X → Y (Σ) as a toric vector bundle over Y (Σ). Throughout we assume that X comes equipped with a Gm(k) action corresponding to a positive grading A = n≥0 An. Klyachko’s result tells us that A corresponds to a filtered vector space A. In ourL setting, the algebra operation on A induces an algebra structure on A, and under this correspondence the filtrations can be seen to define a quasivaluation v : A → OΣ, where OΣ is the semifield of piecewise-linear functions supported on |Σ| and equipped with the tropical operations of pointwise min and +. In particular, for each φ ∈ OΣ there is a finite subdivision of |Σ| into convex rational cones σ such that the restriction φ|σ is linear. When |Σ| = NQ we denote this semifield by ON . When the family X is finite-type, the information in v can always be captured on its values on a certain set of generators B⊂ A. In sympathy with the case when v takes values in a totally ordered group (see [KM19]), we call B a Khovanskii basis of v.

Theorem 1.1. Finite type affine toric flat families X over Y (Σ) are classified by quasivaluations v : A → OΣ with a finite Khovanskii basis such that for each σ ∈ Σ, the restriction v|σ has a linear adapted basis given by a basis of standard monomials in the coordinate ring A of a general fiber of X . Moreover, a family will have reduced and irreducible fibers if and only if the corresponding quasivaluation is a valuation.

See Section 3.5 for the notion of linear adapted basis. Let I be the ideal of forms which present A by B. In Section 4 we define a certain subfan K(I) of the Gröbner fan Σ(I) of I. We define an apartment in K(I) to be a subcomplex τ ∩ K(I), where τ is a maximal face of the Gröbner fan of I. For this and the notion of Σ-adapted map, see Section 4.

Corollary 1.2. [Proposition 4.11] The information of a toric flat family π : X → Y (Σ) with Khovanskii basis B is equivalent to giving a Σ-adapted piecewise-linear map Φv :Σ → K(I). 2 In this way, different ways to fold a fan Σ into K(I) correspond to distinct toric flat families. Moreover, in Section 4 (Proposition 4.11) we show that Φv is captured by its values on the rays of Σ, which we organize into a matrix D(Φv) called the diagram of v. This gives a method for constructing flat toric families over Y (Σ) when Σ is a simplicial fan: for every ̺ ∈ Σ(1) choose a point in w(̺) ∈ K(I) such that all the rays from a face σ ∈ Σ lie in a common apartment of K(I). When v is a valuation, Φv takes values in the tropical variety Trop(I). This happens when the toric family is a toric vector bundle.

Theorem 1.3. Toric vector bundles over Y (Σ) correspond to valuations v : Sym(E) → OΣ ∗ such that for each σ ∈ Σ, the restriction v |σ: Sym (E) → Oσ has a Khovanskii basis given by a basis Bσ ⊂ E. Equivalently, one can think of the information of a toric vector bundle as a point Φ ∈

TropOΣ (L) in the tropical variety over the semifield OΣ, as a piecewise linear map from Σ to Trop(L), or as a matrix with rows in Trop(L). It is well-known that line bundles over Y (Σ) are classified by integral piecewise-linear maps φ :Σ → Z; Corollary 1.2 extends this description to toric vector bundles: Φ : Σ → Trop(L), and toric flat families: Φ : Σ → K(I). 1.2. Global sections and strong Khovanskii bases. Let A be a sheaf of algebras on a smooth toric variety Y (Σ) defined by a quasivaluation v : A → OΣ. In Section 5 we consider v 0 the ring R(A, ˆ)= L∈P ic(Y (Σ)) H (Y (Σ), L⊗Y (Σ) A) of global sections of A. In the case of a toric vector bundleL E, this ring corresponds to the Cox ring R(E) of the associated projective bundle PE. We wish to give conditions on Φ : Σ → K(I), or equivalently the diagram D(Φ), which ensure that R(A, vˆ) is finitely generated. A generating set B ⊂ A is said to be a strong Khovanskii basis if it is the image of a generating set of R(A, vˆ) under a natural quotient map defined in Section 5. A strong Khovanskii basis is always a Khovanskii basis, but not vice-versa, and in general finite strong Khovanskii bases do not exist. Our first contribution to the problem of strong Khovanskii bases is Algorithm 5.6, which constructs a finite strong Khovanskii basis if one does exist. This should be compared to [KM19, Algorithm 2.18]. In particular, when applied to the Cox ring R(E) of a toric vector bundle, Algorithm 5.6 will terminate in finite time if and only if the Cox ring R(E) is finitely generated. In this case, the constructed set B ⊂ Sym(E) will contain as its degree 1 piece a representation of the matroid defined by DiRocco, Jabbusch, and Smith in [DRJS18]. We mention the interesting work of Nødland [Nod], where it is shown that a generating set of R(E) can be taken to be a (possibly infinite) union of (representations of) DiRocco-Jabbusch-Smith matroids. More generally, we characterize the existence of finite strong Khovanskii bases for A with reduced and irreducible fibers (equivalently, for valuations v : A → OΣ). Let n = |Σ(1)| be the number of rays of Σ.

Theorem 1.4. Let (A, v) be a valuation as above with Khovanskii basis B⊂ A, and let IB be v ¯ ¯ ± ± −1 ˆ(bj ) the kernel of the map k[X, Y ] → A⊗k k[t1 ,...,tn ], where Xi → 1⊗ti and Yj → bj ⊗t , then B is a strong Khovanskii basis if and only if the ideal hIB,Xii is prime for all i ∈ [n]. We also give a characterization of strong Khovanskii bases in terms of prime cones in the tropical variety Trop(IB) (see [KM19]). In particular, we show that B is a strong Khovanskii basis if and only if the prime cones containing the rows of the diagram D(Φ) in Trop(I) lift to prime cones in Trop(IB) under a natural projection map π : Trop(IB) → Trop(I). We use Theorem 1.4 to study the Mori dream space property of projectivized toric vector bundles in several cases. When I is a generic linear ideal and the toric variety is projective space, we find a large family of diagrams whose associated bundles PE are Mori dream 3 spaces (Corollary 5.11). We give a classification of those diagrams whose associated bundles are Mori dream spaces when I is generated by a linear hypersurface (Corollary 5.10). We also recover finite generation results of Cox rings of projectivized toric vector bundles from the literature ([Gon], [GHPS], [HS], [Nod] and Corollary 5.9). In Section 5.4 we introduce the set ∆(Σ, Trop⋆(I)) (Definition 5.16) of diagrams D(Φ) for bundles over Y (Σ) with fiber E and Khovanskii basis B ⊂ Sym(E). We show that ∆(Σ, Trop⋆(I)) is the support of a polyhedral fan, and that the subset of diagrams for which B is a strong Khovanskii basis is a union of polyhedral cones (Proposition 5.20).

Conjecture 1.5. The set ∆(Σ, Trop⋆(I)) is the support of a fan F with the property that the set of those diagrams corresponding to bundles for which B is a strong Khovanskii basis is a union of faces of F.

1.3. Tropical geometry over ON . Let N and M be dual lattices as above, and let QM be the ﬁeld of rational functions on the torus TN .

m Deﬁnition 1.6. For p(t)= Cmt ∈ k[TN ] let wN (p) = min{m | Cm 6=0}. In particular wN (p) is the support functionP of the Newton polytope of p. Let wN : QM → ON denote w w p w w the corresponding extension of N to the ﬁeld of quotients: N ( q )= N (p) − N (q).

Well-known properties of Newton polytopes imply that wN respects multiplication: wN (pq)= wN (p)+ wN (q), and addition: wN (p + q) ≥ min{wN (p), wN (q)}. In particular, wN is a valuation from QM onto ON , and the image of k[TN ] under wN is precisely the set of support functions for polyhedra with vertices in M. In [SS09] it was suggested by Sturmfels and Speyer that one could develop tropical geometry over a tropical semiring of polyhedra; we view Theorem 1.3 as a first step in this direction: the points on tropical linear spaces over ON correspond to toric vector bundles. In Section 6 we explore this perspective further by showing that the operation of tropicalizing the solution to a system of equations over QM can be used to define toric vector bundles and flat toric families. Asking if all such bundles arise in this way is the analogue of asking if the Fundamental Theorem of Tropical Geometry [MS15] holds over ON .

1.4. The category of prevalued vector spaces. We let OˆΣ denote a semiring which consists of certain limits of elements of OΣ (see Section 3). In Section 3 we introduce the category VectΣ of vector spaces equipped with a prevaluation into OˆΣ. For the technical core of the paper, we show that there is an adjoint pair of functors L, R which relate VectΣ M to the category ShY (Σ) of TN -equivariant sheaves on the toric variety Y (Σ). Locally for M σ ∈ Σ, the operation R : Vectσ → ShY (σ) is akin to taking the Rees algebra of a filtration, M and L : ShY (σ) → Vectσ is its left adjoint. This is related to work of Perling [Per04] and Klyachko [Kly89] by the natural duality between filtrations and prevaluations (see Section 2). For a general TN -sheaf F, the corresponding prevaluations may take values in functions from OˆΣ which have an infinite number of domains of linearity, or may take infinite values, depending on the algebraic properties of F. For nicely behaved sheaves such as those which are locally projective or locally free, there is no such issue and we get a prevaluation with values in OΣ. On these sheaves and a broader category of eversive sheaves, the functors L and R become an equivalence, and we obtain classification results for toric sheaves by prevaluations. 4 2. Background on valuations In this section we collect the basic definitions and background for prevaluations on vector spaces, and valuations and quasivaluations on algebras. We start by introducing the general notion of a prevaluation with values in a semilattice. The term prevaluation appears in [KK12, Section 2.1] where the notion of a prevaluation with values in a totally ordered group is defined. Let (O, ⊕, ⊗) be an idempotent semialgebra. This means that the operations ⊕ and ⊗ satisfy the same axioms as addition and multiplication in a ring, with the exception that there are not necessarily additive inverses, and a⊕a = a for all a ∈ O. Let Z = Z∪{∞}. All semialgebras we consider are sets of Z-valued functions on (subsets of) a lattice N. Addition ⊕ and multiplication ⊗ are taken to be the pointwise min and + of functions. The additive identity, also denoted ∞, is the function which assigns ∞ to every point. Any idempotent semialgebra O has an intrinsic partial ordering , where a b if a⊕b = b. We denote the neutral element with respect to ⊕ by ∞. Definition 2.1. For a k-algebra A, a quasivaluation v : A → O is a function which satisfies the following: (1) v(fg) v(f) ⊗ v(g), (2) v(f + g) v(f) ⊕ v(g), (3) v(C) = 0, for all C ∈ k \{0}, (4) v(0) = ∞. A quasivaluation v is said to be a valuation if v(fg) = v(f) ⊗ v(g). A prevaluation on a vector space E is a function which satisfies (2) − (4) above. Let A be a k-algebra with finite generating set B⊂ A with d = |B|. We let π : k[x] → A α be the associated presentation, with ideal IB = ker(π). For any monomial x ∈ k[x] with B α = (α1,...,αn) there is a function evxα : O → O defined by sending (ψ1,...,ψd) to d ⊗αi B i=1 ψi . Following [GG16, 5.1], the tropical variety TropO(IB) ⊂ O is defined to be m α(j) Lthe set of tuples (ψ1,...,ψd) such that for any polynomial j=1 Cj x ∈ IB we have: P

(1) evxα(j) = evxα(j) jM∈[m] j∈[Mm]\{i} for any i ∈ [m]. The following is a well-known relationship between valuations and the tropical variety, see [Pay09], [GG16]. Proposition 2.2. Let v : A → O be a valuation and B⊂ A a generating set, then the tuple v v B ( (b1),... (bd)) ∈ O is a point in the tropical variety TropO(IB).

When O = Z we can associate a natural ﬁltration {Gr(v) | r ∈ Z} to a prevalation v : E → Z, where Gr(v) = {f | v(f) ≥ r}. Similarly, we let G>r(v) = {f | v(f) > r}. The associated graded vector space grv(E) is deﬁned to be the direct sum:

(2) grv(E)= Gr(v)/G>r(v). Mr∈Z

If v : A → Z is a quasivaluation, it is straightforward to show that grv(A) is k-algebra. The function v can be recovered as v(f) = max{r | f ∈ Gr(v)}, where this is taken to be ∞ if the maximum is never attained. For the following see [KM19, Section 2.5]. Definition 2.3. A vector space basis B ⊂ E is said to be an adapted basis for a prevaluation v : E → Q¯ if B ∩ Gr(v) is a basis of Gr(v) for each r ∈ Z. 5 v k k v If bi,i =1,...k are from an adapted basis then ( i=1 Cibi)= ⊕i=1 (bi). This identity simplifies computations for prevaluations with adaptedP bases. We finish this section by recalling the notion of Khovanskii basis for a quasivaluation on an algebra. Khovanskii bases for general quasivaluations are the subject of [KM19]. Definition 2.4. Let A be a k-algebra, and v : A → Z be a quasivaluation. A subset B⊂ A is said to be a Khovanskii basis of v if the equivalence classes B¯ ⊂ grv(A) generate grv(A) as a k-algebra.

3. The category VectΣ and the functors L and R

Recall that OΣ is the set of piecewise linear functions with a ﬁnite number of linear domains deﬁned on the set Σ ∩ N, and OˆΣ is the of functions ψ : Σ ∩ N → Z such that ψ(ℓρ)= ℓψ(ρ) for all ℓ ∈ Z≥0 and ρ ∈ Σ ∩ N.

Deﬁnition 3.1. Let VectΣ be the category of pairs (E, v), where v : E → OˆΣ is a prevaluation over k. A morphism φ : (E, v) → (D, w) of prevalued vector spaces is a k-linear map φ : E → D such that v(f) ≤ w(φ(f)) for all f ∈ E.

Strictly speaking, VectΣ only depends on the support of Σ. The fan structure of Σ will be used in Section 3.5 when it becomes necessary to single out special elements of OΣ ⊂ OˆΣ.

3.1. Constructions in VectΣ.

Definition 3.2. For (E, v), (D, w) ∈ VectΣ the direct sum is (E, v)⊕(D, w) = (E⊕D, v⊕w), where (v ⊕ w)(f + g)= v(f) ⊕ w(g) ∈ OˆΣ. The space (0, −∞) is the additive identity for ⊕, and it is straightforward to show that ⊕ is both a product and coproduct in VectΣ (see [nLa19a]). The direct sum operation allows us to extend the notion of adapted basis (Definition 2.3) to the context of prevaluations into OˆΣ. For a vector space E and ψ ∈ OˆΣ we let (E, ψ) ∈ VectΣ denote the vector space with prevaluation which assigns every non-zero element of E the function ψ. Definition 3.3. We say (E, v) has an adapted basis B ⊂ E if the natural maps (k, v(b)) → v v ∼ v (E, ), 1 → b define an isomorphism (E, ) = b∈B(k, (b)). This is said to be a linear adapted basis if v(b) ∈ M ⊂ OΣ ⊂ OˆΣ for each bL∈ B. The notions of epimorphism, monomorphism, kernel, and cokernel all make sense in VectΣ, but it is not always the case that a monomorphism is the kernel of a map. As a consequence, VectΣ is not an abelian category.

Deﬁnition 3.4. Let (E, v) ∈ VectΣ and π : E → D be a surjection. The pushforward prevaluation π∗(v): D → OˆΣ is deﬁned by the formula π∗(v)(g)(ρ) = max{v(f)(ρ) | π(f)= g}, where it is understood that π∗(v)(g)= ∞ if the maximum is never attained. Similarly, given φ : F → E, we have the pullback φ∗(v), where φ∗(v)(f)= v(φ(f)).

A morphism φ : (E, v) → (D, w) is the kernel of a morphism in VectΣ if and only if it is a kernel in Vectk and v = φ∗(w). It is a cokernel if and only if it is a cokernel in Vectk and w = φ (v). The colimit lim(E , v ) of a diagram I is constructed by taking lim v : lim E → ∗ −→ i i −→ i −→ i ˆ v OΣ to be the pushforward of the prevaluation on i∈I (Ei, i) under the quotient map π : E → lim E . In particular, [lim v ](f) = maxL{⊕ℓ v (f ) | ℓ π (f )= f}. I i∈I i −→ i −→ i i=1 i i i=1 I i L 6 P v ˆ ρ v A prevaluation : E → OΣ is captured by a family of subspaces Gr ( ) ⊂ E. For ρ ∈ Σ ∩ N and r ∈ Z let: ρ v v (3) Gr ( )= {f | (f)(ρ) ≥ r}. v ρ v We have (f)(ρ) = max{r | f ∈ Gr ( )}.

Deﬁnition 3.5. The tensor product (E1, v1)⊗(E2, v2) is the pair (E1 ⊗k E2, v1 ⋆v2), where v1 ⋆ v2 : E1 ⊗k E2 → OˆΣ is deﬁned by the following spaces:

ρ v v ρ v ρ v (4) Gr ( 1 ⋆ 2)= Gs ( 1) ⊗k Gt ( 2) ⊂ E1 ⊗k E2, s+Xt=r v v ρ v v in particular 1 ⋆ 2(f)(ρ) = max{r | f ∈ Gr ( 1 ⋆ 2)}.

The object (k, 0) is the “multiplicative identity” in VectΣ. Moreover, for any simple tensor x ⊗ y ∈ E1 ⊗ E2 we have (v1 ⋆ v2)(x ⊗ y)= v1(x)+ v2(y). It is straightforward to show that a commutative algebra object (A, v) ∈ VectΣ is the same information as a commutative k-algebra A equipped with a quasivaluation v : A → OˆΣ. We let AlgΣ denote the category of commutative algebra objects in VectΣ. Tensor product also allows us to make sense of Schur functors in VectΣ.

Deﬁnition 3.6. Let λ be a partition of [n]. The Schur functor Sλ : VectΣ → VectΣ takes ⊗n a prevalued vector space (E, v) to (Sλ(E),sλ(v)) ⊂ (E, v) , where sλ(v) is the pullback of ⋆n ⊗n v under the inclusion map Sλ(E) ⊂ E .

Showing that Sλ : VectΣ → VectΣ is a functor is straightforward. Moreover, the categories we deal with in this paper are symmetric monoidal and Cauchy-complete (see [nLa19b]). It follows that the functor Sλ makes sense in any of these categories, and all strictly monoidal functors we consider commute with any Sλ.

3.2. The functors L and R. A strongly convex rational polyhedral cone σ ⊂ NQ deter- ∨ ∨ mines an affine semigroup σ ∩ M, where σ = {u | hρ,ui ≤ 0 ∀ρ ∈ σ} ⊆ MQ. We let S0 = k[M] be the Laurent polynomial ring determined by M, and Sσ ⊂ S0 denote the affine semigroup algebra associated to σ∨. Remark 3.7. Our definition for the dual cone σ∨ is the negative of the convention found in the literature on toric varieties (e.g [CLS11] and [Stu96]). This is to conform with the min convention for valuations and tropical geometry.

M Let ModSσ denote the category of M-graded Sσ-modules. In what follows we set up the M M functors R : Vectσ → ModSσ and L : ModSσ → Vectσ, and show that they are adjoint (Theorem 3.17). The functor R is recognizable as taking the Rees space (algebra) of a ﬁltration, so L can be characterized as the left adjoint of the Rees construction. ∨ Fix a module R = m∈M Fm(R). For u ∈ σ ∩ M we can view the element χu ∈ Sσ as a family of linear maps χLm,u : Fm(R) → Fm+u(R). In this way R determines a directed system in the category of k-vector spaces (this is a central observation of [Per04], [KS98]). We let E = lim F (R) be the colimit of this system. There is a natural surjection φ : R → E , R −→ m R R and for each m ∈ M there is a subspace φR(Fm(R)) ⊂ ER. Let m ⊂ Sσ be the maximal ∨ ideal generated by the forms χu − 1 for all u ∈ σ ∩ M. Similarly, let m0 ⊂ S0 be the ideal generated by χu for all u ∈ M. 7 Proposition 3.8. The map φR induces an isomorphism ER =∼ R/mR. Furthermore, the ∼ natural map i : R → R ⊗Sσ S0 induces an isomorphism of vector spaces R/mR = R ⊗Sσ

S0/m0R ⊗Sσ S0, and R ⊗Sσ S0 is isomorphic to the induced module ER ⊗k S0.

Proof. The second part follows easily from the flatness of S0 as an Sσ module. To see the first part, note that the vector space ER is defined to be the quotient of R by the ∨ span of the forms {f − g | f ∈ Fm,g ∈ Fn, ∃u, v ∈ σ ∩ M | χuf = χvg}. We have f − g = (f − χuf) − (g − χvg)= f(1 − χu) − g(1 − χv) ∈ mR. Moreover, mR is the span of the forms f − χvf, so for any such element we have (χv)f = (χ0)χvf ∈ R.

v ˆ ρ Now we define a function R : ER → Oσ. For ρ ∈ σ ∩ N and r ∈ Z let Gr (R) ⊂ ER be the sum: ρ (5) Gr (R)= φR(Fm(R)). hρ,mXi≥r ρ ρ Observe that Gr (R) ⊇ Gs (R) whenever r ≤ s, so these spaces define an decreasing filtration on ER for each ρ ∈ σ ∩ N. v v ρ Definition 3.9. For f ∈ ER, we let R(f): σ ∩N → Z be R(f)(ρ) = max{r | f ∈ Gr (R)}. M v Proposition 3.10. There is a functor L : ModSσ → Vectσ with L(R) = (ER, R).

Proof. We must check that vR(f) ∈ Oˆσ, it is then immediate that vR is a prevaluation. Fix v ρ ρ ℓρ ℓ ∈ Z>0, and suppose R(f)(ρ) = r, so f ∈ Gr(R) \ Gr−1(R). By deﬁnition, f ∈ Gℓr (R), ℓρ and if f ∈ Gℓr−1(R), then for some n1,...,nd we have f ∈ Fn1 (R)+ ··· + Fnd (R) where hℓρ,nii > ℓr. But then hρ,nii > r, which contradicts vR(f)(ρ)= r. For ψ : R1 → R2 there is an induced map L(ψ): ER1 → ER2 . For any m ∈ M the restriction map ψm : Fm(R1) → ρ ρ Fm(R2) satisﬁes φR ◦ ψm = L(ψ) ◦ φR. It follows that L(ψ) maps Gr(R1) into Gr (R2), so L(ψ) is a morphism in Vectσ.

Example 3.11. Consider the localization S ⊂ S for τ ⊂ σ a face. We have S = 1 S σ τ τ χu σ ∨ for some u ∈ σ ∩ M such that hρ,ui = 0 for all ρ ∈ τ and ESτ = k. If ρ ∈ τ then ∨ v hρ, vi ≤ 0 for all v ∈ τ , so Sτ (1)(ρ) = 0. But if ρ ∈ σ \ τ then hρ,ui > 0, so in this case v Sτ (1)(ρ)= ∞.

Example 3.12. Consider an Sσ-module R generated by f1,...,fN ∈ R with deg(fi)= λi. ℓ v ℓ If p = i=1 χui fi, then φR(p) = q ∈ φR(Fλi (R)), so R(q) ≥ i=1 λi. Furthermore, N ∨ v Fm(R) P6= 0 only if m ∈ i=1 λi +[σ ∩MP]. It follows that R(q) is theL maximum over a finite ℓ number of expressionsS of the form i=1 λi, and only a finite number of such expressions are possible. As a consequence, we seeL that the image vR(ER) ⊂ Oσ is a finite set. ′ Let τ ⊂ NQ and σ ⊂ NQ be pointed polyhedral cones, and let ι : τ → σ be a linear map induced by a map of lattices ι : N ′ → N. There is an associated map on dual lattices ∗ ′ ∗ ∗ ι : M → M and semigroup algebras ι : Sσ → Sτ . The map ι gives an extension functor ′ M M † −⊗Sσ Sτ : ModSσ → ModSτ . We also have a functor ι : Vectσ → Vectτ obtained by ♯ composing v : E → Oˆσ with the map on semialgebras ι : Oˆσ → Oˆτ given by precomposition with ι. We show that these two functors coincide under L. As a consequence, the restriction v of L(R) = (ER, R) to a facet τ ⊂ σ is L(R ⊗Sσ Sτ ). Proposition 3.13. The following diagram commutes. 8 ′ M −⊗Sσ Sτ M ModSσ −−−−−−→ ModSτ

Lσ Lτ    ι†  Vecty σ −−−−→ Vecty τ

′ ′ ′ ∼ ′ Proof. Let R = R ⊗Sσ Sτ and R0 = R ⊗Sσ k[M]. We have R ⊗Sτ k[M ] = R0 ⊗k[M] k[M ]. ′ ′ ∗ By setting χm′ = 1 for each m ∈ M we obtain ER =∼ ER′ , so φR′ ◦ (ι ⊗ 1) = φR. Now ′ ι(ρ) let ρ ∈ τ ∩ N , and consider Gr (R) = hι(ρ),mi≥r φR(Fm(R)). This space is the image ρ ′ ′ of hι(ρ),mi≥r Fm(R) ⊂ R under φR. Similarly,P Gr (R ) is the image of hρ,mi≥r Fm(R ) underL φR′ where L

′ (6) Fm(R ) = ( Fn(R) ⊗k kχu)/ ∼ . n∈M,u∈τ ∨∩MM ′,ι∗(n)+u=m ∗ ′ We have hι(ρ),mi ≥ r if and only if hρ, ι (m)i ≥ r, so φR(Fm(R)) ⊂ φR′ (Fι∗(m)(R )) and ι(ρ) ρ ′ ∼ ρ ′ Gr (R) ⊆ Gr (R ) in ER = ER′ . Similarly, if f ∈ Gr (R ) then we can write f = φR(gi), ′ ∗ ∗ where gi ∈ φR(Fni (R)) = φR ((ι ⊗ 1)(Fni (R) ⊗k kχu)), where ι (ni) + u =Pmi with ∗ ι(ρ) hρ,mii≥ r. But then hρ, ι (ni)i≥ r, so that hι(ρ),nii≥ r. It follows that f ∈ Gr (R) and ι(ρ) ρ ′ ′ ♯v v Gr (R)= Gr(R ). Now ﬁx ρ ∈ τ ∩ N and f ∈ AR, then ι R(f)(ρ)= R(f)(ι(ρ)), which † ♯ is equal to vR′ (f)(ρ) by the above calculation. This shows that ι ◦ Lσ(R) = (ER, ι vR) is ′ v ′ ′ isomorphic to (ER , R )= Lτ ◦ (R ⊗Sσ Sτ ) by the map which identiﬁes ER with ER .

For any (E, v) ∈ Vectσ and m ∈ Mσ we can consider the space Fm(v)= {f | v(f) ≥ m}. ∨ If u ∈ σ ∩ M then m ≥ m + u in Oˆσ, so Fm(v) ⊆ Fm+u(v). v v M Deﬁnition 3.14. For (E, ) an object of Vectσ we deﬁne the Rees module R(E, ) ∈ ModSσ to be the following M-graded vector space:

(7) R(E, v)= Fm(v). mM∈M

We let χu ∈ Sσ act by the inclusion map: Fm(v) ⊆ Fm+u(v). M Proposition 3.15. The Rees module construction extends to a functor R : Vectσ → ModSσ .

Proof. It remains to deﬁne the morphism R(ψ): R(E1, v1) → R(E2, v2) associated to ψ : (E1, v1) → (E2, v2). We have v1(f) ≥ m implies v2(ψ(f)) ≥ v1(f) ≥ m, so ψ(Fm(v1)) ⊂ ∨ Fm(v2). For any u ∈ σ ∩ M there is a commuting square:

χm,u Fm(v1) −−−−→ Fm+u(v1)

ψ ψ    χm,u  Fmy(v2) −−−−→ Fm+yu(v2) M ′ ′ So R(ψ) is a morphism in ModSσ . It is straightforward to show R(ψ◦ψ )= R(ψ)◦R(ψ ). v v Example 3.16. We compute R(k, Sτ ) for Sτ from Example 3.11. We must determine if v Sτ (1) ≥ u for each u ∈ M. This inequality always holds for ρ∈ / τ, therefore it holds every- v where if and only if hρ,ui≤ 0 for all ρ ∈ τ. It follows that R(k, Sτ )= u∈τ ∨∩M kχu = Sτ . 9 L Now we show that L and R form an adjunction.

M Theorem 3.17. The (bi)functors HomMod (−, R(−)) and HomVectσ (L(−), −) are natu- Sσ rally isomorphic.

Recall the universal property of colimit: if we are given a system of maps ψm : Fm(R) → E which commute with χ for u ∈ σ∨ ∩M, then there is a unique map ℓ(ψ) = lim ψ : E → E u −→ m R such that ψm = (ℓ(ψ) ◦ φR)m.

Proof. Let φ : L(R) → (E, v) be a morphism in Vectσ. The image φR(Fm(R)) is a subspace of Fm(vR), so we let r(φ): R → R(E, v) be deﬁned by the maps r(φ)m = φm ◦φR : Fm(R) → Fm(v). The following diagram commutes:

φR φm Fm(R) −−−−→ Fm(vR) −−−−→ Fm(v)

χu     φR  φm+u  Fm+yu(R) −−−−→ Fm+yu(vR) −−−−→ Fm+yu(v), as the right two vertical arrows are inclusions, with the middle occuring in the colimit ER. v As a consequence, r(φ) ∈ HomModM (R, R(A, )). Sσ Given ψ : R → R(E, v), we have a family of maps ψm : Fm(R) → Fm(v) which commute with each χu. We get ℓ(ψ): ER → E, where ψm = ℓ(ψ) ◦ φR on Fm(R). Furthermore, ρ v ρ v v Gr (R) is mapped into hρ,mi≥r Fm( ) ⊂ Gr ( ) so ℓ(ψ) ∈ HomVectσ (L(R), (E, )). Now we show that Pℓ and r are inverses of each other. Given φ : L(R) → (E, v) and f ∈ ER, pick fˆ ∈ Fm(R) so that φR(fˆ) = f. Then r(φ)(fˆ) = φ ◦ φR(fˆ) = φ(f) ∈ Fm(v). It follows that ℓ(r(φ))(f) = φ(f). If ψ : R → R(E, v), ψm = (ℓ(ψ) ◦ φR)m, so r(ℓ(ψ))m = (ℓ(ψ)◦φR)m = ψm. We omit the proof that r and ℓ are natural, as it is straightforward.

3.3. Eversive objects. For (E, v) ∈ Vectσ, the space LR(E, v) has underlying space lim F (v) ⊆ E and prevaluation deﬁned by the spaces Gρ(R(E, v)) = F (v) ⊂ −→ m r hρ,mi≥r m ρ v v v Gr ( ). It follows that there is a map η : LR(E, ) → (E, ). The objectP RL(R) is the v v ρ M-graded module m∈M Fm( R), where Fm( R) = ρ∈σ∩N Ghρ,mi(R). There is a map ǫ : R → RL(R) whichL takes Fm(R) to φR(Fm(R)) ⊆ FTm(vR).

Example 3.18. Let I ⊂ Sσ be an M-graded ideal spanned by the characters Ω(I) ⊂ M, then RL(I) ⊂ Sσ is the ideal with Ω(RL(I)) = conv(Ω(I)) ∩ M, i.e. the lattice points in the convex hull of Ω(I). v ρ v Definition 3.19 (Eversive objects). We say that (E, ) ∈ Vectσ is eversive if Gr(R(E, )) = ρ v M ∼ v ev Gr ( ). We say that R ∈ ModSσ is eversive if Fm(R) = Fm( R). We let Vectσ ⊂ Vectσ and M,ev M ModSσ ⊂ ModSσ denote the full subcategories on eversive objects. Proposition 3.20. M v v For any R ∈ ModSσ and (E, ) ∈ Vectσ, both R(E, ) and L(R) are eversive. Furthermore, ǫ and η are isomorphisms if and only if the corresponding objects ev ∼ M,ev are eversive. The functors R and L define an equivalence Vectσ = ModSσ . M M M We let AlgSσ be the category of commutative algebras in ModSσ . If R ∈ AlgSσ then R/mR =∼ ER is a k−algebra, and vR is a quasivaluation on ER. Similarly, it is easy to check v v that R(A, ) is an Sσ-algebra if (A, ) ∈ Algσ. We obtain a slightly stronger result below. Recall that a functor F : (C, ⊗) → (D, ⊠) between symmetric monoidal categories is said to 10 be strictly monoidal if F (A⊗B) is naturally isomorphic to F (A)⊠F (B). It is said to be lax- monoidal if there is a natural transformation (of bifunctors) m : F (A) ⊠ F (B) → F (A ⊗ B). The following is a straightforward consequence of definitions. Proposition 3.21. The functor L is strictly monoidal, and the functor R is lax-monoidal. ev ∼ M,ev Furthermore, L and R define an equivalence Algσ = AlgSσ . M 3.4. Global equivalence. Now we consider the category ShY (Σ) of TN -equivariant quasi- M coherent sheaves on the toric variety Y (Σ) (see [Per04]). If F ∈ ShY (Σ), then for any t ∈ TN ∗ ∗ there is an isomorphism φt : t F =∼ F, where t F is the pullback along the automorphism t : Y (Σ) → Y (Σ) defined by t, such that the following diagram commutes:

φt t t∗(t∗F) (t t )∗F 1 2 2 1 =∼ 1 2 F

∗ t t2φt1 φ 2

∗ t2F

M A morphism φ : E →F in ShY (Σ) is a map of sheaves which respects these diagrams. For a face σ ∈ Σ let Y (σ) ⊂ Y (Σ) be the corresponding open subvariety. By applying the M section functor Γ(Y (σ), −) we obtain objects and morphisms in ModSσ . M,ev M Deﬁnition 3.22. Let ShY (Σ) ⊂ ShY (Σ) be the full subcategory on quasicoherent sheaves F M,ev such that Γ(Y (σ), F) ∈ ModSσ for each σ ∈ Σ. M M,ev Theorem 3.23. There is a monoidal functor L : ShY (Σ) → VectΣ. Under L, ShY (Σ) is equivalent to a full subcategory of VectΣ.

Proof. Proposition 3.8 implies that the fiber EF over Spec(S0/m0) = id ∈ TN ⊂ Y (Σ) carries a prevaluation vF,σ : EF → Oˆσ for each face σ ∈ Σ. By Proposition 3.13, for v v any f ∈ EF , and τ = σ1 ∩ σ2, the restrictions F,σ1 (f) |τ and F,σ2 (f) |τ coincide. It follows that we obtain an object (EF , vF ) ∈ VectΣ. Proposition 3.8 also implies that a morphism φ : F → G gives a map L(φ)|σ : (EF , vF|σ) → (EG , vG|σ) whose underlying vector M space map does not depend on σ, so we have a functor L : ShY (Σ) → VectΣ. We observe M ([Sta19, Lemma 17.15.4]) that for any σ ∈ Σ, and F, E ∈ ShY (Σ), L(F ⊗Y (Σ) E) restricted to σ is L◦ Γ(Y (σ), E |Y (σ) ⊗Y (σ)F |Y (σ)) = L(Γ(Y (σ), E) ⊗Sσ Γ(Y (σ), F)) since E, F are quasicoherent and Y (σ) is affine. The latter is L(Γ(Y (σ), E)) ⊗ L(Γ(Y (σ), F)), so it follows that L(E ⊗Y (Σ) F)= L(E) ⊗ L(F). M,ev v v Now let F, G ∈ ShY (Σ), and consider a map ψ : (EF , F ) → (EG , G ). For any σ ∈ Σ we obtain a map on modules R(ψ|σ): R(EF , vF |σ) → R(EG , vG|σ), which in turn induces a map on quasicoherent Y (σ) sheaves: R˜(ψ|σ): R˜(EF , vF |σ) → R˜(EG , vG|σ). The sheaves M,ev ˜ v ˜ v F, G were chosen in ShY (Σ), so R(EF , F|σ) and R(EG , G|σ) coincide with the restrictions F |Y (σ) and G |Y (σ), and we obtain an induced map on descent data for the open cover of ˜ Y (Σ) by the Y (σ) for σ ∈ Σ: R(ψ): {(Y (σ), F |Y (σ)), σ ∈ Σ}→{(Y (σ), G |Y (σ)), σ ∈ Σ} 11 (see e.g. [Sta19, Lemma 68.3.4]). This map induces a unique map between F and G. It’s M,ev straightforward to check that this construction is inverse to L on ShY (Σ), in particular L(φ) : (EF , vF ) → (EG , vG ) is an isomorphism if and only if φ : F =∼ G. In Definition 3.22 we require that Γ(Y (σ), F) has the additional property that any localization Sτ ⊗Sσ Γ(Y (σ), F) for τ ⊂ σ a face is also an eversive module. Proposition 3.26 shows that this holds if F is locally free or projective. Additionally, the following proposition shows that this is the case for modules which are eversive and finitely generated or eversive and flat when it is combined with Propositions 3.34 and 3.28 Lemma 3.24. Let R be eversive with the property that F (R)= ℓ Gρi (R) for some m i=1 hρi,mi

ﬁnite set ρ1,...,ρℓ ∈ N ∩ σ, then Sτ ⊗Sσ R is eversive as well forT any face τ ⊂ σ. ′ Proof. If R is eversive, it is torsion free, and R = Sτ ⊗Sσ R is torsion-free as well. In this case, ∼ ′ ER = ER′ and the graded component Fn(R ) is isomorphic to the sum u∈τ ⊥∩N Fn+u(R) ρ ′ ′ in ER . We must show that the inclusion Fn(R ) ⊆ ρ∈τ∩N Ghρ,ni(R)P is an equality. For each facet containing τ we select a normal vector uj , andT we let u = uj. For all ρ ∈ τ ∩ N ρ we have hρ,ui = 0 and hρi,ui < 0 for each ρi ∈/ τ ∩ N. If f ∈ Pρ∈τ∩N Ghρ,ni(R) then vR(f)(ρi) ≥ hρi,ni for each ρi ∈ τ ∩ N. We select an appropriateT multiple su so that ′ vR(f)(ρi) ≥ hρi,n + sui for each ρi ∈/ τ ∩ N. It follows that f ∈ Fn+su(R) ⊂ Fn(R ).

By Proposition 3.34 it follows that if R is flat and eversive, Fm(R) is the intersection of the spaces Gρi (R) for ρ generators of the rays of σ. Similarly, if R is finitely generated hρi,mi i ℓ and eversive, Proposition 3.28 shows that there is a finite decomposition σ = i=1 σi such v that the restrction R(f)|σi is linear for each f ∈ ER, so we get a similar decompositionS in this case in terms of the generators of the rays of the σi. 3.5. Free and projective modules. It will be useful to know how the eversive condition behaves with respect to direct sum. We leave the following to the reader. M Proposition 3.25. The functors L and R respect direct sums. Furthermore, in ModSσ and Vectσ, a direct sum is eversive if and only if its components are eversive. Let P be a free M-graded S -module, then we can write P = S [b ] for B an σ bi∈B σ i M-homogeneous basis. We let deg(bi) = λi ∈ M. For any m ∈ M Lwe have a direct sum decomposition: F (P )= kχ [b ]. m ui+λi=m ui i Proposition 3.26. The functorsL L and R give an equivalence between M-homogeneous free modules and prevalued spaces with a linear adapted basis (Definition 3.3). Furthermore, if P = S [b ] is free with deg(b ) = λ , then P is eversive, L(P ) ∼ (k, λ ), and bi∈B σ i i i = bi∈B i ∨ Fm(PL) is isomorphic to the subspace of EP with basis Bm = {bi | m − λi L∈ σ ∩ M}. Proof. By Proposition 3.25 and Theorem 3.23 it suffices to note that if deg(b) = λ then L(Sσ[b]) = (k, λ) and ǫ : RL(Sσ [b]) =∼ Sσ[b]. M Propositions 3.26 and 3.25 imply that projective modules in ModSσ are eversive and correspond under L to summands of spaces with a linear adapted basis. We identify the category of TN -vector bundles on Y (Σ) with the full subcategory of locally-free TN -sheaves.

Corollary 3.27. The category of locally projective TN -sheaves on Y (Σ) is equivalent to the full subcategory of VectΣ on spaces (E, v) such that (E, v|σ) is a summand of space with a linear adapted basis for each σ ∈ Σ. The category of finite rank TN -vector bundles on 12 Y (Σ) is equivalent to the full subcategory of VectΣ on spaces (E, v) such that E is finite dimensional and (E, v|σ ) has a linear adapted basis for each σ ∈ Σ. Proof. If F is locally projective then Theorem 3.23 implies that it is classified by L(F) = (E, v). The latter must have the property that each restriction (E, v|σ) is a summand of a space with a linear adapted basis by Proposition 3.26. Given (E, v) with this property, v v v R(E, |σ) is always a projective module, so R(E, |τ ) = R(E, |σ) ⊗Sσ Sτ by Proposition 3.13. As in the proof of Theorem 3.23, we can glue the corresponding quasi-coherent sheaves R˜(E, v|σ ) to produce F, a locally projective on Y (Σ) such that L(F) = (E, v). Now, in the special case that E is the sheaf of sections of a finite rank vector bundle, Γ(Y (σ), E) is an M-homogeneous free module, so each (E, v|σ ) has a linear adapted basis.

Corollary 3.28. Let (E, v) ∈ VectN with E a ﬁnite dimensional vector space, then the following are equivalent.

(1) The image v(E \{0}) is a finite subset of ON \ {∞}. (2) There is a finite complete fan Σ ⊂ NQ with (E, v)= L(E) for E a toric vector bundle on Y (Σ). v M (3) There is a finite complete fan Σ ⊂ NQ with (E, )= L(F) for F ∈ ShY (Σ) coherent.

Proof. Given v : E → ON with v(E) ⊂ ON finite we find any fan Σ such that v(f)|σ∈ Mσ for all σ ∈ Σ and all f ∈ E. The image v|σ (E \{0}) ⊂ Oσ is then a finite set λ1,...,λk ∈ Mσ ⊂ Oσ. By construction, the partial order on Oσ must restrict to a total ordering on the ˆ λi. We let grσ(E) be the associated graded vector space, and we choose a basis Bσ ⊂ grσ(E). Any lift Bσ ⊂ E of this basis is adapted to v|σ, this proves (1) → (2). The case (2) → (3) is immediate, and (3) → (1) is a consequence of Example 3.12.

Example 3.29. If (E, v) and (F, w) are equipped with linear adapted bases B1 ⊂ E, B2 ⊂ F , one checks that B1 × B2 =∼ {b1 ⊗ b2 | bi ∈ B2}⊂ E ⊗k F gives a linear adapted basis ρ v ρ v ρ w ρ w as follows. For any ρ ∈ Σ ∩ N, Gs ( ) ∩ B1 ⊂ Gs ( ) and Gt ( ) ∩ B2 ⊂ Gt ( ) are bases, so ρ v w ρ v w v w Gs+t( ⋆ ) ∩ B1 × B2 is a basis of Gs+t( ⋆ ). This shows that B1 × B2 is adapted to ⋆ . Moreover, v ⋆ w(b1 ⊗ b2)= v(b1)+ w(b2) for any b1 × b2 ∈ B1 × B2, so B1 × B2 is linear.

Example 3.30. Example 3.29 allows us to find adapted bases of Sλ(E, v) for any Schur functor Sλ and (E, v) with a linear adapted basis B. For any λ we obtain a basis {...,bτ ,...} of Sλ(E) by applying the symmetrizers sτ corresponding to semi-standard fillings τ of λ to B. It is straightforward to check that if v is adapted to B ⊂ E, then every simple tensor of ⊗|λ| bτ has the same value. The simple tensors with entries in B are an adapted basis of E , so we conclude that sλ(v)(bτ ) is linear if v(b) is linear for all b ∈ B. For any ρ ∈ σ ∩ N, we v v have sλ( )ρ( Cτ bτ ) ≥ sλ( )ρ(bτ ). If the sum of the minimum contributions min Cτ bτ is non-zero, thenP some simpleL tensor in the elements of B in this sum achieves theP top value, so this is an equality. If min Cτ bτ = 0, then Cτ = 0, as the bτ form a basis. It follows that the bτ are an adapted basisP of Sλ(E, v). ∗ Example 3.31. For a TN -vector bundle E, the dual E corresponds to the locally free sheaf ∗ r ∗ ∗ ∗ with Γ(E , Y (σ)) = i=1 Sσ[bi ], where deg(bi ) = − deg(bi) = −λi. We have L(E ) = ∗ v∗ ∗ v∗ r (E , ), where locallyL (E , |σ)= i=1(k, −λi). 3.6. Flat modules. Our interest inL flat families leads us to consider the image L(R) of a flat Sσ-module R. We make use of Lazard’s Theorem ([Sta19, Theorem 10.80.4]). M Proposition 3.32. Let R ∈ ModSσ , then the following are equivalent:

(1) The functor −⊗Sσ R is exact. 13 (2) R is a direct limit of M-homogeneous free modules. (3) For any φ : P → R, where P is M-homogeneous and free, and any p ∈ ker(φ), there are maps φ′ : P ′ → R and π : P → P ′ such that φ = φ′ ◦ π with P ′ M-homogeneous and free, and p ∈ ker(π). M For module R ∈ ModSσ there are two k-vector spaces associated to a point ρ ∈ σ∩N. We have the ﬁber over a point in the orbit corresponding to the face τ containing ρ in its relative ⊥ interior by taking the quotient R/mτ R, where mτ = h{χu − 1,χv | u ∈ τ , hρ, vi < 0 ∀ρ ∈ ρ ρ τ}i. Alternatively, we have the associated graded space grρ(ER) = r∈Z Gr (R)/G>r(R). These two spaces are naturally isomorphic if R is ﬂat. L ˆ ρ ρ Proposition 3.33. The map φρ : R → grρ(ER) sending Fm(R) to Ghρ,mi(R)/G

χvj gj + (χui − 1)hi for hρ, vj i < 0 and hρ,uii = 0, where gj and hi are homogeneous. ˆ ρ PLet deg(g)P = m − v so that χvg ∈ Fm(R), then φρ(χvg) ∈ φR(Fm−v(R)) ⊂ Ghρ,mi(R). It ˆ ˆ ˆ M follows that φρ(χvg) = 0. Similarly, since hρ,ui = 0, φρ(χuh) = φρ(h). If R ∈ AlgSσ then L(R) ∈ Algσ; it is straightforward to check that grρ(ER) is a graded algebra, and φρ is a map of algebras.

Now we show that φρ is a injective for a free module P . Let pi ∈ Fmi (P ) with hρ,mii = r, n and suppose that φρ( i=1 pi)=0. Let B ⊂ EP be a linear adapted basis with deg(bj )= λj k for bj ∈ B. Then pi =P j=1 cij χvij bj with vij + λj = mi. If hρ, vij i < 0 then cij χvij bj ∈ mτ P , so without loss ofP generality we assume that hρ, vij i = 0. This implies that hρ, λj i = r ρ ρ for each bj . Since the bj with hρ, λj i = r form a basis of Gr (P )/G>r(P ), we conclude that n n k n k n i=1 cij = 0 for each j. Then i=1 pi = j=1( i=1 cij χvij )bj = j=1( i=1 cij (χvij − P1))bj ∈ mτ P . Finally, observe thatP grρ : VectPσ →PVectk is a functorP whichP commutes with colimits (it is a direct sum of cokernels). The functor L is a left adjoint, so it also commutes with colimits. For any flat module R, we can write lim P = R for P free. It follows that −→ i i R/m R ∼ lim P /m P ∼ lim gr (L(P )) ∼ gr (lim L(P )) ∼ gr (L(R)) = gr (E ). τ = −→ i τ i = −→ ρ i = ρ −→ i = ρ ρ R ˆ ℓ A function φ ∈ Oσ is said to be concave if for any ρ1,...,ρℓ ∈ σ we have φ( i=1 ρi) ≥ ℓ v ˆ i=1 φ(ρi). Next we show that R : ER → Oσ takes concave values when R is flat.P P Proposition 3.34. Let R be a flat Sσ module, then for any f ∈ ER, vR(f) is concave. Moreover, for any m ∈ M we have F (v )= Gρi (R), where ̺ = Z ρ . m R ̺i∈σ(1) hρi,mi i ≥0 i T v M ′ ρ+ρ′ Proof. If (E, ) = L(R), where R ∈ ModSσ , let ρ,ρ ∈ σ ∩ N, and consider f ∈ Gr (R) ℓ ′ for some r ∈ Z. We can write f = i=1 fi for fi ∈ φR(Fmi (R)) with hρ + ρ ,mii≥ r. We ′ ′ P ρ ρ let si = hρ,mii and ti = hρ ,mii with si + ti = r, so that fi ∈ Gsi (R) ∩ Gti (R). It follows ′ ′ ρ ρ ρ+ρ′ ρ ρ that f ∈ s+t≥r Gs (R) ∩ Gt (R), and Gr (R) ⊆ s+t≥r Gs (R) ∩ Gt (R). We show that ′ P P ρ ρ containment holds in the other direction when R is flat. Let f ∈ Gs (R)∩Gt (R) for s+t ≤ r, ′ then f = gi = hj for gi ∈ Fmi (R) and hj ∈ Fnj (R) with hρ,mii≥ s and hρ ,nj i≥ t. Since R isP flat it isP torsion-free; we choose an appropriate m and identify f ∈ Fm(R) with ∨ f = χui gi = χvj hj for ui + mi = vj + nj = m. This is always possible as σ is a full dimensionalP coneP in MQ. Now we choose a presentationπ ˆ : P → R with pi ∈ Fmi (P ) and 14 ′ ′ qj ∈ Fnj (P ) such thatπ ˆ(pi)= gi,π ˆ(hj )= qj . We can find φ : P → P , π : P → R such that ′ ′ ρ ′ ρ ′ ρ+ρ′ ′ π ◦ φ =π ˆ with P free and π( χui pi) = π( χvj qj )= p ∈ Gs (P ) ∩ Gt (P ) ⊆ Gr (P ). ρ+ρ′ ′ ρ+ρ′ ρ+ρ′ We have L(π)(Gr (P )) ⊆ GPr (R), so LP(π)(p) = f ∈ Gr (R). In conclusion, if R is flat then: ℓ Pi=1 ρi ρ1 ρℓ (8) Gr (R)= Gt1 (R) ∩···∩ Gtℓ (R). ℓX Pi=1 ti≥r Pℓ ρ v ρ1 ρℓ i=1 i ℓ If (f)(ρi)= ti, it follows that f ∈ Gt1 (R)∩···∩Gtℓ (R) ⊂ Gr (R), where r = i=1 ti, v ℓ ℓ v v so that (f)( i=1 ρi) ≥ r = i=1 (f)(ρi). This proves that (f) is concave. P Now we consider F (v)= Gρ (R) ⊆ Gρi (R). Pick ρ ∈ σ ∩ N and P m Pρ∈σ∩N hρ,mi ̺i∈σ(1) hρi,mi d ∈ Z so that dρ = Td ρ for some d ∈ ZT . Now suppose that v(f)(ρ ) ≥ hρ ,mi ≥0 ̺i∈σ(1) i i i ≥0 i i for m ∈ M. It follows that v(f)(dρ)= v(f)( d ρ ) ≥ v(f)(d ρ ). We have P ̺i∈σ(1) i i ̺i∈σ(1) i i v(f) ∈ Oˆ , so dv(f)(ρ) = v(f)(dρ) ≥ P v(f)(d ρ ) ≥ P hd ρ ,mi = dhρ,mi, σ ̺i∈σ(1) i i ̺i∈σ(1) i i and v(f)(ρ) ≥ hρ,mi. As ρ was arbitrary,P it follows that f ∈ FPm(v). 3.7. Klyachko spaces. We say a function ψ :Σ ∩ N → Z is piecewise-linear on the fan Σ (abbreviated PL) if the restriction ψ |σ is in Mσ for each face σ ∈ Σ. It is well-known (see [Ful93, Chapter 3]) that PL functions on Σ correspond to TN -Cartier divisors on Y (Σ). For such a ψ, we let O(ψ) denote the associated invertible sheaf. In particular, for m ∈ M, O(m) denotes the linearization of the structure sheaf of Y (Σ) corresponding to the TN -character v v v v χm. For any (E, ) ∈ VectΣ we let Fψ( )= {f | (f) ≥ ψ} = σ∈Σ Fψ|σ ( |σ ) ⊆ E. T M,ev v Proposition 3.35. Let ψ be a PL function on Σ, and let F ∈ ShY (Σ) with L(F) = (E, ), then Hom M (O(ψ), F) is isomorphic to Fψ(v). ShY (Σ) Proof. Since O(ψ) is locally free and L(O(ψ)) = (k, ψ) we may use Theorem 3.23

For each ray ̺i ∈ Σ(1), we can consider the first integral point ρi ∈ ̺i. We call the ρi v spaces Gr ( ) ⊂ E the Klyachko spaces. The filtrations of E defined by these spaces play a prominent role in Klyachko’s main theorem [Kly89]. With the following proposition we recover the fact that a vector bundle E is captured by its Klyachko spaces. In particular, notice that the spaces Fm(v|σ ) ⊂ E (which determine the modules Γ(Y (σ), E)) are a special case of the hypothesis. M,ev Proposition 3.36. Let F ∈ ShY (Σ) be flat and let ψ be a PL function on Σ, then:

(9) F (v)= Gρi (v). ψ ψ(ρi) ̺i∈\Σ(1)

Proof. By definition, F (v) ⊂ Gρi (v), and if v(f)(ρ ) ≥ hρ ,m i for each ̺ ∈ ψ ̺i∈Σ(1) ψ(ρi) i i σ i σ(1) then Proposition 3.34 impliesT that v(f)| ≥ m , so F (v| )= Gρi (v). σ σ mσ σ ̺i∈σ(1) ψ(ρi) T 3.8. Quasivaluations on graded algebras. We prove a technical theorem classifying TN - equivariant degenerations of k-algebras using quasivaluations into OˆΣ. In Section 4 we refine this result by describing the Khovanskii bases of these functions. Fix a k-algebra A, and a direct sum decomposition A = i∈I Ai (not necessarily a grading) into finite dimensional k-vector spaces. A homogeneousL Y (Σ)-degeneration of this information is defined to be 15 M a flat sheaf of algebras A ∈ AlgSΣ which can be written A = i∈I Ai where each Ai is v ˆ coherent, such that EA = i∈I EAi = i∈I Ai = A. A quasivaluationL : A → OΣ is said v ∼ v to be homogeneous if (A, L) = i∈I (AiL, |Ai ) in VectΣ. Theorem 3.37. Any homogeneousL degeneration is eversive, L(A) = (A, v), where v : A → OˆΣ is homogeneous, and each v|σ has a linear adapted basis. In particular, v takes values in OΣ. Moreover, all such quasivaluations arise in this way.

Proof. A degeneration A = i∈I Ai is ﬂat if and only if each Ai is ﬂat, and therefore locally projective, and therefore locallyL free. It follows that A is eversive and (A, v|σ ) has a linear adapted basis. Such a homogeneous quasivaluation v : A → OΣ likewise corresponds under Theorem 3.23 to a unique homogeneous degeneration.

4. Khovanskii bases

In this section we define the notion of Khovanskii basis for a valuation v : A → OˆΣ, and we prove Theorems 1.1 and 1.3. For a generating set B ⊂ A with B = {b1,...,bd} we let π : k[x] → A be the surjection with π(xi)= bi ∈B and I = ker(π). B We let NB = Z and MB = Hom(NB, Z). For w ∈ NB there is a valuation vˆw : k[x] → Z α where vw(x )= hw, αi. We let vw = π∗(ˆvw). As in [KM19, Definition 3.1], we refer to vw as the weight quasivaluation on A associated to w ∈ NB. We let grw(A) denote the associated graded algebra of vw for w ∈ NB. For any w ∈ NB there is a corresponding initial ideal ∼ inw(I) ⊂ k[x]. By [KM19, Lemma 3.4] we have grw(A) = k[x]/ inw(I), and π(x) = B ⊂ A is a Khovanskii basis of vw. Let Σ(I) ⊂ NB denote the Gröbner fan of I (see e.g. [Stu96, Proposition 2.4]). Recall ′ that if w, w ∈ NB lie in the same face of Σ(I) then inw(I) = inw′ (I). For a maximal cone τ ∈ Σ(I), let Bτ ⊂ A be the associated standard monomial basis, see [Stu96, Proposition 1.1]. We make use of the following proposition from [KM19]. Proposition 4.1. [KM19, Proposition 4.9] Let v : A → Z have Khovanskii basis B ⊂ A and w = (v(b1),..., v(bd)) ∈ Σ(I). Then v = vw, and if Bτ ⊂ A is a standard monomial α α basis with w ∈ τ¯, then Bτ is adapted to v, and v(b )= hw, αi for any b ∈ Bτ . ′ It can happen that vw = vw′ for distinct w, w ∈ NB. To account for this, the map ι : NB → NB where ι(w) = (vw(b1),..., vw(bd)) was introduced in [KM19, Section 3.2]. By [KM19, Proposition 3.7], if I is homogeneous with respect to a positive grading, then ι2 = ι, ′ and ι(w)= ι(w ) if and only if vw = vw′ . It is not hard to show that ι is a piecewise-linear map on each face of Σ(I), and that the image of ι is the support of a subfan K(I) ⊂ Σ(I) which contains the tropical variety Trop(I). Points w ∈ K(I) correspond precisely to the distinct quasivaluations with Khovanskii basis B ⊂ A.

4.1. Khovanskii bases in AlgΣ. v Definition 4.2. A generating set B ⊂ A is said to be a Khovanskii basis of (A, ) ∈ AlgΣ if it is a Khovanskii basis of vρ : A → Z for every ρ ∈ |Σ| in the sense of Definition 2.4. v For (A, ) ∈ AlgΣ and a subset B ⊂ A we define ΦB : |Σ| → NB to be the function v v v v ˆd ΦB(ρ) = ( ρ(b1),..., ρ(bd)). We can also think of ΦB as the tuple ( (b1),..., (bd)) ∈ ON . When it is clear from context we will drop the subscript B. Definition 4.3. Let TropOΣ (I), KOΣ (I), and ΣOΣ (I) be the sets of piecewise-linear maps on |Σ| with values in Trop(I), K(I), and Σ(I), respectively. The sets Trop (I), K (I), OˆΣ OˆΣ and Σ (I) are defined similarly. OˆΣ 16 For any function Φ = {φ1,...,φd} ⊂ OˆΣ there is a canonical valuation ˆvΦ : k[x] → OˆΣ α α defined by letting the monomials x be an adapted basis and setting ˆvΦ(x ) = αiφi. The pushforward π∗vˆΦ : A → OˆΣ is then a quasivaluation on A. We use this constructionP to give an alternative characterization of Khovanskii bases. v v v Proposition 4.4. Let (A, ) ∈ AlgΣ, {b1,...,bd} = B⊂ A, and suppose ( (b1),..., (bd)) = Φ ∈ Σ (I), then B is a Khovanskii basis for v if and only if v = π ˆv . Moreover, in this OˆΣ ∗ Φ case Φ ∈ K (I). OˆΣ

Proof. By deﬁnition of pushforward, [π∗ˆvΦ]ρ for ρ ∈ |Σ| is the weight quasivaluation vΦ(ρ). If B⊂ A is a Khovanskii basis of v, it is a Khovanskii basis of vρ in the sense of Deﬁnition 2.4. By Proposition 4.1, vρ = vw, where wi = v(bi)(ρ). This means vρ = vΦ(ρ) and Φ(ρ) ∈ K(I) for all ρ ∈ Σ ∩ N, so v = π∗ˆvΦ. Conversely, if v = π∗ˆvΦ, then vρ = vΦ(ρ), so each vρ has Khovanskii basis B.

Recall that if I is homogeneous with respect to a positive grading on k[x] then the support of Σ(I) is N , so the condition Φ ∈ Σ (I) is always satisﬁed. B OˆΣ Proposition 4.5. Let A be a positively graded k-algebra. For any quasivaluation v : A → ON with Φ = (v(b ),..., v(bn)) we have Φ ∈ K (I). The set K (I) is in bijection with the 1 OΣ OˆΣ set of quasivaluations with ﬁnite Khovanskii basis B. If v is a valuation then Φ ∈ Trop (I). OˆΣ

Proof. For any quasivaluation v : A → OΣ with Φ as above and ρ ∈ |Σ| we have Φ(ρ) ∈ K(I). Now let v, v′ be quasivaluations with Khovanskii basis B with tuples Φ and Φ′, and let ′ v v′ ρ ∈ Σ ∩ N. IfΦ=Φ then ρ(b) = ρ(b) for all b ∈ B. By Proposition 4.1 we must have v v′ v v′ ρ = ρ for all ρ ∈ |Σ|, so = . Moreover, the ﬁbers of A coincide with the associated graded algebras of the vρ by Proposition 3.33, this shows that ﬁbers are domains if and only if the vρ are valuations, so we may use Proposition 2.2.

By abuse of notation we let ι : ΣOΣ (I) → ΣOΣ (I) denote the map Φ → ι ◦ Φ. It is 2 straightforward to check that ι = ι and that the image of ι is KOΣ (I).

4.2. Proof of Theorem 1.1. Recall that π : X → Y (Σ) is a toric flat family constructed as the relative spectrum: X = SpecY (Σ)(A), for A a flat sheaf of algebras over Y (Σ) with general fiber A. First we show that finite Khovanskii bases appear when X is of finite type.

Proposition 4.6. Let R be a finitely generated Sσ-algebra, then L(R) ∈ Algσ has a finite Khovanskii basis. Moreover, if R is flat and positively graded then it is sufficient to check that the special fiber R/m0R is a finitely generated k-algebra. In this case, Φ is linear on σ, Φ(σ) is contained in a face of K(I), and v has a linear adapted basis of standard monomials with respect to any maximal face of Σ(I) which contains the image Φ(σ). ˆ ˆ Proof. There is a presentationπ ˆ : Sσ[x] → R, let bi =π ˆ(xi) ∈ Fλi (R), and bi = φR(bi) ∈ ˆ φR(Fλi (R)) ⊂ ER = A. By assumption, for each face τ ⊂ σ, the images of the bi in R/mτ R generate as a k-algebra, and since φρ : R/mτ R → grρ(A) is a surjection of algebras (Proposition 3.33), this is the case for their images in grρ(A) as well. By definition, the image φ (ˆb ) is in the image of φ (F (R)) in Gρ (R)/Gρ (R). As a consequence, ρ i R λi hρ,λii >hρ,λii ˆ ˜ for each bi ∈ A, φρ(bi) coincides with the equivalence class bi ∈ grρ(ER). Let R be a flat, positively graded Sσ-algebra, R = n≥0 Rn, where each Rn is a coherent Sσ module. Then each Rn is free as an Sσ-module,L so it follows that L(R) has a linear 17 adapted basis B, and we may write R = b∈B Sσ ⊗ kb. Let B ⊂ B be such that the equivalence classes B˜ ⊂ R/m0R generate as aLk-algebra, then a graded version of Nakayama’s lemma implies that B generates R. ◦ For any ρ ∈ σ , B˜ ⊂ grρ(A) is a generating set, so [KM19, Algorithm 2.18] implies that B ⊂ A is a Khovanskii basis of vρ. We can also apply this argument to the restriction of L(R) to a face of σ, so it follows that B ⊂ A is a Khovanskii basis of L(R). From above, the presentation of R/mτ R by the image of the Khovanskii basis can be obtained by composing the inclusion k[x] ⊂ Sσ[x] withπ ˆ and the quotient map. The kernel of this presentation is ′ ◦ inρ(I). It follows that if ρ,ρ ∈ τ then inρ(I) = inρ′ (I), so Φ(ρ) and Φ(ρ) must lie in the same face of K(I). By construction, the set B⊂˜ A is k-linearly independent, and each v(b) b ∈ B˜ defines a linear function on σ. Let τ ∈ Σ(I) be any maximal cone which contains Φ(σ), then for any ρ ∈ σ, the standard monomials for τ are an adapted basis of vΦ(ρ). It follows that the corresponding monomials in B⊂ R form an Sσ basis of R.

Proposition 4.7. Suppose for each σ ∈ Σ the restriction (A, v|σ ) has a ﬁnite Khovanskii v basis Bσ, and that the image ΦBσ (σ∩N) ⊂ NBσ lies in a face of Σ(IBσ ), then (A, |σ ) ∈ Algσ has an adapted basis. Moreover, if v(bi)|σ∈ Mσ ⊂ Oσ for each bi ∈Bσ then (A, v)= L(A) for a locally free family of algebras A on Y (Σ). v v B v Proof. By Proposition 4.1, ρ = ΦBσ (ρ). It follows that τ is an adapted basis of ρ for v v α any ρ ∈ σ ∩ N, where ΦBσ (σ) ⊂ τ ⊂ ΣBσ . If (bi)|σ ∈ Mσ for bi ∈Bσ, then (b )|σ∈ Mσ for α any b ∈ Bτ by Proposition 4.1, so Bτ ⊂ A is a linear adapted basis for (A, v|σ ) ∈ Vectσ.

Proof of Theorem 1.1. Suppose π : X → Y (Σ) is flat and finite-type with X = SpecY (Σ)(A), with A positively graded. Let Rσ be the coordinate ring of the induced family over Y (σ) ⊂ Y (Σ). Proposition 4.6 implies that (A, v|σ) has a finite Khovanskii basis Bσ ⊂ A for each

σ ∈ Σ, ΦBσ is linear, and ΦBσ (σ) lies in a face of K(IBσ ). Taking the union B = σ∈Σ Bσ gives a finite Khovanskii basis of L(A). Let I be the ideal associated to B.S For any ρ ∈ σ, the point Φ(ρ) ∈ K(I) is computed by finding a standard monomial expression α for each b ∈ B with respect to monomials in Bσ: b = Cαb , then taking the minimum min{hΦBσ (ρ), αi | Cα 6= 0}. Let ≺σ be any monomialP order corresponding to a maximal face of Σ(IBσ ) which contains ΦBσ (σ), and let ≺ be any extension of ≺σ to I ⊂ k[x] which makes elements of Bσ strictly larger than elements of B\Bσ. By considering the Bσ standard monomial expression for b ∈ B, we see that b ∈B\Bσ and the monomials in in≺σ (IBσ ) generate the initial ideal in≺(I). It follows that the standard monomials of ≺σ are the standard monomials of ≺, and that Φ(σ) lies in the closure of the corresponding maximal face of Σ(I). v v By Proposition 4.7, if (A, ) ∈ AlgΣ, each (A, |σ ) has a finite Khovanskii basis, ΦBσ is v linear, and ΦBσ (σ) is contained in a face of K(IBσ ), then (A, )= L(A), where A is a locally free sheaf of algebras of finite type. Finally, Theorem 3.37 implies that A and L(A) can be recovered from one another. If Σ is finite, then B = σ∈Σ Bσ is a finite Khovanskii basis of L(A). Moreover, v is a valuation if and only if X hasS reduced, irreducible fibers; in this case Φ ∈ TropOΣ (I).

4.3. The Diagram of a toric ﬂat family. Proposition 4.5 implies that any v : A → OΣ with Khovanskii basis B is determined by its values Φ ∈ KOΣ (I) on B. Theorem 1.1 shows that the points Φ(ρi) for ρi ∈ ̺i ∈ σ(1) for σ ∈ Σ a face must all lie in a common maximal face τ ∈ Σ(I). Moreover, it is immediate that the restriction of v(bα) to σ is a linear function, for any member of the standard monomial basis associated to τ. These observations suggest 18 the main idea of this section: the values of Φ on ray generators of Σ are enough information to determine the ﬂat family. To be compatible with the fan Σ, these values must satisfy combatibility conditions with certain subcomplexes of K(I).

Deﬁnition 4.8. An apartment Aτ ⊂ K(I) is a subcomplex of the form K(I) ∩ τ, where τ is a maximal face of the Gr¨obner fan Σ(I).

Let Bτ ⊂ A be the standard monomial basis associated to a facet τ ∈ Σ(I), and let B∩Bτ πτ : K(I) → Q be the projection along the components associated to B ∩ Bτ . For any α α monomial x ∈ Bτ , xi | x only if xi corresponds to an element of B ∩ Bτ . It follows that B∩Bτ B w ∈ Aτ is determined by the projection πτ (w). Let φτ : Q → Q be the map which α takes the value min{hw, αi | b = Cαb , Cα 6= 0} on the component corresponding to b, α where b = Cαb is the expressionP of b in the basis Bτ . DefinitionP 4.9. We say Φ : |Σ| → K(I) is adapted to Σ if for every σ ∈ Σ there is an B∩Bτ apartment Aτ such that Φ(σ) ⊂ Aτ and πτ ◦ Φ: σ → Q is a linear map. Definition 4.10 (The Diagram). Let Φ : |Σ| → K(I) be as above, then the diagram D(Φ) v is defined to be the matrix whose j, i-th entry is Φ(ρj )(bi), where ρj ∈ Σ(1) and bi ∈B. Proposition 4.11. [Proof of Corollary 1.2] If Φ, Φ′ : |Σ| → K(I) and D(Φ) = D(Φ′) then ′ Φ=Φ . Moreover, if Φ: NQ → K(I) is a Σ-adapted map for Σ ⊂ NQ, then Φ defines a flat toric family with general fiber Spec(A) over Y (Σ). ′ Proof. By assumption, for any σ ∈ Σ, Φ(σ) is in some apartment Aτ ⊂ K(I). For any ρ,ρ ∈ ′ α σ and b ∈B we have vΦ(ρ+ρ′ )(b) = min{hΦ(ρ + ρ ), αi | Cα 6=0}, where b = Cαb is the v α v α standard monomial expression for b with respect to τ. We have Φ(ρ+ρ′ (b )= PΦ(ρ)+Φ(ρ′ )(b ) on standard monomials, so vΦ(ρ+ρ′ )(b)= vι(Φ(ρ)+Φ(ρ′ ))(b). It follows that Φ(σ) is determined by the image of the ray generators ρi ∈ ̺i ∈ σ(1). Now, if Φ : NQ → K(I) is as above, then for any σ ∈ Σ, the quasivaluations vΦ(ρ), ρ ∈ σ have a common linear adapted basis given by the standard monomials associated to τ, so we may use Theorem 1.1.

Given a simplicial fan Σ ⊂ NQ and a generating set B ⊂ A, we can produce toric flat families with general fiber Spec(A) by constructing the corresponding diagrams. We choose wi ∈ K(I) in bijection with the rays of Σ such that for each σ ∈ Σ, the wi corresponding to ̺i ∈ σ(1) lie in a common apartment Aτ . Now we use Proposition 4.11 to define adapted piecewise-linear maps Φ|σ: σ → K(I), and glue them to form Φ : |Σ| → K(I). To extend this construction to arbitrary fans, the wi must be chosen to satisfy any linear relations that hold among the ray generators ρi. Remark 4.12. The term “apartment” is used in relation to the theory of buildings (see [KM]). If L ⊂ k[x] is a linear ideal, the maximal faces τB ⊂ Σ(L) correspond to bases B ⊂ B. The set τB ∩ Trop(L) is then identified with the set of valuations on k[x]/L with adapted basis given by the monomials in B. The valuation vρ corresponding to a point m ρ ∈ τB is entirely determined by its values on B, so τB ∩ Trop(L) =∼ Q , where m = |B|. Under the natural identification of the spherical building of GLm(k) with the valuations on m k[x]/L framed by bases, φB(Q ) is mapped onto the apartment of the spherical building corresponding to B.

4.4. Proof of Theorem 1.3. For any TN -vector bundle E on Y (Σ) there is a Z≥0-graded ∗ locally-free, TN -sheaf OE = Sym(E ). If (E, v) = L(E), then by Theorem 3.37, this is the degeneration corresponding to the algebra Sym(E∗, v∗) = (Sym(E∗),s(v∗)). In this case, ∗ ∗ s(v ) : Sym(E ) → OΣ ⊂ OˆΣ is a valuation. 19 Proof of Theorem 1.3. Given a toric vector bundle E, we consider the prevalued vector space (E∗, v∗)= L(E∗) associated to the dual bundle (Example 3.31). Theorem 3.23 implies that the restriction of v∗ to each face σ ∈ Σ has a linear adapted basis. In turn, we obtain a n ∗ n ∗ linear adapted basis of monomials in each (Sym (E ),s (v )|σ ) (Example 3.30). It follows ∗ that Bσ is a Khovanskii basis for each s(v )|σ and B = ∪σ∈ΣBσ is a Khovanskii basis for s(v∗). By construction, E can be recovered from (E∗, v∗), so two distinct bundles yield ∗ ∗ distinct valuations on Sym(E ). Conversely, if v : Sym(E ) → OΣ has Khovanskii basis B with linear adapted bases Bσ, then (E, w) classiﬁes a toric vector bundle, where w is the dual of the restriction of v to E∗. Now, s(w∗) and v take the same values on a common Khovanskii basis B, so they must coincide by Corollary 4.5.

Recall that a set of polynomials {f1,...,fm}⊂ I is said to be a tropical basis of I [MS15, m Definition 2.6.3] if Trop(I)= i=1 Trop(hfii). T Proposition 4.13. The set TropON (I) in Definition 4.3 coincides with the tropical variety of I over ON as defined in Section 2. Moreover, if {f1,...,fm} ⊂ I is a tropical basis for Trop(I) then it is also a tropical basis over ON . d Proof. A function Φ ∈ ON is in the tropical variety over ON as defined in Section 2 if and only if equation 1 holds for all f ∈ I. This happens if and only if for all ρ ∈ N, the point Φ(ρ) satisfies the 1 for all f over Q¯ , which in turn holds if and only if 1 is satisfied for each element m of a tropical basis of I. The latter equivalence implies that TropON (I)= i=1 TropON (hfii) for {f1,...,fm}⊂ I a tropical basis. T Lemma 4.14. If I is a linear ideal then K(I) = Trop(I) and KON (I) = TropON (I).

Proof. It is always the case that Trop(I) ⊂ K(I). Let w ∈ K(I) and Cixi ∈ I be a circuit. It is known that the circuits of a linear ideal form a tropical basis.P Suppose for some j, wj < wk for all k 6= j, Ck 6= 0. We obtain a contradiction: vw(xj ) ≥ max{wk | k 6= j, Ck 6= 0} > wj = vw(xj ). As a consequence w ∈ Trop(I). v v If (A, ) ∈ AlgΣ has a finite Khovanskii basis B and is a valuation, the image of p Φ: |Σ| → Trop(I) must lie in the locus Trop (I) of points w ∈ Σ(I) for which inw(I) is a prime ideal. This condition depends on the initial ideal, so Tropp(I) is a union of relatively open faces of K(I) (in fact, if I is positively graded, the closure of an open face in Tropp(I) is in Tropp(I)). We say that an ideal I is well-poised (see [IM]) if Tropp(I) = Trop(I). Proposition 4.15. Let B ⊂ E∗ and let I ⊂ k[x] be the corresponding linear ideal. Valua- v ∗ tions : Sym(E ) → OΣ with Khovanskii basis B are in bijection with points Φ ∈ TropOΣ (I). ˆ Moreover, for any Φ ∈ TropOΣ (I) there is a refinement π : Σ → Σ such that the corresponding valuation corresponds to a vector bundle E over Y (Σ)ˆ . v ∗ Proof. Valuations : Sym(E ) → OΣ are classified by their values Φ ∈ KOΣ (I) = TropOΣ (I). v Moreover, any point Φ ∈ TropOΣ (I) determines a quasi-valuation Φ with Khovanskii basis B which must be a valuation since I is well-poised. By construction, the restriction of v ∗ ∗ ℓ to E is computed on f ∈ E by taking max of expressions of the form i=1 φi where Φ = (φ1,...,φd). There are only a finite number of functions of this type,L so we conclude that v takes a finite number of values on E∗. By Corollary 3.28, there is a fan Σ such that B contains a linear adapted basis for each restriction v|σ , σ ∈ Σ. In lieu of taking a refinement Σ,ˆ one can check if Φ : N → Trop(I) is Σ-adapted using Proposition 4.11. Moreover, if Σ is simplicial, any matrix D with n = |Σ(1)| rows in Trop(I) 20 such that the rows coming from σ ∈ Σ lie in a common apartment Aτ ⊂ Trop(I) defines a toric vector bundle over Y (Σ).

4.5. Connection with the work of Di Rocco, Jabbusch, and Smith. We finish this section by connecting the notion of Khovanskii basis to the matroid defined by Di Rocco, Jabbusch, and Smith in [DRJS18]. Proposition 4.16. With notation as above, let B ⊂ E be a representation of the matroid defined in [DRJS18], then B is a Khovanskii basis of v.

ρi v Proof. In [DRJS18, Section 3] it is observed that the Klyachko spaces Gr ( ) for Z≥ρi = ηi ∩ N, ηi ∈ σ(1), generate a distributive lattice under intersection, so there is a “compatible ρi v basis” Bσ ⊂ B. This means that Bσ ∩ Gr ( ) is a basis for each Klyachko space. By Proposition 3.36, Bσ ∩ Fm(v|σ ) is also a basis for m ∈ M and σ ∈ Σ, and by Proposition ρ v 3.26, Bσ ∩Gr ( ) is a basis for all ρ ∈ σ ∩N. Take b ∈ Bσ and write it as a linear combination ℓ ′ ′ v ′ i=1 Cibi of members of a linear adapted basis B of |σ with deg(bi)= λi. It follows that ℓ v v ℓ bP∈ i=1 Fλi ( ). The set Bi = Bσ ∩ Fλi ( ) is a basis, and b ∈ i=1 Bi, so for some j, v ′ ′ v b ∈ FPλj ( ). As B is adapted, it follows that each bi ∈ Fλj ( ) as well,S so λi ≤ λj for each i. v ℓ As a consequence, (b)= i=1 λi = λj . It follows that Bσ is a linear adapted basis. L 5. Strong Khovanskii bases

Let Σ be a smooth fan with toric variety Y (Σ). The TN -linearized line bundles on Y (Σ) correspond to functions ψ : |Σ| → Q whose restriction to each facet σ ∈ Σ is linear. In turn, such functions are determined by their values ψ(ρi) for ρi ∈ ̺i ∈ Σ(1). Thus, letting n |Σ(1)| = n, we identify the TN -linearized line bundles on Y (Σ) with the free group Z . Let A be a ﬂat toric sheaf of algebras corresponding to v : A → OΣ, and let L be a line bundle on Y (Σ). The space of maps from L to A decomposes into a direct sum of the TN -maps from the TN -linearized line bundles whose underlying line bundle is L:

∼ HomY (Σ)(L, A) = Hom M (O(ψ), A). ShY (Σ) O(Mψ)=∼L

By Propositions 3.34 and 3.35, each space Hom M (O(ψ), A) can be identiﬁed with the ShY (Σ) Klyachko space Fψ(v) ⊆ A. The ring of global sections of A is deﬁned as the following direct sum:

∼ R(A, vˆ)= HomY (Σ)(L, A) = Fψ(v). L∈Pic(MY (Σ)) ψM∈Zn When A = Sym(E) for a toric vector bundle E, the global section ring (denoted R(E)) is isomorphic to the Cox ring of the projective bundle PE of rank 1 quotients of E [GHPS]. The main result of this section is Theorem 1.4, which which characterizes the finite generation of R(A, vˆ) in terms of the map Φv, or equivalently the diagram D(Φv). Let SΣ be the Cox ring of Y (Σ); recall that SΣ is isomorphic to the polynomial ring k[X̺ | ̺ ∈ Σ(1)]. The ring R(A, vˆ) can be recognized as the Cox module of A, where sections n v Γ(O(φ)) ⊂ SΣ for an effective class φ ∈ Z≤0 act by shifting Fψ( ) onto the corresponding subspace of Fψ+φ(v). By Proposition 3.10, the total section ring defines a quasivaluation ˆv n vˆ : A → OCΣ , where CΣ = Q≤0; whence the notation R(A, ). 21 5.1. Total section algebras as Rees algebras. Corollary 1.2 implies that the total sec- v v ¯ tion ring R(A, ˆ) is determined by the valuations ρi : A → Z, or equivalently the rows of the diagram D(Φv). With this in mind, we broaden our perspective and consider finite generation properties of general multi-Rees algebras.

Definition 5.1. Let A be an algebra of finite type over k, and let vi : A → Z¯ 1 ≤ i ≤ n n be quasivaluations with a common Khovanskii basis B. For r = (r1,...,rn) ∈ Z , let n r v F = i=1 Gri ( i). The multi-Rees algebra is the direct sum: T R(A, v1,..., vn)= Fr. rM∈Zn For a toric flat algebra A, the total section ring R(A, vˆ) is the multi-Rees algebra defined v v n by i = ρi . Moreover, if we let Σ(n) ⊂ Q be the fan composed only of the rays Q≥0ei for 1 ≤ i ≤ n, any multi-Rees algebra R(A, v1,..., vn) can be realized as a total section ring by defining Φ : Σ(n) → Trop(IB) to be the map which sends ei to the weight of vi. Total section rings are then those multi-Rees algebras whose valuations vi satisfy the apartment compatibility conditions which allows extension to higher dimensional fans. From this perspective, Cox rings R(E) of projectivized toric vector bundles correspond to those multi-Rees algebras where A is a polynomial ring, and the vi are weight valuations emerging from points on a tropicalized linear space. The next Lemma determines properties of multi-Rees algebras, and sharpens the descrip- j tion of a total section algebra R(A, vˆ) as a Cox module. For v : A → OΣ, j ≤ n and r ∈ Z , j v v j v let Fr ( ) = 1≤i≤j Gri ( i). We define the partial multi-Rees algebras Rj = r∈Zj Fr ( ), ± ± and let Sj =T k[X1,...,Xj ], and Tj be the Laurent polynomial ring k[t1 L,...tj ]. Let mj = hX1 − 1,...,Xj − 1i⊂ Sj .

Lemma 5.2. For each j, Rj is an algebra over Sj , and is naturally identified with a sub- v j algebra of A ⊗k Tj . For each j, ρj+1 defines a Z -homogeneous quasivaluation on Rj , and the Rees algebra of this valuation is Rj+1. In particular, Rj =∼ Rj+1/hXj+1 − 1i, grv (R ) ∼ R /hX i, and R(A, vˆ) = R . Moreover, R(A, vˆ) is eversive, and if ρj+1 j = j+1 j+1 n w ∈ Cσ ⊆ CΣ for σ ∈ Σ, then ˆvw = vw. ± Proof. The algebra R1 is clearly a Rees algebra of A, and a subalgebra of A ⊗k k[t1 ], where −1 we view X1 as t1 . If this statement holds for j, then we use Rj ⊂ A ⊗k Tj and the fact v j that ρj+1 defines a Z -homogeneous quasivaluation on A ⊗k Tj . It is then straightforward j v to check that the rj+1 − st filtration level of Fr1,...,rj ( ) with respect to this quasivaluation j+1 v is Fr1,...,rj ,rj+1 ( ). It is straightforward to check that the graded components of Rj /hXj − 1i are the spaces j−1 ρi ∼ vˆ m vˆ i=1 Gri . Consequently, Rj /hXj − 1i = Rj−1, and the algebra R(A, )/ nR(A, ) is ∼ isomorphicT to R1/hX1 −1i = A. It follows that L(R(A, vˆ)) = (A, vˆ) for some quasivaluation ˆv : A → O . Moreover, if w ∈ C ⊆ C , where w = w ρˆ for ray generators CΣ σ Σ ρi∈σ(1) i i ρî ∈ Cσ, we have: P

w v v π(w) v Gr (ˆ)= Fψ( )= GR ( ). P Xwiψ(ρi)≥r ρi∈σ(1) v v In particular, this implies that ˆρˆi = ρi as quasivaluations on A. Now, by construction, the quasivaluation vˆ is concave with respect to the rays of CΣ, so we have: 22 Gρ˜ (ˆv)= Gρ (v)= F (v) ⊆ Gw (ˆv) ⊆ Gρ˜ (ˆv). ψ(ρ) ψ(ρ) ψ P wiψ(ρi) ψ(ρ) ρ˜∈C\Σ(1) ρ∈\Σ(1) w\∈CΣ ρ˜∈C\Σ(1)

The middle term Fψ(v) is a graded component of R(A, vˆ), and the inclusion into the space Gw (ˆv) is a consequence of the definition of the functor L. The equality on w∈CΣ P wiψ(ρi) theT left, and the inclusion on the right then shows that Definition 3.19 is satisfied. 5.2. Strong Khovanskii bases and proof of Theorem 1.4. By Proposition 4.6 and Lemma 5.2 above, if R(A, vˆ) is finitely generated as a k-algebra, then the image B ⊂ A of this set of generators is a finite Khovanskii basis for both vˆ and v.

Definition 5.3. We say a set B ⊂ A is a strong Khovanskii basis for v : A → OΣ if it is the image of a generating set of R(A, vˆ). The following proposition gives the general picture of what is going on when there is a strong Khovanskii basis. It has also been discovered independently by Nødland, [Nod] in the case of toric vector bundles. Proposition 5.4. Let (A, v) be as above, then B ⊂ A is a strong Khovanskii basis if and only if each space Fψ(v) is spanned by monomials in B. Proof. Any strong Khovanskii basis must have the stated property. For sufficiency, we note v ρ v that B will be a Khovanskii basis for , indeed, B monomials span each Gr ( ) space. If we let π : k[x] → A → 0 be the presentation by B, it follows that v = π∗w, where w(xi)= v(bi), bi ∈ B. We get also get an induced map πψ : Fψ(w) → Fψ(v) for each ψ ∈ Div(Σ). By w ± ± assumption, each πψ is surjective. The algebra R(k[x], ˆ ) ⊆ k[x][t1 ,...,tn ] has a basis of those monomials in x,t1,...,tn which satisfy certain integral inequalities coming from the rows of D(Φv). It follows that R(k[x], wˆ ) is a saturated affine semigroup algebra, and is therefore finitely generated. v Unfortunately, it is not enough to show that ˆ : A → OCΣ has a finite Khovanskii basis. Recall that this means that the associated graded grρ(A) is finitely generated for all n v v ρ ∈ CΣ ∩ Z . Finite generation of the toric fixed point fiber R(A, ˆ)/mCΣ R(A, ˆ) is the v ˆ v v obstruction to finite generation of R(A, ˆ), however the maps φρ : R(A, ˆ)/mCΣ R(A, ˆ) → grρ(A) from Proposition 4.6 are only surjections in general. It could be the case that the v associated graded algebras grρ(A) are finitely generated when R(A, ˆ) is not. In particular, R(A, vˆ) may not be flat over SΣ. Moreover, if R(A, vˆ) is flat as an SΣ module and A is positively graded then R(A, vˆ) must be free. in this case, v has a global adapted basis, and A splits as a direct sum of line bundles. This special circumstance holds in the cluster algebra example in Section 6.4. The next Proposition shows that we can guarantee this circumstance holds if we assume that the image of Φ is small.

Proposition 5.5. Let v : A → OΣ have a ﬁnite Khovanskii basis B⊂ A, and suppose that the image of the induced map Φ:Σ → K(I) lies in a single apartment, then the associated sheaf A splits as a direct sum of line bundles, and B is a strong Khovanskii basis.

Proof. By assumption, v is the pushforward of a valuation w : k[x] → OΣ determined α by w(x ) = αiφi, where φi = v(bi) for bi ∈ B. For each ψ ∈ Div(Σ) there is a map α πψ : Fψ(w) →P Fψ(v). The space Fψ(w) has as a basis those monomials x such that w(xα) ≥ ψ. Let B ⊂ A be the image of the set of standard monomials for the face τ, then B is a global adapted basis of v, so A splits as a sum of line bundles. Moreover, for each ψ 23 we must have Fψ(v) ∩ B ⊂ Fψ(v) is a basis. It follows that πψ is always surjective, so we may use Proposition 5.4. Now we show that it is always possible to ﬁnd a strong Khovanskii basis if one exists by adapting [KM19, Algorithm 2.18]. Let v : A → Z¯ be a quasivaluation with associated graded algebra grv(A). Let B = {b1,...,bd} ⊂ A be a generating set with ideal IB, and let JB be the ideal of polynomials which vanish on the image B⊂¯ grv(A). We have inv(IB) ⊆ JB, and these ideals are equal if and only if B is a Khovanskii basis of v [KM19, Proposition 2.17]. Let JB = hGi⊂ k[x].

(1) For an element g ∈ G, compute h = g(b1,...,bd) ∈ A and h¯ ∈ grv(A). (2) Check if h¯ can be written as a polynomial in B⊂¯ grv(A). (3) If h¯ cannot be written as a polynomial in B¯ then add h to B. (4) Repeat steps (1) − (3) for each element of G. (5) If B is unchanged then JB = inv(IB) so return B, otherwise go back to step (1). v v ± The Rees algebra R(A, ) = r∈Z Fr( ) is naturally a subalgebra of A ⊗k k[t ]. A −1 generating set B¯ ⊂ grv(A) then liftsL to a k[t ]-algebra generating set Bˆ ⊂ R(A, v), where v(bi) −1 ˆbi = bi ⊗ t . In particular R(A, v) is generated by B∪{ˆ t } as a k-algebra. As a consequence, one knows that R(A, v) has a finite generating set if and only if (A, v) has a finite Khovanskii basis. We use this perspective to prove finiteness properties for R(A, vˆ).

Algorithm 5.6. Let R0 = A, and B⊂ A be a ﬁnite generating set.

(1) Starting with a generating set Bj ⊂ Rj , use [KM19, Algorithm 2.18] to construct a ′ v Khovanskii basis Bj of ρj+1 . ˆ′ −1 (2) Set Bj+1 = Bj ∪{tj+1}⊂ Rj+1 ⊂ A ⊗k Tj+1, and go to (1). Proposition 5.7. A pair (A, v) has a strong Khovanskii basis if and only if 5.6 stops. Proof. If 5.6 stops, (A, v) has a strong Khovanskii basis. Conversely, if B ⊂ R is a ﬁnite strong Khovanskii basis, then its specialization in Rj is a generating set. It follows that the j-th instance of [KM19, Algorithm 2.18] terminates for each j. Now we give a proof of Theorem 1.4.

Proof of Theorem 1.4. If B is a strong Khovanskii basis, then k[X,¯ Y¯ ]/hI,X i ∼ grv (R ) i = ρi n−1 v for some ordering on [n]. The latter algebra is a domain since ρi is a valuation, so Lemma 5.2 implies that hI,Xii is prime for all i ∈ [n]. We prove the converse by induction. We let Rm ⊆ A ⊗k Tm be as in Algorithm 5.6. We show that Rm is finitely generated by the images of X1,...,Xm and Y¯ for each m ∈ [n]. The base case R0 = A is handled by the fact that B is a weak Khovanskii basis. We assume this holds for Rm, and we let Im denote the kernel of φm : k[X1,...,Xm, Y¯ ] → Rm. Let in(Im) denote the initial ideal of Im with respect to the term weighting induced by the v ¯ ¯ valuation ρm+1 : A ⊗k Tm+1 → Z, and let Jm ⊂ k[X1,...,Xm, Y ] denote the kernel of the map ψ : k[X ,...,X , Y¯ ] → grv (R ) ⊆ grv (A) ⊗k T . We observe that m 1 m ρm+1 m ρm+1 m v m+1 at each stage ρm+1 is homogeneous with respect to the Z grading on Rm. We have in(Im) ⊆ Jm as above, and moreover, the images of X1,...,Xm, Y¯ are a Khovanskii basis v for ρm+1 if and only if this is an equality. In this case, the Rees algebra Rm+1 is finitely generated by the images of X1,...,Xm+1, Y¯ as in the statement of the theorem above, so to prove the induction step, it suffices to show that in(Im) = Jm. The quotient algebra of Jm is a domain. Moreover, since B is a weak Khovanskii basis, the algebra obtained from 24 this quotient by inverting the X is grv (A) ⊗k T . Therefore, it suffices to show that i ρm+1 m the quotient algebra of in(Im) is also a domain of the same dimension. We prove this statement with two general observations about algebras with gradings. Let S = k[x, y¯]/J, where J is prime and homogeneous with respect to a Z-grading, and deg(x)= −1, then the kernel J0 of the map k[¯y] → S/hx − 1i is prime. To see this, let φ be an isomorphism between a polynomial rings k[..., y ,...] and k[¯y] obtained by setting xdeg(yi ) x to 1. Let [ 1 J] ⊂ k[..., y ,...] be the prime ideal of degree 0 forms. Each element of x 0 xdeg(yi) 1 [ x J]0 is mapped into J0 by φ, and both ideals have the same height, so it follows that the 1 image φ([ x J]0) is J0. Moreover, the dimension of S/hx − 1i is one less than the dimension of S. By repeatedly applying this observation to the prime ideal hI,Xm+1i, we see that the ′ ¯ ¯ ¯ kernel Im of the map k[X1,...,Xm, Y ] → k[X, Y ]/hXm+1,Xm+2 − 1,...,Xn − 1i is prime, and the image algebra has the same dimension as grv (A) ⊗k T . ρm+1 m ∼ Next we consider the general case of an algebra S = k[x, y¯]/J as above. If S = m∈Z Sm, then the images Fm of the spaces Sm in S/hx − 1i form an algebra filtration, andL xFm ⊂ Fm−1. The image B of the sety ¯ in S/hx−1i generates, and the same holds for the image B⊂¯ S/hxi =∼ grF (S). It follows that B is a Khovanskii basis for the filtration F . If J0 denotes the ′ kernel of the presentation of S/hx−1i and J0 denotes the kernel of the presentation of S/hxi, ′ we conclude that J0 = inF (J0). We apply this observation to the ideal J = Im+1, the kernel of the map φm+1 : k[X1,...,Xm+1, Y¯ ] → k[X,¯ Y¯ ]/hI,Xm+2 −1,...,Xn −1i and x = Xm+1. ′ ′ ′ In this case, J0 = Im, and J0 = Im from above. It follows that Im = in(Im) ⊆ Jm. But we ′ have seen that the quotient algebra of Im = in(Im) has the same dimension as the quotient algebra of Jm. This proves the induction step. ′ The localization of R = k[X,¯ Y¯ ]/IB obtained by inverting φ(Xi), 1 ≤ i ≤ n is isomorphic n to A ⊗k Tn. As a consequence, there is an isomorphismπ ˜ : Trop(IB) → Trop(IB) × Q , v whereπ ˜(w1,...,wd,m1,...,mn) = (...,wj + ρi (bj )mi,..., −m1,..., −mn). We let π : Trop(IB) → Trop(IB) be the compositionP ofπ ˜ with the projection to the Trop(IB) component, and we let s : Trop(IB) → Trop(IB) be the natural section s(w) = (w, 0,..., 0). n The ideal IB is homogeneous with respect to the Z grading on k[X,¯ Y¯ ], so for any d+n m = (0,..., 0,m1,...,mn) ∈ Q and u ∈ Trop(IB) we have inu+m(IB) = inu(IB). In particular, the fan structure on Trop(IB) induced from the Gröbner fan of IB is stable under n d+n ′ addition by m ∈ {(0,..., 0)}× Q ⊂ Q . Moreover, if u, u ∈ Trop(IB) have the same ′ initial ideal, then the same holds for π(u) and π(u ). We give give Trop(IB) the coarsest fan ′ structure such that π(u) and π(u ) share the same face whenever inu(IB) = inu′ (IB). This is a refinement of the Gröbner fan structure on Trop(IB). Faces C˜ ∈ Trop(IB) are of the n form C˜ =∼ C × Q for C ∈ Trop(IB). We say that C˜ ∈ Trop(IB) is the lift of C ∈ Trop(IB). We recall the notion of prime cone from [KM19]. A face C ∈ Trop(IB) is said to be a prime cone if its corresponding initial ideal is a prime ideal. If Ψ : |Σ| → Trop(IB) corresponds to a valuation, then we must have that the face Ci ∈ Trop(IB) containing Ψ(ρi) is a prime cone for 1 ≤ i ≤ n. We let C˜i ∈ Trop(IB) be the corresponding lift.

Corollary 5.8. The set B ⊂ A is a strong Khovanskii basis if and only if each lift C˜i is a prime cone. In particular, this holds if and only if s ◦ Ψ(ρi) ∈ Trop(IB) is a prime point for each ̺i ∈ Σ(1). v ¯ Proof. For each ray ρi, the valuation ρi : A → Z defines a valuation on A ⊗k Tn, which ′ w ′ ¯ in turn induces a valuation on R . We let ρi : R → Z denote the weight quasivaluation v ¯ ¯ ′ induced by the values of ρi on the generating set {φ(X), φ(Y )} ⊂ R . In particular, we w w v w ˜ ˜ have ρi (φ(Xj )) = 0 and ρi (φ(Yj )) = ρi (bj ), so the weight of ρi is Ψ(ρi) ∈ Ci. 25 ′ ′ There is an inclusion of algebras R ⊆ R(A, vˆ). We let ordi : R → Z¯ be the quasivaluation ′ ¯ obtained by the hφ(Xi)i-adic filtration. Moreover, we let degi : R → Z be the valuation obtained by reporting the homogeneous degree in the i-th direction. We claim that ordi + w ′ degi = ρi as homogeneous quasivaluations on R . This is a consequence of the fact that both sides of this equation take the same values on a shared Khovanskii basis {φ(X¯), φ(Y¯ )}. ′ ′ Now, if B is a strong Khovanskii basis, then R = R(A, vˆ), and hφ(Xi)i⊂ R is a prime ideal. It follows that ordi is a valuation. Since degi is a homogeneous grading function, this w ˜ ˜ implies that ρi is a valuation as well, so Ci is a prime cone. Similarly, if Ci is a prime w cone, then ρi is a valuation, and ordi must also be a valuation; this implies that hφ(Xi)i is prime. Theorem 1.4 completes the proof. 5.3. Cox rings of projectivized toric vector bundles. Next we apply Theorem 1.4 to several classes of projectivized toric vector bundles. If E and E′ are toric vector bundles such ′ that E = E⊗Y (Σ) L for some TN -linearized line bundle L, then the projectivizations agree: PE′ =∼ PE, and R(E) =∼ R(E′). If D is a diagram of E, then we obtain a diagram D′ for E′ as follows. We let H be the |Σ(1)| × |Σ(1)| Z-matrix whose i-th diagonal entry is the degree ri of L along the divisor of Y (Σ) associated to the i-th ray ̺i ∈ Σ(1), with off-diagonal entries equal to ∞. Then D′ = D ⊙ H, where ⊙ is tropical matrix multiplication. In particular, we ′ get the i-th row of D from the i-th row of D by adding ri to each entry. ′ We say that E is sparse if it can be written as E⊗Y (Σ) L, where L is a line bundle, and the diagram of E is stuctured as follows. The rows of D come in groups S1,...,Sr, where all of the rows from Sj are dij ej for a non-negative tuple dj = (d1j ,...,dkj ,j ). In other words, each row of D has at most one non-zero entry. Corollary 5.9. If E is sparse, then PE is a Mori dream space.

Proof. Let L ⊆ k[y1,...,yd] be the linear ideal. We consider φ : k[Y1,...,Yd,X1,...,Xn] → dj −1 R(E), φ(Yj ) = yj ⊗ t , φ(Xi) = ti . Suppose that hℓ1(¯y),...,ℓm(¯y)i = L, and let dj pi(Y,¯ X¯) be the polynomial obtained by substituting Yj X for yj in ℓi, then the ideal J = hp1,...,pmi ⊂ k[Y,¯ X¯] is of the form considered in Lemma 5.12. It follows that this ideal is prime of height m. But J ⊆ IB, and IB is also height m. Moreover, the ring k[Y,¯ X¯]/hIB,Xii can be presented as a quotient of k[Y,X¯ 1,... Xˆi,...,Xn] by an ideal Ji of height m which contains the ideal hp1|Xi=0,...,pm|Xi=0i. Another application of Lemma

5.12 then shows that Ji = hp1|Xi=0,...,pm|Xi=0i is prime, so hIB,Xii is prime as well. The Proposition now follows from Theorem 1.4.

We can apply Corollary 5.9 to rank 2 vector bundles, which were considered in [Gon], and [Nod]. If D is the diagram of a rank 2 vector bundle, then there are at most two distinct entries in each row of D. To see this, note that the entries of the i-th row of D are the values of a valuation vi : Sym(E) → Z¯ on a spanning set B ⊂ E. It is an elementary consequence of the valuation axioms that members of B with distinct values must be linearly independent. It follows that after an appropriate transformation, D is of the type considered in Corollary 5.9. The bundles considered in [GHPS, Section 6.2] and [Nod, Proposition 5.2] are sparse by a similar argument. Next we classify the global coordinate rings of toric vector bundles presented by hypersurfaces. Corollary 5.10. Let E be a toric vector bundle with diagram D adapted to Σ, with rows d w1,...,wn ∈ Trop( j=1 yj). Then the corresponding Khovanskii basis B is strong if and only if for each pairP of rows wi, wj there is a column which contains the minimal entries of wi and wj . 26 Proof. Without loss of generality, we may assume that all entries of D are non-negative, and d αj that the minimal entry in each row is 0. Let D = [α1,...,αd], then F = j=1 Yj X ∈ IB. A polynomial of this form is irreducible if and only if no Xi dividesP all the monomials appearing in F . Each row of D has some 0 entry, so we conclude that F is irreducible, and

IB = hF i. Now, observe that D satisﬁes the row conditions above if and only if ins◦Ψ(ρj )(F ) is irreducible for 1 ≤ j ≤ n. The Corollary is therefore a consequence of Corollary 5.8.

We say a bundle E is uniform if we can ﬁnd a Khovanskii basis B ⊂ E such that LB is generated by the rows of a matrix with non-zero maximal minors. Observe that if E is a toric vector bundle of rank r over projective space Pn−1 with diagram D, matrix of size n × r + 1, then the fact that D is adapted to the fan of Pn−1 implies conditions (1) and (2) in Corollary 5.10 so that PE is a Mori dream space. We show a similar result for uniform bundles over PE which satisfy a certain technical assumption. Corollary 5.11. Let E be a rank r uniform toric vector bundle over Pn−1 with Khovanskii basis B such that |B| = d and d − r

The next lemma is used in the proofs of Corollaries 5.9 and 5.11.

Lemma 5.12. Let ψ : k[y1,...,yd] → k[z1,...,zm] have the property that ψ(yi) is a monomial for all i, the supports of ψ(yi), ψ(yj ) are disjoint for i 6= j, and every monomial ψ(yi) contains a variable with exponent 1. If L ⊂ k[y1,...,yd] is a monomial-free linear ideal, then J = hψ(L)i⊆ k[z1,...,zm] is a prime complete intersection.

Proof. The supports of the ψ(yi) are disjoint, so ψ is a tensor product of maps ψi : k[yi] → d d k[zj | j ∈ supp(ψ(yi))], where ψi(yi) = ψ(yi). It follows that ψ is flat. Let T ⊂ A (k) be torus of points with all coordinates non-zero, then T d ∩V (L) is a non-empty open subscheme 27 of V (L) ⊂ Ad(k). For any closed point p ∈ V (L) ∩ T d ⊆ T d, the fiber over p is irreducible of dimension m − d. The Jacobian of J can be shown to have full rank, so the varieties V (J) ⊂ Am(k), V (J) ∩ T m are generically reduced. The dimension of the fibers implies that V (J) is a complete intersection, so V (J) and V (J) ∩ T m are reduced. Now V (J) ∩ T m has irreducible fibers over V (L), so V (J) ∩ T m is irreducible. If V (J) had more than one m irreducible component, say V1, V2, then the generic point of V (J) ∩ T is in one of the ∗ components, say V1. The map ψ : V (J) → V (L) is open by flatness, so it maps V2 \ V1 to an open subset of V (L), but also has image in the open subset formed by taking the complement of the image of V1. It follows that the image of V2 \ V1 is empty, and V1 = V (J) is irreducible.

Example 5.13. We consider the hypersurface bundle E over P2 deﬁned by the ideal L = 6 h i=1 yii⊂ k[y] and the following diagram: P 400321 (10) D = 040213 004132  

Let B = {b1,...,b6}⊂ E be the corresponding arrangement. If ψi is the i-th column, then v 2 Fψi ( )= hbii⊂ E. The rays ̺0,̺1,̺2 of the fan of P correspond to the ﬁrst, second, and third rows of D, respectively. Each pair of rows lives in a common apartment of Trop(L). In particular, we have adapted bases over each face: B\{b1} is adapted over σ0 = R≥0{̺1,̺2}, B\{b2} is adapted over σ1 = R≥0{̺0,̺2}, and B\{b3} is adapted over σ2 = R≥0{̺0,̺1}. This diagram satisﬁes Corollary 5.10, so R(E) is presented by the following polynomial:

4 4 4 3 2 2 3 3 2 Y1X1 + Y2X2 + Y3X3 + Y4X1 X2 X3 + Y5X1 X2X3 + Y6X1X2 X3 .

The next example shows a hypersurface bundle over P1 ×k P1 whose Khovanskii basis is not strong.

Example 5.14. Let E be the rank 3 bundle over P1 ×k P1 deﬁned by the following diagram.

2 10 0 0 21 0 (11) D = 0 02 1   1 00 2  

In particular, the Khovanskii basis B = {b1,b2,b3,b4}⊂ E ⊂ Sym(E) satisﬁes b1 + b2 + b3 + b4 = 0. The corresponding ideal IB ⊂ k[X1,X2,X3,X4, Y1, Y2, Y3, Y4] is generated by 2 2 2 2 the polynomial Y1X1 X4 + Y2X1X2 + Y3X2X3 + Y4X3X4 . The ideal hIB,X1i is not prime, whence B is not a strong Khovanskii basis by Theorem 1.4. Algorithm 5.6 creates a strong ′ Khovanskii basis from B by adding two new elements: B = B∪{c1,c2}, where c1 = b3 + b4 ′ and c2 = b1 + b4. The diagram with respect to B is obtained by adding two new columns, computed by applying the weight valuations represented by the rows of D to c1 and c2: 28 210010 021001 (12) D′ =   002110   100201  

The ideal IB′ ⊂ k[X1,X2,X3,X4, Y1, Y2, Y3, Y4,Z1,Z2] is generated by 5 polynomials:

2 2 X1X3Y3 + X4 Y4 + X1Z1, X1X2Y2 + X3 Y3 − X4Z2, 2 2 X1X4Y1 + X2 Y2 − X3Z1, X3X4Y4 + X1 Y1 + X2Z2, X1X3Y1Y3 − X2X4Y2Y4 − Z1Z2.

We conclude that R(E) =∼ k[X,¯ Y,¯ Z¯]/IB′ . We ﬁnish this section with an example of a rank 3 bundle over P2 whose projectivization is not a Mori dream space. Example 5.15. We take E to be the rank 3 bundle deﬁned by the diagram:

00021 2 2 D = 31010 ∈ ∆(P , Trop(hy4 − y1 − y2,y5 − y2 − y3i), P ) 01301  ± ± Let R2 ⊂ Sym(E) ⊗k k[t1 ,t2 ] be the 2nd Rees algebra in Algorithm 5.6. This algebra 2 2 is presented as a quotient of k[Y1 ...Y5,X1,X2] by hY5X1 − Y2X2 − Y3, Y4X1 − Y1X2 − Y2i. 2 This ideal is the kernel of the map determined by sending Y1 → ω1t2, Y2 → ω2t2, Y3 → ω3, 2 Y4 → (ω1 + ω2)t1t2, Y5 → (ω2 + ω3)t1, where ω1,ω2,ω3 ∈ Sym(E) are a set of polynomial generators. We show that the associated graded algebra gr3(R2) of R2 with respect to the weight v valuation ρ3 defined by the third row of D is not finitely generated. Let F0,n;n ⊂ R2 be the n subspace Fn,0 ∩ Sym (E), and let Γ = n≥0 F0,n;n. We sketch an argument to show that gr3(Γ) is not finitely generated. As gr3(Γ)L is an invariant subring of gr3(R2) by the action of a torus, its failure to be finitely generated implies that gr3(R2) cannot be finitely generated, and so neither can the Rees algebra R3 = R(E). An invariant theory computation shows that Γ is generated by F0,n;n for n =1, 2, 3. These components have the following elements as basis members:

2 2 {ω1,ω2}⊂ F0,1;1 {ω1,ω1ω2,ω2,ω1ω3}⊂ F0,2;2 3 2 2 3 2 2 2 2 {ω1,ω1ω2,ω1ω2,ω2,ω1ω3,ω1ω2ω3,ω1ω3, (2ω2ω3 + ω2ω3)}⊂ F0,3;3

It can be shown by induction that the graded components F0,3n−2;3n−2, F0,3n−1;3n−1, F0,3n;3n have bases depicted in Figure 1. All monomials in the quadrillateral shaded region are 3n 3n−1 3n−2 2 included, as is ω2 (resp. ω2 ,ω2 ). Also included are the polynomials (2ω2ω3 + 2 k p ω2ω3) ω2 , where p +3k = 3n (resp. 3n − 1, 3n − 2). This basis can be shown to be v adapted to the valuation ρ3 . The components of the associated graded algebra gr3(Γ) ⊂ 2 2 k p gr3(R2) are similarly described, except the polynomials (2ω2ω3 + ω2ω3) ω2 are replaced 2 k p with the monomials (2ω2ω3) ω2 . This is the source of the inﬁnite generation. In the n−1 2n component F0,3n;3n, the corner element ω1ω2 ω3 is only reachable as a diﬀerence of 2 2 n−1 2 (2ω2ω3 + ω2ω3) ω1ω2 ∈ F0,3(n−1);3(n−1)F0,3;3 and monomials obtained from the product 29 3n ω2 3n−1 2 2 3(n−1) ω1ω2 (2ω2ω3 + ω2ω3)ω2

2 2 n−1 3 (2ω2ω3 + ω2ω3) ω2

2 2 n (2ω2ω3 + ω2ω3)

n−1 2n ω1ω2 ω3

3n n 2n ω1 ω1 ω2 3n−1 ω2 3n−2 2 2 3(n−1)−1 ω1ω2 (2ω2ω3 + ω2ω3)ω2

2 2 n−1 2 (2ω2ω3 + ω2ω3) ω2

n−1 2n−1 ω1ω2 ω3

3n−1 n 2n−1 ω1 ω1 ω2 3n−2 ω2 3n−3 2 2 3(n−1)−2 ω1ω2 (2ω2ω3 + ω2ω3)ω2

2 2 n−1 (2ω2ω3 + ω2ω3) ω2

n−1 2(n−1) ω1ω2 ω3

ω3n−2 n 2(n−1) 1 ω1 ω2

Figure 1. Graded components F0,3n;3n (top), F0,3n−1;3n−1 (middle) and F0,3n−2;3n−2 (bottom) of Γ.

F0,3n−1;3n−1F0,1;1. However, after taking associated graded, no terms exist in lower degrees with monomials supported in the degrees necessary to reach this point. It follows that for n−1 2n every n, ω1ω2 ω3 ∈ F0,3n;3n is an essential generator of gr3(Γ).

5.4. Diagram alterations and the fan ∆(Σ, Trop⋆(IB)). We finish this section by showing that the language of prime cones also provides a mechanism to make new Mori dream 30 space bundles from old. In principle, these results can be formulated for any multi-Rees algebra, but we focus on the projectivized bundle case. To prepare, we make several definitions. Let B ⊂ Sym(E) be a (not necessarily linear) Khovanskii basis for a toric vector bundle bundle E defined by Ψ : |Σ| → Trop(IB). By construction, the rows of the corresponding diagram D are in the subset Trop⋆(IB) ⊆ Trop(IB) of those points which come from the tropical variety Trop(LB), where LB is the ideal which vanishes on the linear part of B. In particular, w ∈ Trop(LB) defines a weight valuation vw : Sym(E) → Z¯, which in v v turn defines the point ( w(b1),..., w(br)) ∈ Trop⋆(IB). n n×r Definition 5.16. Let ∆(Σ, Trop∗(IB)) be the subset of Trop∗(IB) ⊂ Q consisting those matrices D which are adapted to Σ.

By Proposition 4.11, any D ∈ ∆(Σ, Trop⋆(IB)) deﬁnes a toric vector bundle over Y (Σ) with general ﬁber E.

n×r Proposition 5.17. The set ∆(Σ, Trop⋆(IB)) ⊂ Q is the support of a polyhedral fan.

Proof. Give Trop⋆(IB) the Gr¨obner fan structure. If (w1,...,wn) ∈ ∆(Σ, Trop⋆(IB)), and wi ∈ Ci, where Ci is the face of Trop⋆(IB) containing wi, then C1 × ... × Cn ⊂ ∆(Σ, Trop⋆(IB)) as well.

Next we describe alterations to the rows of a diagram D ∈ ∆(Σ, Trop⋆(IB)) which preserve the ﬁnite generation of the corresponding total section ring.

Proposition 5.18. Let E and D be as above, with R(E) finitely generated. Let w ∈ Trop(IB) ′ be a prime point with π(w) ∈ Trop⋆(IB), and let D be obtained by appending π(w) to D. If ′ ′ ′ ′ ′ Σ is a fan such that D ∈ ∆(Σ , Trop⋆(IB)), then the associated bundle E has R(E ) finitely ′′ ′′ ′′ generated. If D is obtained from D by deleting a row, and D ∈ ∆(Σ , Trop⋆(IB)), then the associated bundle E′′ has R(E′′) finitely generated. Proof. By the construction in Lemma 5.2, the algebra R(E′) is the Rees algebra of R(E) and the valuation defined by π(w). As π(w) is the image of a prime point, R(E′) is finitely generated. In the second case, R(E′′) is a quotient of R(E).

By Corollary 5.8, each row wi of D is in the image π(C˜i) ⊂ Trop(IB) of a prime cone ˜ ˜ Ci ⊂ Trop(IB). Let Ki be the polyhedral complex π(Ci) ∩ Trop⋆(IB). By Proposition 5.18, ′ we can append wi ∈ Ki to D, then delete wi while maintaining a diagram adapted to Σ and preserving finite generation of the associated total section ring. Next we show that given D ∈ ∆(Σ, Trop⋆(IB)), with finitely generated total section ring, there is a polyhedral family of diagrams containing D all of whose total section rings are also finitely generated. To prepare for this construction, let w1,...,wn be the rows of D ∈ ∆(Σ, Trop⋆(IB)). We show the existence of a generating set F ⊂ IB with well-behaved initial forms.

Lemma 5.19. There exists a generating set {f1,...,fℓ} = F ⊂ IB with the property that for each wi, the initial forms of F generate ins(wi)(IB), and ins(wi)(fj )= fj|Xi=0.

Proof. We let Gi ⊂ IB be a Gröbner basis with respect to wi. We assume without loss of generality that the elements of Gi are irreducible and homogeneous with respect to the Zn+1 n grading on IB. We let F = i=1 Gi. It remains to show that ins(wi)(f)= f|Xj =0 for each f ∈F. This is a consequenceS of the homogeneity and irreducibility of f. ′ We define CF ⊂ ∆(Σ, Trop⋆(IB)) to be the set of those diagrams D with the property ′ ′ that in ′ (f) = in (f) for each f ∈F, where w is the i-th row of of D . s(wi) s(wi) i 31 ′ ′ ′ Proposition 5.20. Let E be the bundle defined by D ∈ CF , then R(E ) is finitely generated. ˆ ′ ′ Proof. We will make a diagram D for Σ(2n) whose rows are w1, w1,...,wn, wn and show that the associated total section ring is finitely generated. The proposition then follows from the fact that R(E′) is a specialization of this section ring. ¯ ¯ ¯ ′ ′ ¯ Starting with I0 = IB ⊂ k[X, Y ], we build a sequence of ideals Ii ⊂ k[X, x1,...,xi, Y ] each satisfying Theorem 1.4. By construction, in ′ (I )= hf | ,...,f | i is prime, s(w1) 0 1 X1=0 ℓ X1=0 it follows that the Rees algebra of v ′ and R(E) can be presented as a quotient of s(w1) ¯ ′ ¯ ˜ ˜ ˜ ˜ k[X, x1, Y ] by the ideal I1 = hf1,..., fℓi, where the fi are obtained as follows. The form f α β ′ p(α,β) is constructed from f = Cα,βX Y by appending a power (x1) , where p(α, β) is the β difference between v ′ (b ) and the minimum such value that appears among the monomials w1 P of f. By construction, f˜ is in the kernel of the presentation of the Rees algebra, and the initial forms in ′ (f˜) = in (f) generate a prime ideal of height equal to the height s(w1) s(w1) of I1. It follows that the lifts f˜ generate I1, and constitute a set F˜ ⊂ I1 satisfying the ′ properties of Lemma 5.19. We may now repeat this procedure with w2, and so on.

Proposition 5.20 shows that the locus of diagrams in ∆(Σ, Trop⋆(IB)) which correspond to bundles E with R(E) ﬁnitely generated is a union of polyhedral cones. It is natural to ask if these cones are faces of a fan structure on ∆(Σ, Trop⋆(IB)) (Conjecture 1.5). Example 5.21. We apply Propositions 5.20 and 5.18 to the bundle from Example 5.14. To ﬁnd an appropriate set F ⊂ IB we must add two polynomials to the generating set:

2 2 X1X3Y3 + X4 Y4 + X1Z1, X1X2Y2 + X3 Y3 − X4Z2, 2 2 X1X4Y1 + X2 Y2 − X3Z1, X3X4Y4 + X1 Y1 + X2Z2,

X1X3Y1Y3 − X2X4Y2Y4 − Z1Z2 3 2 2 2 X2 Y2 + X4 Y1Z1 − X2X3Y2Z1 − X3 X4Y1Y3 3 2 2 2 X3 Y3 − X1 Y2Z1 − X3X4Y3Z2 − X1X4 Y2Y4

′ In this case B is composed of linear forms, so IB′ = LB′ . The cone CF ⊂ ∆(Σ, Trop(LB′ )) is composed of diagrams of the form:

ℓ1 +2a ℓ1 + a ℓ1 ℓ1 ℓ1 + a ℓ1  ℓ2 ℓ2 +2b ℓ2 + b ℓ2 ℓ2 ℓ2 + b (13) D = ℓ ℓ ℓ + c + d ℓ + c ℓ + c ℓ  3 3 3 3 3 3   ℓ4 + e ℓ4 ℓ4 ℓ4 + e + f ℓ4 ℓ4 + e  

Here ℓ1,...,ℓ4 ∈ Z correspond to addition by elements of the lineality space of Trop(LB′ ) (equivalently tensoring E by TN -linearized line bundles), and a,b,c,d,e,f ∈ Z≥0. The tropical variety Trop(IB′ ) contains one prime cone C˜0 in addition to the 4 prime ′ cones containing the rows of the matrix D from Example 5.14. The image π(C˜0) is C0 = ′ Q(1, 1, 1, 1, 1, 1) + Q≥0(0, 0, 0, 0, 1, 0) + Q≥0(0, 0, 0, 0, 0, 1). Let Σ be the fan of the blow- 1 1 up BLp(P ×k P ), where p is the toric fixed point corresponding to the cone spanned by {(1, 0), (0, −1)} in the fan of P1 ×k P1. It can be shown that the following diagram is adapted to Σ′, where the bottom row is associated to the new ray in Σ′. 32 210010 021001 (14) D′′ = 002110   100201   000011   By Proposition 5.18, the algebra R(E′′) corresponding to the bundle E′′ defined by this diagram is finitely generated.

6. Examples

In this section we discuss several examples of ON valuations on the coordinate rings of Grassmannian varieties, and more general cluster varieties. We also present general methods to construct toric vector bundles. 6.1. Vector bundles from tropicalization. By Theorem 1.3, one can construct a toric vector bundle by ﬁnding a point Φ ∈ TropON (L), where L is a linear ideal. The following proposition is a consequence of the fact that the circuits of a linear ideal are a tropical basis. Proposition 6.1. Let Ψ= ψ¯ be an n-tuple of piecewise linear functions, and suppose that for each minimal circuit i∈J aixi ∈ L, i∈J ψi = i∈J\{j} ψi for each j ∈ J, then ¯ ¯ ψ ∈ TropON (L). Moreover,P suppose P is aL collection ofLn integral polyhedra in MQ such that Pj is in the convex hull of the union i∈J\{j} Pi for each circuit as above, then the ¯ support functions of the polyhedra P deﬁneS a point in TropON (L). One can use pad and paper to sketch collections of polyhedral P¯ which correspond to toric vector bundles by Lemma 6.1. We describe another method using linear algebra.

Proposition 6.2. Let λ¯(t) = (λ1(t),...,λn(t)) with λi(t) ∈ QM be in the kernel of the matrix MI whose rows are given by the coeﬃcients of the minimal circuits of L, then w w ( N (λ1),..., N (λn)) ∈ TropON (L). Moreover, if λi(t) ∈ S0 ⊂ QM for each 1 ≤ i ≤ n, then each wn(λi) is the support function of a polytope as in Lemma 6.1. w w Proof. Proposition 2.2 immediately implies that ( N (λ1),..., N (λn)) ∈ TropON (L).

6.2. ON tropical points on Grassmannian varieties. Let I2,n be the Plücker ideal 2 n n which cuts out the affine cone X2,n ⊂ (k ) over the Grassmannian variety Gr2(k ) of 2-planes in n-space. In general, the idealV I2,n is well-poised. It follows that, similar to tropicalized linear spaces, flat TN families π : G → Y (Σ) with reduced, irreducible fibers, 2 n general fiber X2,n, and a TN -equivariant embedding into the trivial bundle (k ) × Y (Σ) ¯ correspond to piecewise-linear maps Ψ : Σ → Trop(I2,n), or equivalentlyV a point ψ ∈

TropON (I2,n). Moreover, the Pl¨ucker generators pij pkℓ − pikpjℓ + piℓ4pjk for 1 ≤ i

Lemma 6.3. The support functions ψij of the edges Pij deﬁne a point on TropON (I2,4).

Proposition 6.4. Let m = {m1,...,mn}⊂ M, with ∆(m) and Pij defined as above, then the support functions ψij of Pij define a point on TropON (I2,n). 33 Proof. Plücker relations are a tropical basis, and by Lemma 6.3 we have ψ¯ = (...ψij ,...) ∈ TropON (hpij pkℓ − pikpjℓ + piℓ4pjki) for each 1 ≤ i

Theorem 6.5. Let m and ∆(m) be as above, and let PI = conv{mi | i ∈ I} for I ⊂ [n], ¯ | I |= m, then the tuple ψ = (...ψI ,...) deﬁnes a point on TropON (Im,n), where ψI is the support function of PI . Proof. We use a technique similar to the proof of Proposition 6.2. Let L(m, t) be the following matrix with entries in QM :

tm1 ··· tmn  a2,1 ··· a2,n  . . .  . .. .    a ··· a   m,1 m,n where the aij are chosen generically in k. By sending the Pl¨ucker generator pI to the I-minor δI of L(m, t), we obtain a ring map φm : Am,n → QM , and a valuation wN ◦ φm : Am,n → mi ON . One checks that δI (L(m, t)) = i∈I dit , where di is a non-zero m − 1 × m − 1 determinant in the aij . It follows thatPwN ◦ φm(pI ) is the support function of PI . The theorem is then a consequence of Proposition 2.2.

2 4 Example 6.6. We return to the case X2,4 ⊂ (k ), and its corresponding Pl¨ucker ideal hp12p34 − p13p24 + p14p23i. Let m1 = (0, 0),m2V= (1, 0),m3 = (0, 1), and m4 = (0, 0). We 2 view the convex hull of the mi as a degenerate tetrahedron in Q . Following Theorem 6.5, there is an associated ring map A2,4 → Q(Z2)∗ given by the minors of a matrix:

1 t10 t01 1 a2,1 a2,2 a2,3 a2,4 2 with generic aij . We obtain an associated valuation w : A2,4 → OZ2 . Letting x : Z → Z and y : Z2 → Z be projections onto the x and y axes, respectively, we have:

w(p12) = min{x, 0} w(p13) = min{0,y} w(p14)=0

w(p23) = min{x, y} w(p24) = min{x, 0} w(p34) = min{y, 0}

The common reﬁnement of the domains of linearity of these generators is the fan Σ ⊂ Q2. There is a corresponding piecewise-linear map Φ : Σ → Trop(I2,4). Recall that Trop(I2,4) can be identiﬁed with the space of metric trees with four leaves. When used with the min convention the lengths are the non-leaf edges of these trees become negative, which is somewhat counterintuitive. To compensate we record the negatives of the values of Φ.

0 0 0 .5 −.5 0 0 −Φ(1, 0) = 0 0 −Φ(1, 1) = 0 −.5 −Φ(0, 1) = 0 0

0 1 0 .5 .5 0 0 −Φ(−1, 0) = 0 0 −Φ(−1, −1) = 0 .5 −Φ(0, −1) = 0 1

34 σ2

σ3 • •

σ1 • •

σ4

•• σ6

σ5

Figure 2. The fan Σ

The map Φ maps the positive x and y axes to the origin, and maps the ray through (1, 1) and σ4 and σ5 to the face τ ⊂ Trop(I2,4) corresponding to the initial ideal hp12p34 − p13p24i. The family G(Φ) → Y (Σ) has fibers X2,4, except over the divisors D(−1, −1),D(1, 1) ⊂ 2 4 Y (Σ), where the fiber is the toric variety V (hp12p34 − p13p24i) ⊂ (k ). V Example 6.7. In the previous example we see a simple toric family defined by a map Φ:Σ → Trop(I2,4), which hits only one non-trivial face of the tropical variety. Now we consideramap Ψ : Σ → Trop(I2,4) which folds Σ into all three non-trivial faces of Trop(I2,4):

0 0 1 1 0 1 0 1 1 1 0 0 −Ψσ1∪σ2 (x, y) = max{x, y} +max{x − y, 0} 0 0 +max{y − x, 0}

1 1 0 1 0 0 1 0 −Ψ (x, y) = (−x + max{−y,y}) 1 1 +max{y, 0} 0 0 +max{−y, 0} 0 0 σ3∪σ4 .

0 0 1 1 0 1 0 1 1 1 0 0 −Ψσ5∪σ6 (x, y) = (−y + max{−x, x}) +max{x, 0} 0 0 +max{−x, 0}

On the ray generators of Σ we have:

0 0 1 1 1 1 1 1 0 1 0 1 1 1 1 1 1 1 0 0 −Ψ(1, 0) = + 0 0 −Ψ(1, 1) = −Ψ(0, 1) = +

1 1 1 1 0 1 0 1 1 −Ψ(−1, 0) = 1 1 −Ψ(−1, −1)=2 1 1 + 0 0 −Ψ(0, −1) = 1 1

We let w : A2,4 → OΣ denote the valuation associated to Ψ. By Theorem 1.4, we compute w −1 w the map ψ : k[X1,...,X6, Y12,...,Y34] → R(A2,4, ˆ), where ψ(Xi)= ti ∈ F−ei ( ), and:

ψ(Y12)= p12 ∈ F−2,−2,−3,−2,−5,−2(w) ψ(Y13)= p13 ∈ F−3,−2,−3,−2,−4,−2(w)

ψ(Y14)= p14 ∈ F−3,−2,−2,−2,−5,−2(w) ψ(Y23)= p23 ∈ F−3,−2,−2,−2,−5,−2(w)

ψ(Y24)= p24 ∈ F−3,−2,−3,−2,−4,−2(w) ψ(Y34)= p34 ∈ F−2,−2,−3,−2,−5,−2(w) 35 w Here pij ∈ Fr1,...,r6 ( ) means that the value of the valuation associated to the k-th ray is rk on pij ; in particular the length of the unique path from i to j in the metric tree deﬁned 2 2 2 by the image of the k-th ray is −rk. We have ker(ψ)= hX1 Y12Y34 −X5 Y13Y24 +X3 Y14Y23i. The pij are a Khovanskii basis for w, and the ideal hker(ψ),Xii is prime for all i ∈ [6], so

w ∼ 2 2 2 R(A2,4, ˆ ) = k[X1,...,X6, Y12,...,Y34]/hX1 Y12Y34 − X5 Y13Y24 + X3 Y14Y23i.

In particular, the Pl¨ucker generators are a strong Khovanskii basis of w. We can recover w the algebra R(A2,4, |σ1 ) obtained by restricting the coordinate sheaf of the family G(Ψ) → Y (Σ) to the open toric subvariety Y (σ1) ⊂ Y (Σ) by setting X3 = X4 = X5 = X6 = 1:

w ∼ 2 R(A2,4, |σ1 ) = k[X1,X2, Y12,...,Y34]/hX1 Y12Y34 − Y13Y24 + Y14Y23i. 6.3. Prime cones. In [KM19] the authors show that a full rank discrete valuation can be constructed on a domain A with Khovanskii basis B⊂ A by choosing a linearly independent set u = {u1,...,ud} in a prime cone C ⊂ Trop(I). These points are then arranged into the rows of a d × n matrix M, where n = |B|. The valuation vu then sends bi to the i-th column of M. The collection u deﬁnes a linear map Φu : Cd → C ⊂ Trop(I), where Cd is a smooth cone with d rays. In this case, the matrix M is the diagram of the corresponding ﬂat family over Ad(k). This family is described in [KM19, Section 6].

6.4. Examples from cluster algebras. As we have seen in this section, ON valuations appear on the coordinate rings of rational varieties. In particular, any map φ : A → k(TN ) can be composed with the canonical valuation wN : k(TN ) → ON . Cluster varieties are equipped with large combinatorial families of such maps, in particular there is one for each cluster chart. Following [GHKK18], we consider an A-type cluster variety A associated to a non-degenerate skew-symmetric, bilinear form { , } on a lattice N. Let P ∗ : N → M be the map P ∗(n)= {n, ·} with dual map −P ∗ = P ∗∗ : N → M. We make technical assumptions T that A has no frozen variables, and that Θ = Aprin(Z ) (see [GHKK18, Section 7]). ∨ A seed s = {e1,...,en} ⊂ N determines a cone Q≥0s = σs ⊂ NQ and a dual cone − ∨ − σs ⊂ MQ called a chamber. The monoid Ns = σs ∩ N defines a polynomial ring k[Ns ] equipped with an action by the torus TN , in particular the TN character corresponding to n − ∗ a monomial z ∈ k[Ns ] is P (n) ∈ M. Each seed s determines a chart is : TN → A. ′ A mutation of seeds µk : s → s along the k-th element corresponds to a birational map µk : TN 99K TN and a piecewise linear map µk : N → N. There is also a corresponding + dual piecewise-linear map Tk : M → M. Let Hk = {m | hek,mi = 0} ⊂ M, with Hk − − + and Hk the two corresponding half-spaces. On Hk , Tk is the identity map, and on Hk , ∗ Tk(m)= m + hek,miP (ek). The mutation maps µk : N → N and Tk : M → M commute with P ∗ : N → M. One of the central constructions of [GHKK18] is the canonical basis Θ ⊂ k[A]. The elements of Θ are linearly independent, and the vector space can(A) = kθ ⊂ k[A] is shown to be a subalgebra of k[A] under the conditions we have assumed here.L A choice of seed s defines for each θ ∈ Θ an element gs(θ) ∈ M called the g-vector. For a mutation ′ µk : s → s we have gs′ (θ) = Tk(gs(θ)). There are distinguished subspaces of can(A) each ∗ ∨ with a basis inherited from Θ. For each m ∈ M we have Fm(s)= h{θ | m−gs(θ) ∈ P (σs )}i. We let Ss = m∈M Fm(s) be the Rees space of this system of filtrations. L 36 − Proposition 6.8. The space Ss is a free, M-graded k[Ns ] module with generators given by the canonical basis Θ. Furthermore, Ss is an algebra, and L(Ss) = (can(A), vs), where vs : can(A) → Oσs has linear adapted basis Θ. n − Proof. Once again, the M-grading on Ss is given by assigning z θ for n ∈ Ns the weight P ∗(n)+ gs(θ). The fact that Ss is an algebra and vs is a valuation are both deep results from [GHKK18, Section 9]. The freeness property is a consequence of Proposition 3.26.

∗ Letσ ˆs ⊂ NQ be the cone that maps to σs ⊂ MQ under P . For ρ ∈ σs ∩ M with ρ = P ∗(ρ′) we have vs(θ)[ρ]= hρ′,gs(θ)i. Once again, we fix a seed s. The g-fan ∆s ⊂ MQ for the seed s is a simplicial fan composed ′ of cones σs,s′ , where s runs over all other seeds in the cluster complex and σs,s = σs. Here one thinks of σs has the positive orthant. For a seed s′ connected to s through a composite series of mutations µk, σs,s′ ⊂ MQ is defined by the property Tk(σs,s′ )= σs′ . Remarkably, ′ whenever µ : s → s is a simple mutation, σs,s′ ∩ σs,s share the common facet Hk ∩ σs. It ′ is shown in [GHKK18] that a mutation µk : s → s takes ∆s to ∆s′ , and each restriction Tk : σs,s′′ → σs′,s′′ is linear. For σs,s′ ∈ ∆s, we use the map Tk : σs,s′ → σs to define a vs s′ vs s′ vs′ k valuation , : can(A) → Oσs,s′ , where , (f)[ρ]= (f)[T (ρ)].

vs s′ Proposition 6.9. The valuations , : can(A) → Oσs,s′ fit together to define a valuation w ˆ s : can(A) → O∆s . Furthermore, for each θ ∈ Θ, there is a line bundle Ds(θ) on the toric w variety Y∆s , and L( θ∈Θ Ds(θ)) = (can(A), s). L Proof. We show that for any σs,s′ and σs,s′′ meeting at a facet in ∆s, the restrictions of vs,s′ and vs,s′′ agree. Since Θ is an adapted basis of all vs,s′ , it suffices to show the restrictions ′ of vs,s′ (θ) and vs,s′′ (θ) coincide. If this already holds when s = s (ie there is a simple ′′ mutation µ : s → s ), then for ρ ∈ σs,s′ ∩ σs,s′′ we have vs,s′ (f)[ρ] = vs′ (f)[Tk(ρ)] = ′ ′ vs′,s′′ (f)[Tk(ρ)] = vs,s′′ (f)[ρ], where µk : s → s . So it remains to check the case when s is adjacent to s. Let Tk : σs,s′ → σs′ ; then be definition T is identity on the shared face of σs,s′ and σs, as this is contained in Hk. Now let ρ be in this face, then vs,s′ (θ)[ρ] = vs′ (θ)[ρ] = ∗ ′ ′ ∗ hP (ρ ),gs′ (θ)i, for ρ ∈ NQ in the hyperplane which maps to Hk under P . This is equal to ′ ′ ∗ ′ ∗ ′ hρ ,Tk(gs(θ))i = hρ ,gs(θ)+ hek,gs(θ)iP (ek)i. But by assumption hρ , P (ek)i = {ρ ,ek} = ′ ∗ ′ ∗ ′ −{ek,ρ } = −hek, P (ρ )i = 0, since P (ρ ) ∈ Hk. This implies that the vs,s′ fit together w ˆ into a valuation s : can(A) → O∆s with adapted basis Θ. For each θ ∈ Θ, the restriction w of s(θ) to a face of ∆s is linear; it follows that there is a toric line bundle Ds(θ) on Y∆s corresponding to the piecewise linear function ws(θ). Theorem 3.23 implies the rest.

Observe that while ws depends on the choice of seed s, the ring of global coordinates R(can(A), ws) does not, as the cluster fans ∆s and the valuations ws are all related by piecewise-linear maps.

References

[And13] Dave Anderson. Okounkov bodies and toric degenerations. Math. Ann., 356(3):1183–1202, 2013. [CLS11] D.A. Cox, J.B. Little, and H.K. Schenck. Toric Varieties, volume 124 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2011. [DRJS18] Sandra Di Rocco, Kelly Jabbusch, and Gregory G. Smith. Toric vector bundles and parliaments of polytopes. Trans. Amer. Math. Soc., 370(11):7715–7741, 2018. [Ful93] William Fulton. Introduction to toric varieties, volume 131 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 1993. The William H. Roever Lectures in Geometry. [GG16] Jeffrey Giansiracusa and Noah Giansiracusa. Equations of tropical varieties. Duke Math. J., 165(18):3379–3433, 2016. 37 [GHKK18] Mark Gross, Paul Hacking, Sean Keel, and Maxim Kontsevich. Canonical bases for cluster algebras. J. Amer. Math. Soc., 31(2):497–608, 2018. [GHPS] Jose Gonzalez, Milena Hering, Sam Payne, and Hendrik Süß. Cox rings and pseudoeffective cones of projectivized toric vector bundles. J. Algebra and Number Theory, 6 (2012), no. 5, 995-1017. [Gon] J Gonzalez. Projectivized rank two toric vector bundles are mori dream spaces. Communications in Algebra, 40 (2012) no. 4, 1456-1465. [HS] Jürgen Hausen and Hendrik Süß. The cox ring of an algebraic variety with torus action. Adv. Math. 225 (2010), no. 2, 977–1012. [IM] Nathan Ilten and Christopher Manon. Rational complexity-one T-varieties are well-poised. Int. Math. Res. Not, rnx254, 2017. [KK12] Kiumars Kaveh and A. G. Khovanskii. Newton-okounkov bodies, semigroups of integral points, graded algebras and intersection theory. Ann. of Math., 176(2):925–978, 2012. [Kly89] A. A. Klyachko. Equivariant bundles over toric varieties. Izv. Akad. Nauk SSSR Ser. Mat., 53(5):1001–1039, 1135, 1989. [KM] Kiumars Kaveh and Christopher Manon. Toric principal bundles, piecewise linear maps and buildings. arXiv:1806.05613 [math.AG]. [KM19] Kiumars Kaveh and Christopher Manon. Khovanskii bases, higher rank valuations, and tropical geometry. SIAM J. Appl. Algebra Geom., 3(2):292–336, 2019. [KS98] Allen Knutson and Eric Sharpe. Sheaves on toric varieties for physics. Adv. Theor. Math. Phys., 2(4):873–961, 1998. [MS15] Diane Maclagan and Bernd Sturmfels. Introduction to tropical geometry, volume 161 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2015. [nLa19a] nLab authors. biproduct. http://ncatlab.org/nlab/show/biproduct, April 2019. Revision 33. [nLa19b] nLab authors. Schur functor. http://ncatlab.org/nlab/show/Schur%20functor, April 2019. Revision 92. [Nod] Bernt Nodland. Some positivity results for toric vector bundles. arXiv:2002.00604 [math.AG]. [Pay09] Sam Payne. Analytification is the limit of all tropicalizations. Math. Res. Lett., 16(3):543–556, 2009. [Per04] Markus Perling. Graded rings and equivariant sheaves on toric varieties. Math. Nachr., 263/264:181–197, 2004. [SS09] David Speyer and Bernd Sturmfels. Tropical mathematics. Math. Mag., 82(3):163–173, 2009. [Sta19] The Stacks project authors. The stacks project. https://stacks.math.columbia.edu, 2019. [Stu96] Bernd Sturmfels. Gröbner bases and convex polytopes, volume 8 of University Lecture Series. American Mathematical Society, Providence, RI, 1996.

Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, USA. Email address: [email protected]

Department of Mathematics, University of Kentucky, Lexington, KY, USA Email address: [email protected]