This paper introduces a valuation-based framework to classify torus equivariant flat families and toric vector bundles, linking tropical geometry with algebraic geometry to analyze Mori dream space properties.
Contribution
It develops a valuation approach using piecewise linear functions to classify flat families and toric vector bundles, connecting tropical geometry with classical algebraic geometry.
Findings
01
Classification of torus equivariant flat families via piecewise linear maps
02
Tropicalized linear spaces characterize toric vector bundles
03
Criteria for Mori dream space property in projectivized toric bundles
Abstract
Using the notion of a valuation into the semifield of piecewise linear functions, we give a classification of torus equivariant flat families of finite type over a toric variety base, by certain piecewise linear maps between fans. As a consequence we derive a classification of toric vector bundles phrased in terms of tropicalized linear spaces. We use these tools to give a characterization of the Mori dream space property for a projectivized toric vector bundle.
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TopicsNonlinear Waves and Solitons · Algebraic structures and combinatorial models · Polynomial and algebraic computation
Full text
Toric flat families, valuations, and applications to projectivized toric vector bundles
Kiumars Kaveh
Department of Mathematics, University of Pittsburgh,
Pittsburgh, PA, USA.
Using the notion of a valuation into the semifield of piecewise linear functions, we give a classification of torus equivariant flat families of finite type over a toric variety base by certain piecewise linear maps between fans. As a consequence, we derive a classification of toric vector bundles phrased in terms of tropicalized linear spaces. We use these tools to give a characterization of the Mori dream space property for a projectivized toric vector bundle.
The first author is partially supported by National Science Foundation Grant DMS-2101843 and a Simons Collaboration Grant (award number 714052). The second author is partially supported by National Science Foundation Grant DMS-2101911 and a Simons Collaboration Grant (award number 587209).
We use ideas from Gröbner theory and tropical geometry to give a classification of torus equivariant flat families over a toric variety base (toric flat families), far extending Klaychko’s classification of equivariant vector bundles on toric varieties (toric vector bundles). We use our techniques to give an algorithm to establish that a given projectivized toric vector bundle is a Mori dream space. In particular, this recovers many known constructions of such Mori dream spaces in the literature ([Gon12], [GHPS12], [HS10], [Nod]) and gives a systematic way to produce many new classes of examples. Below we first give an overview of the paper and then explain the main results in more detail.
Let R be a positively graded finitely generated k-domain with X=Proj(R) the corresponding irreducible projective variety with dim(X)=n. Given a full rank valuation v:R∖{0}→Zn+1, the theory of Newton-Okounkov bodies assigns to R a convex body Δ(R,v)⊂Rn which encodes asymptotic information about the Hilbert function of R and hence geometric information about (X,O(1)). When the value semigroup S(R,v):=v(R∖{0}) is finitely generated, we say that (R,v) has a finite Khovanskii basis. In this case one has a flat family over the affine line A1 with generic fiber X and the special fiber X0, the projective toric variety associated to the semigroup S(R,v). Such a family is called a toric degeneration (see [And13]). The Khovanskii basis theory is concerned with the computational side of the Newton-Okounkov bodies theory.
It provides a means to generalize Gröbner basis theory for polynomial rings to much larger classes of finitely generated algebras R (see [KM19, Ehr20]).
In this paper, we replace Zn+1-valued valuations with valuations with values in the semifield ON of integral piecewise linear functions on a finite rank lattice N. We introduce a notion of Khovanskii basis in this setting. We see that, extending the construction of toric degenerations over an affine line from Zn+1-valued valuations, finite Khovanskii bases in this setting give rise to flat families with general fiber X and over a toric variety base Y(Σ). In particular, this applies to vector bundles over a toric variety Y(Σ) and yields a systematic and algorithmic study of global coordinate rings (Cox rings) of projectivized toric vector bundles. We summarize our results in this regard as follows:
∙
After fixing some algebraic data, we classify toric vector bundles by points on a certain polyhedral fan (Proposition 6.12), and we show that special faces of this fan characterize natural families of bundles with Mori dream space property (Proposition 6.2).
2. ∙
We give an algorithm (Algorithm 5.6) which constructs a finite generating set of the Cox ring of a projectivized toric vector bundle, provided one exists. The algorithm does not terminate if one does not exist.
3. ∙
We characterize the generating sets of Cox rings of projectivized toric vector bundles using prime points on an associated tropical variety (Theorem 1.5).
The above recover several finiteness results from the literature ([Gon12], [GHPS12], [HS10], [Nod]), and obtain new classification results for bundles with the Mori dream space property.
In a future work, we plan to address applications of our approach to geometry of matroids. It seems that the point of view, introduced in the present paper, of regarding a toric vector bundle as a tropical point (over the semifield of piecewise linear functions), gives a very natural way to construct and extend the tautological classes of matroids as introduced in [BEST].
To prove the above results, we develop a theory of Khovanskii bases for valuations with values in the semialgebra of piecewise linear functions. This requires us to carefully study the category of vector spaces equipped with prevaluations into the piecewise linear semialgebra. An important ingredient is establishing an adjunction between this category and the category of T-equivariant quasi-coherent sheaves on a toric variety.
1.1. Classification of toric flat families
We take the ground field k to be an algebraically closed field. We let TN to be the algebraic torus with cocharacter lattice N and character lattice M=Hom(N,Z). We denote by Y(Σ) the toric variety associated to a fan Σ in NQ=N⊗Q. We let OΣ denote the set of all piecewise linear functions on ∣Σ∣, the support of Σ, that are integral with respect to N. The set OΣ together with the tropical operations of min and + is a semifield which we refer to as the piecewise linear semifield on Σ. 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.
Definition 1.1**.**
By a toric flat family over Y(Σ) we mean a flat affine family π:X→Y(Σ) equipped with an action of the torus TN lifting its action on the toric variety Y(Σ). In particular, the fibers of π are affine schemes.
Toric flat families include many important and well-studied classes of varieties such as torus equivariant vector bundles and principal bundles on toric varieties (toric vector bundles and toric principal bundles), see [Kly89, BDP20, KM22]. Another class of toric flat families are toric degenerations, i.e. toric flat families over the affine line. Toric degenerations have a close connection with the theory of Newton-Okounkov bodies (see [And13, KMM22]) and toric degenerations of special classes of varieties have been studied extensively by various authors.
Before stating our main results, we give a brief review of some previous work in the literature. Toric vector bundles were first classified by Kaneyama in [Kan75] using certain cocycles. Klyachko then gave a classification [Kly89] in terms of compatible Z-filtrations in a vector space. More recently, the authors [KM22], and Biswas, Dey, and Poddar [BDP20, BDP16] have given different classifications of toric principal bundles. The classification of Biswas, Dey, and Poddar is along the lines of Kaneyama’s, utilizing certain cocycles. The authors’ classification in [KM22] is in terms of piecewise-linear maps to Tits buildings, and should be viewed as an extension of Klyachko’s result.
Klyachko’s work has lead to a number of powerful results and generalizations. In [Pay09a], Payne uses Klyachko’s classification to study moduli space of toric vector bundles over Y(Σ) with fixed total Chern class. In [Per04], Perling gives a Klyachko-like classification of torus-equivariant sheaves on a toric variety in terms of representations of certain quivers derived from the fan Σ. We also mention the work of DiRocco, Jabbusch and Smith [DRJS18] where they give a criterion for global generation of a toric vector bundle in terms of an associated collection of convex polytopes called a parliament of polytopes.
The general approach in the present paper encompasses and extends many previous works in the literature. We describe the connection between [DRJS18] in Section 4.5. We also consider torus-equivariant sheaves in Section 3, where we attach a prevalued vector space to any such sheaf. This leads to an adjunction of categories which is not an equivalence in general. Nevertheless, it is an equivalence on a subcategory which we call eversive sheaves, and in particular contains reflexive sheaves.
Klyachko’s classification is very close to the classification of toric vector bundles by tropical points (Theorem 1.4) that we obtain as a special case of our classification of toric flat families (Theorem 1.2). We remark that Theorem 1.4 is essentially contained in Payne’s observation in [Pay08, Pay09b] that the Klyachko data of a toric vector bundle can be used to construct a filtration-valued function on the support ∣Σ∣ of the fan.
Throughout we assume that X comes equipped with a Gm-action preserving the fibers and commuting with the TN-action. From the TN-equivariance it follows that the fibers of X over the open orbit in Y(Σ) (the general fibers) are all isomorphic to Spec(A), for a finitely generated positively graded k-domain A. Specifically, we fix a point x0 in the open orbit in Y(Σ) and we take A to be the coordinate ring of the fiber over x0. One can think of x0 as the identity element in the torus TN.
Let A be the sheaf of OY(Σ)-algebras such that SpecY(Σ)(A)=X.
The Gm-action on X corresponds to a positive grading A=⨁n≥0An. Each An can be regarded as the sheaf of sections of a TN-equivariant vector bundle on Y(Σ). We thus can regard A as an infinite dimensional TN-equviariant vector bundle on Y(Σ). The Klyachko classification of toric vector bundles then implies that A corresponds to a system of filtrations on the k-algebra A, regarded as an infinite dimensional vector space. In our setting, this system of filtrations gives rise to a map:
[TABLE]
When the family has reduced and irreducible fibers, it is straightforward to verify that the map v satisfies the axioms of a valuation from an algebra to a semifield.
We point out that in the valuation theory literature in commutative algebra, a valuation takes values in a totally ordered abelian group such as Zd.
Given a domain A and a valuation v:A∖{0}→Zd, the theory of Khovanskii bases attempts to generalize basic constructions and techniques from Gröbner theory of polynomial ring to (A,v) (see [KM19, Ehr20]). One of the purposes of the present paper is to show that valuations with values in the piecewise linear semifield are also important in algebraic geometry and are the right gadgets to classify toric flat families.
When the family X is finite-type, the information in v can be captured in its values on a certain set of generators B⊂A. In sympathy with the case when v takes values in Zd, we call B a Khovanskii basis of v.
The following is our main theorem summarizing these ideas (see Section 4).
Theorem 1.2** (Toric flat families as valuations).**
*Let π:X→Y(Σ) be a toric flat family of finite type with reduced, irreducible fibers and let A denote the coordinate ring of general fibers. Such families are classified by valuations
v:A→OΣ having a finite Khovanskii basis B with the following property: for each σ∈Σ, A has a certain vector space basis Bσ consisting of monomials in B such that v(b)∣σ is a linear function, for all b∈Bσ. We refer to Bσ as a linear adapated basis (see Section 3.4).
*
Next we interpret the data of a valuation v:A→OΣ as a piecewise linear map between fans. This point of view is motivated by the classification of toric vector bundles in terms of piecewise linear maps to the Tits buildings of general linear groups (see [KM22], also [Kly89]). As above let B be a finite Khovanskii basis for (A,v). Let I be the ideal of relations among the generating set B. It gives a presentation of A as a quotient of a polynomial ring by I. In Section 4 we define a certain subfan K(I) of the Gröbner fan of I. Motivated by the theory of buildings we define an apartment in K(I) to be the subcomplex obtained by intersecting K(I) with a maximal face τ of the Gröbner fan of I.
Our next result, roughly speaking, states that different ways to “fold” the fan Σ into the fan K(I) correspond to distinct toric flat families over Y(Σ) with general fiber Spec(A) and Khovanskii basis B. More precisely, we have the following (Proposition 4.11).
Corollary 1.3** (Toric flat families as peicewise linear maps).**
The information of a toric flat family π:X→Y(Σ) with a finite Khovanskii basis B is equivalent to a
piecewise linear map Φ:∣Σ∣→K(I) which maps each cone in Σ to an apartment in K(I). We call such a piecewise linear map a Σ-adapted map (see Section 4).
The above corollary says that K(I) can be thought of as a “classifying space” for the toric flat families with general fiber Spec(A) and Khovanskii basis B.
In Section 4 (Proposition 4.11) we show that Φ is captured by its values on the rays of Σ, which we organize into a matrix D(Φ) called the diagram of Φ. This gives a systematic combinatorial method for constructing toric flat families over Y(Σ). For example when Σ is a simplicial fan, a diagram corresponds to choosing a point w(ϱ)∈K(I), for every ray ϱ∈Σ(1), such that for all ϱ from a face σ∈Σ the w(ϱ) land in a common apartment of K(I).
When the family has reduced and irreducible fibers, and hence v is a valuation, Φ=Φv in fact takes values in the tropical variety Trop(I). In the particular case when the toric family is a toric vector bundle we get the following.
Theorem 1.4** (Toric vector bundles as tropical points).**
Let E be an r-dimensional vector space.
The toric vector bundles over Y(Σ), with E as general fiber, 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 the monomials of a vector space basis Bσ⊂E.
In this case, the ideal I is a linear ideal L and one can think of the information of a toric vector bundle in three differnt ways:
(i)
As a point Φ∈TropOΣ(L) in the tropical variety over the semifield OΣ (see Section 2 for the notion of tropical variety over a semifield).
(ii)
As a piecewise linear map from ∣Σ∣ to Trop(L) (see Section 4).
(iii)
As a matrix with rows in Trop(L) (see Section 4).
The idea in part (ii) of Theorem 1.4, to encode the data of a toric vector bundle as a family of filtrations of a vector space indexed by the points in the support of the fan ∣Σ∣, is not new and first appeared in the work of Payne [Pay09b]. This construction was used in the work of Hering, Payne, and Mustaţă [HMP10] to give a combinatorial characterization of ampleness for toric vector bundles. Corollary 1.3 and Theorem 1.4 can be thought of as extensions of the well-known fact that toric line bundles over Y(Σ) are classified by integral piecewise linear maps ϕ:∣Σ∣→Z, to toric vector bundles: Φ:∣Σ∣→Trop(L), and toric flat families: Φ:∣Σ∣→K(I).
1.2. Mori dream space property and strong Khovanskii bases
Next we address the problem of finite generation of the total coordinate ring of a toric flat family. In general, this is a notoriously difficult problem. For example, a very special case of this problem that asks which projectivized toric vector bundles are Mori dream spaces is already very difficult and has been subject of much recent research. Our valuation theory approach gives a systematic way to attack this problem and to recover many of the previously known examples of such Mori dream spaces as well as many new examples and constructions.
The question of when a projectivized toric vector bundle is a Mori dream space goes back to the work of Hering, Mustaţă, and Payne [HMP10]. The first results along these lines are found in the works of González [Gon12] and Hausen and Süß [HS10]. In [Gon12], González shows that the projectivization of every rank 2 vector bundle over a projective toric variety is a Mori dream space. In [HS10], Hausen and Süß show that the projectization of the tangent bundle of any smooth, projective toric variety is a Mori dream space. The common thread of these two results is that the Mori dream space property can be obtained by restricting the number of steps in the Klyachko filtrations of the associated bundle to at most two. Such bundles are called sparse bundles in Section 6. Sparse bundles are shown to always have Mori dream space projectivizations by González, Hering, Payne, and Süß in [GHPS12] and by Nödland in [Nod]. We recover this result as Corollary 6.7 of Proposition 6.2. In particular, the sparse condition is a special case of membership in certain faces of the fan which we introduce (see Proposition 6.17). Using this tropical approach, we can produce new families of bundles whose projectivizations are Mori dream spaces. For example, the uniform bundles analyzed in Corollary 6.5 are defined by requiring the ideal I to be a generic linear ideal.
In the usual Khovanskii basis theory, one considers a finitely generated domain A equipped with a valuation v:A∖{0}→Zd. A set B⊂A is then a Khovanskii basis for (A,v) if the images of the elements in B generate the associated graded algebra grv(A). Under mild assumptions, it follows that B is a generating set for A as well. When B is finite, essentially one can reduce computational problems regarding A and ideals in it to computations in grv(A). There is an algorithm to extend a given subset of A to a finite Khovanskii basis. This algorithm terminates in finite time if and only (A,v) has a finite Khovanskii basis ([KM19, Algorithm 2.18 and Corollary 2.19]).
We apply these ideas to the problem of finite generation of the total coordinate ring of a toric flat family. We give an analogue of the algorithm above for a valuation v:A→OΣ to find a finite Khovanskii basis for the total coordinate ring of the family.
More precisely, let A be a sheaf of algebras on a smooth toric variety Y(Σ). In Section 5 we consider the total coordinate ring
[TABLE]
of global sections of A. In the case of a toric vector bundle E, this ring corresponds to the Cox ring R(E) of the associated projective bundle PE. Let v:A→OΣ, Φ:∣Σ∣→K(I) and diagram D(Φ) be the quasivaluation, piecewise linear map and the diagram corresponding to the family respectively. We wish to give conditions on Φ:∣Σ∣→K(I), or equivalently the diagram D(Φ), which ensure that R(A) is finitely generated, and understand the piecewise linear geometry of the corresponding space of maps Φ.
This leads us to define the notion of a strong Khovanskii basis. 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) (Section 5). A strong Khovanskii basis is always a Khovanskii basis, but not vice-versa. In general, finite strong Khovanskii bases do not exist. Our first contribution in this regard 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 (that is, for valuations v:A→OΣ) using the notion of prime points on a tropical variety introduced in [KM19]. Let n=∣Σ(1)∣ be the number of rays in Σ.
Theorem 1.5**.**
Let (A,v) be a valuation as above with Khovanskii basis B⊂A and diagram D, and let IB be the ideal as in Proposition 5.8. Let w~1,…,w~n∈Trop(IB) be the lifts of the rows of D (see Theorem 5.9). Then B is a strong Khovanskii basis if and only if each w~i is a prime point, that is, the initial ideal corresponding to w~i is prime.
We use Theorem 1.5 to study the Mori dream space property of projectivized toric vector bundles in several cases. As an application, we give general combinatorial conditions on the diagram which characterize those Cox rings R(E) which are presented as a complete intersection (Proposition 6.2).
In Section 6.1 we introduce the set Δ(Σ,Trop⋆(I)) (Definition 6.11) of diagrams D(Φ) for bundles over Y(Σ) with fiber E and Khovanskii basis B⊂Sym(E). We show that Δ(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 6.16).
Conjecture 1.6**.**
The set Δ(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.
Proposition 6.17 establishes a weak form of this conjecture. We use a related result (Proposition 6.13) to describe alterations to the diagram D(Φ) which preserve the Mori dream space property. In particular, we give polyhedral conditions implying that the pullback of a Mori dream space bundle along a toric blow-up is a Mori dream space bundle (Corollary 6.14).
We finish the introduction by looking at two ingredients of our valuation theoretic/tropical approach to study toric flat families which we believe are of interest as their own separate topics.
1.3. Tropical geometry over ON
As before, let N and M be dual lattices. Let k(TN) denote the field of rational functions on the torus TN.
Definition 1.7**.**
For p(t)=∑Cmtm∈k[TN] let wN(p)=min{m∣Cm=0}. In particular wN(p) is the support function of the Newton polytope of p. Let wN:k(TN)→ON denote the corresponding extension of wN to the field of quotients: wN(qp)=wN(p)−wN(q).
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 k(TN) to ON, and the image of k[TN]⊂k(TN) under wN is precisely the set of support functions for polyhedra with vertices in M. Moreover, it is well-known that any piecewise-linear function can be represented as the difference of convex functions, it follows that wN is onto. In [SS09] it was suggested by Sturmfels and Speyer that one could develop tropical geometry over a tropical semiring of polyhedra; we view Theorem 1.4 as a first step in this direction: the points on tropical linear spaces over ON correspond to toric vector bundles. In Section 7 we explore this perspective further by showing that the operation of tropicalizing the solution to a system of equations over k(TN) 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 O^Σ 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 O^Σ. For the technical core of the paper, we show that there is an adjoint pair of functors L,R which relate VectΣ to the category ShY(Σ)M of TN-equivariant sheaves on the toric variety Y(Σ). Locally for σ∈Σ, the operation R:Vectσ→ShY(σ)M is akin to taking the Rees algebra of a filtration, and L:ShY(σ)M→Vectσ is its left adjoint. This is related to work of Perling [Per04] and Klyachko [Kly89] by the natural relationship between filtrations and prevaluations (see Section 2). For a general TN-sheaf F, the corresponding prevaluations may take values in functions from O^Σ 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 sheaves which we call eversive sheaves, the functors L and R become an equivalence, and we obtain classification results for toric sheaves by prevaluations. It is shown in Example 3.17 that reflexive sheaves are eversive but not vice versa. Here the term “eversive” comes from the word “eversion,” meaning the act of turning something inside out. The term is meant to suggest taking the prevaluation associated to a sheaf is like revealing inside of the sheaf as in turning it inside out.
Acknowledgements: We would like to thank Laura Escobar and Megumi Harada for patiently reading the paper and giving valuable comments. We also thank Sam Payne for useful conversations. We thank Courtney George for a much clearer formulation of Corollary 6.5.
Notation:
•
N, a finite rank lattice with dual lattice M. We let NR=N⊗R and MR=M⊗R.
•
ρ, an element of N.
•
Σ, a rational polyhedral fan in NQ.
•
σ, a rational polyhedral cone in NQ, usually a face of Σ.
•
ϱ, a ray in Σ, namely an element of Σ(1).
•
ON, the semifield of integral piecewise linear functions on NR with the operations min and +.
•
OΣ, the semifield of piecewise linear functions on ∣Σ∣⊂NR.
•
Y(σ), the affine toric variety corresponding to a polyhedral cone σ⊂NR.
•
Sσ, the coordinate ring of Y(σ).
•
Y(Σ), the toric variety corresponding to a rational polyhedral fan Σ.
Δ(Σ,L), the set of diagrams adapted to Σ with rows in Trop(L) (Section 5).
•
R(A,v^), the total section ring associated to a valuation v:A→OΣ (Section 5).
•
R(E), the Cox ring of the projectivized toric vector bundle PE (Section 6).
2. Background on valuations
In this section we collect the basic definitions and background material for prevaluations on vector spaces, and valuations and quasivaluations on algebras. We start by introducing the notion of a valuation on an algebra with values in an idempotent semialgebra. We call a valuation on a vector space a prevaluation.
The term prevaluation appears in [KK12, Section 2.1] where the notion of a prevaluation with values in a totally ordered group is defined.
Definition 2.1**.**
A semialgebra(O,⊕,⊙) is a set O with two binary operations ⊕ and ⊙ that satsify the same axioms as addition and multiplication in a ring, with the exception that there are not necessarily additive inverses. We denote the identity element with respect to ⊕ by ∞. If all the elements beside the additive inverse ∞ have multiplicative inverses then O is called a semifield.
A semialgebra O is called idempotent if a⊕a=a for all a∈O.
First important example of an idempotent semifield is the set Q=Q∪{∞} together with the operations a⊕b=min(a,b) and a⊙b=a+b. It is usually referred to as the tropical semifield. Clearly Z=Z∪{∞} is a subsemifield of Q.
The following are important examples of (idempotent) semifields and semialgebra in this paper.
Example 2.2** (Semifield of functions on a set).**
Let OX be the set of Q-valued functions on a set X with addition ⊕ and multiplication ⊙ to be the pointwise min and + of functions respectively. The additive identity, also denoted ∞, is the function which assigns ∞ to every point.
Example 2.3** (Semifield of piecewise linear functions on a lattice).**
Let N be a lattice, that is, a free abelian group of finite rank. Ley NQ=N⊗Q. Recall that a piecewise linear function on NQ is a function f:NQ→Q for which there is a complete fan in NQ such that f is linear restricted to each cone in the fan. A piecewise linear function f is called integral if it maps N to Z. The set of all piecewise linear functions on NQ together with operations of min and + of functions is a semifield. The set of integral piecewise linear functions is a subsemifield which we denote by ON.
More generally, if Σ is a (not necessarily complete) fan in NQ, we define OΣ to be the semifield of all integral piecewise linear function on ∣Σ∣, the support of Σ.
This semifield plays a most important role in this paper. We will also need the semialgebra O^Σ obtained by taking limits of elements in OΣ. More precisely, O^Σ is the semialgebra of functions ψ:∣Σ∣→Q such that ψ(ℓρ)=ℓψ(ρ) for all ℓ∈Q≥0 and ρ∈∣Σ∣. Note that a function ψ∈O^Σ is allowed to attain infinite values on points of Σ. Hence O^Σ is not a semifield.
Example 2.4** (Semialgebra of convex polytopes).**
Let M be the dual lattice to N. The set of all convex polytopes in MR=M⊗R has a natural structure of a semialgebra. The sum of two polyotpes is defined to be the convex hull of their union, and the product of two polytopes is defined to be their Minkowski sum. The subset of lattice polytopes is a subsemialgebra. Recall that a lattice polytopes is a polytope whose vertices lie in M. The semialgebra of convex polytopes in MR can be identified with the semialgebra of concave piecewise linear functions. The identification is given by sending a convex polytope to its support function.
Any idempotent semialgebra O has an intrinsic partial ordering ⪰ defined by a⪰b if a⊕b=b.
Definition 2.5**.**
For a k-algebra A, a quasivaluationv:A→O is a function which satisfies the following:
(1)
v(fg)⪰v(f)⊙v(g),
2. (2)
v(f+g)⪰v(f)⊕v(g),
3. (3)
v(C)=0, for all C∈k∖{0},
4. (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={b1,…,bd}⊂A. We let π:k[x]→A be the associated presentation, with ideal IB=ker(π). For any monomial xα∈k[x] with α=(α1,…,αd) there is a function evxα:OB→O defined by
[TABLE]
Following [GG16, 5.1], the tropical variety TropO(IB)⊂OB is defined to be the set of tuples (ψ1,…,ψd) such that for any polynomial ∑j=1mCjxα(j)∈IB we have:
[TABLE]
for any 1≤i≤m. The following is a well-known relationship between valuations and the tropical variety (see [Pay09a, GG16]).
Proposition 2.6**.**
Let v:A→O be a valuation and B={b1,…,bd}⊂A a generating set. Then the tuple (v(b1),…v(bd))∈OB is a point in the tropical variety TropO(IB).
If v:E→O is a prevaluation and ψ∈O, we let Fψ(v)={f∣v(f)≥ψ}. It is straightforward to verify that Fψ(v) is a subspace of E. We use a different notation to distinguish the special case O=Q, setting Gr(v)={f∣v(f)≥r} for r∈Q. Similarly, we let G>r(v)={f∣v(f)>r}. The spaces Gr(v) fit together to form a Q filtration of E by k vector spaces. The associated graded vector space grv(E) is defined to be the direct sum:
[TABLE]
If v:A→Q 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 v(f) is taken to be ∞ if the maximum is never attained. The following definition is from [KM19, Section 2.5].
Definition 2.7**.**
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∈Q.
If bi, i=1,…k are from an adapted basis then v(∑i=1kCibi)=⊕i=1kv(bi). This identity simplifies computations for prevaluations with adapted bases.
We finish this section by recalling the notion of Khovanskii basis for a quasivaluation on an algebra with values in Q. Khovanskii bases for quasivaluations with values in a free abelian group are the subject of [KM19].
Definition 2.8**.**
Let A be a k-algebra, and v:A→Q 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. Equivariant sheaves and the category VectΣ
Recall that OΣ is the semifield of piecewise linear functions with a finite number of linear domains defined on ∣Σ∣, and O^Σ is the semialgebra of functions ψ:∣Σ∣→Q such that ψ(ℓρ)=ℓψ(ρ) for all ℓ∈Q≥0 and ρ∈∣Σ∣. A function ψ∈O^Σ is allowed to attain infinite values on points of Σ and thus O^Σ is not a semifield. We regard elements in O^Σ as limits of elements in OΣ (see the end of Example 2.3).
In this section, we define a category VectΣ of vector spaces equipped with a prevaluation into O^Σ.
Definition 3.1**.**
Let VectΣ be the category whose objects are pairs (E,v), where E is k-vector space and v:E→O^Σ is a prevaluation over k. We call a pair (E,v) a prevalued vector space. 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. We note that, strictly speaking, VectΣ only depends on the support of Σ.
Definition 3.2**.**
For any prevalued vector space (E,v) and ρ∈∣Σ∣ we let vρ:E→Q denote the prevaluation obtained by composition with ρ:
[TABLE]
We let Grρ(v)={f∣v(f)(ρ)≥r}.
Observe that Grρ(v) is the r-th filtration space Gr(vρ) of the Q-filtration associated to vρ. The spaces Grρ(v) can also be used to recover the prevaluation v as v(f)(ρ)=max{r∣f∈Grρ(v)}. We sometimes refer to the spaces Grρ(v) as the Klyachko spaces of v. The filtrations defined by these subspaces are the main players in Klyachko’s classification of torus equivariant vector bundles on toric varieties [Kly89].
Definition 3.3**.**
We let ShY(Σ)M denote the category of TN-equivariant quasi-coherent sheaves of OY(Σ)-modules on the toric variety Y(Σ).
We will construct a functor L:ShY(Σ)M→VectΣ, and show
that a sheaf F∈ShY(Σ)M is determined by its image, the prevalued vector space L(F), if it is from a certain full subcategory ShY(Σ)M,ev. We do this by constructing an adjoint functor R to L over each affine toric chart (Section 3.2). This construction enables us to prove Theorem 1.2 and show that a toric flat family π:X→Y(Σ) is determined by a finite combinatorial data which we call the diagram of π in Section 4.
3.1. Properties of VectΣ
We need to define a few constructions and operations on prevalued vector spaces.
Definition 3.4**.**
For a vector space E and ψ∈O^Σ, we let (E,ψ)∈VectΣ denote the vector space equipped with the prevaluation which assigns to every non-zero element of E the function ψ.
Definition 3.5**.**
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)∈O^Σ.
It is straightforward to show that ⊕ is both a product and a coproduct in VectΣ with the trivial prevalued vetor space ({0},∞) as the identity object.
We use ⊕ to extend the notion of adapted basis for pervaluations with values in Q (Definition 2.7) to prevaluations with values in O^Σ.
Definition 3.6**.**
We say (E,v) has an adapted basisB⊂E if the natural maps (k,v(b))→(E,v), 1→b define an isomorphism (E,v)≅⨁b∈B(k,v(b)). Here (k,v(b)) denotes the 1-dimensional k-vector space equipped with the prevaluation that assigns v(b) to every nonzero element. The set B is said to be a linear adapted basis if, for every b∈B, v(b) is the restriction of a linear function in M to ∣Σ∣.
By Definition 3.5, B is an adapted basis of (E,v) if and only if v(∑iCibi)=min{v(bi)∣Ci=0} for any vector ∑iCibi∈E.
Definition 3.7**.**
Let (E,v)∈VectΣ and π:E→D be a surjection of vector spaces. The pushforward prevaluation π∗(v):D→O^Σ is defined by:
[TABLE]
where it is understood that π∗(v)(g)(ρ)=∞ if the maximum is never attained. Similarly, given ϕ:F→E, we have the pullbackϕ∗(v), defined by ϕ∗(v)(f)=v(ϕ(f)).
The colimit lim(Ei,vi), where i is from an index category I, is constructed by taking limvi:limEi→O^Σ to be the pushforward of the prevaluation on ⨁i∈I(Ei,vi) under the quotient map πI:⨁i∈IEi→limEi. In particular, (limvi)(f)=max{⊕i=1ℓvi(fi)∣∑i=1ℓπI(fi)=f}.
Definition 3.8**.**
The tensor product(E1,v1)⊗(E2,v2) is the pair (E1⊗kE2,v1⋆v2), where v1⋆v2:E1⊗kE2→O^Σ is the prevaluation with associated subspaces Grρ(v1⋆v2) defined as follows:
[TABLE]
That is, (v1⋆v2)(f)(ρ)=max{r∣f∈Grρ(v1⋆v2)}, for ρ∈∣Σ∣, f∈E1⊗kE2.
The prevalued vector space (k,0) is the multiplicative identity object in VectΣ. Also, 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→O^Σ. 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 3.9**.**
Let λ be a partition of {1,…,n}. The Schur functorSλ:VectΣ→VectΣ takes a prevalued vector space (E,v) to (Sλ(E),sλ(v))⊂(E,v)⊗n,
where sλ(v) is the pullback of v⋆n under the inclusion map Sλ(E)⊂E⊗n.
The categories we deal with in this paper are symmetric monoidal and Cauchy-complete (see [nLa19]). 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
In this section, we introduce a functor L from the category of TN-equivariant sheaves to the category of prevalued vector spaces. This construction is a generalization of Klyachko’s construction which assigns a collection of filtrations to a toric vector bundle. Klyachko’s construction itself is a generalization of assigning a piecewise linear function to a toric line bundle.
When the base toric variety is affine, we also define an adjoint functor R from prevalued vector spaces to TN- equivariant sheaves. That L and R are adjoint functors is a main result of this section (Theorem 3.13). The proof is postponed to Section 8. We use this to show that L gives an equivalence of categories between certain subcategory of TN-equivariant sheaves on a toric variety (not necessarily affine) and a full subcategroy of prevalued vector spaces (Theorem 3.19).
As usual Y(Σ) denotes a toric variety with fan Σ. We fix a point in the open orbit and identify the open orbit with the torus TN. For σ∈Σ we denote the corresponding toric affine chart by Y(σ).
First we consider the affine case. A cone σ∈Σ determines an affine semigroup σ∨∩M, where σ∨={u∣⟨ρ,u⟩≤0∀ρ∈σ}⊂MQ is the dual cone of σ. We let S0=k[TN] denote the coordinate ring of TN, and Sσ⊂S0 denotes the affine semigroup algebra associated to σ∨.
Remark 3.10**.**
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.
Let ModSσM denote the category of M-graded Sσ-modules. Let Rm⊂R denote the TN-isotypical component associated to a character m∈M, so that R=⨁m∈MRm. If σ∈Σ is a cone with affine open toric chart Y(σ)⊂Y(Σ) and F∈ShY(Σ)M, then Γ(Y(σ),F)∈ModSσM.
The maximal ideal m⊂Sσ defining the identity element in the open orbit TN⊂Y(σ) is the ideal generated by the forms χu−1 for u∈σ∨∩M. For a module R∈ModSσM let ER=R/mR, and let ϕR:R→ER be the natural surjection.
We now explain the construction of a valuation vR associated to a module R∈ModSσM.
We define Fm(R)⊂ER to be the image of Rm under ϕR. For ρ∈σ∩N and r∈Q let Grρ(R)⊂ER be the subspace defined by:
[TABLE]
Observe that Grρ(R)⊇Gsρ(R) whenever r≤s. So for each ρ∈σ∩N we have a decreasing filtration {Grρ(R)}r∈Q.
Definition 3.11** (Functor L).**
For f∈ER, we let vR(f):σ∩N→Q be the prevaluation defined by
[TABLE]
We define L(R) to be the prevalued vector space (ER,vR)∈Vectσ.
Next, for any (E,v)∈Vectσ and m∈M we can consider the space Fm(v)={f∣v(f)≥m}. If u∈σ∨∩M then m≥m+u as functions on σ, so Fm(v)⊂Fm+u(v).
Definition 3.12** (Functor R).**
For an object (E,v) of Vectσ we define the Rees module R(E,v)∈ModSσM to be the following M-graded vector space:
[TABLE]
We let χu∈Sσ act on R(E,v) by the inclusion map: Fm(v)⊂Fm+u(v).
Theorem 3.13**.**
The constructions L:ModSσM→Vectσ and R:Vectσ→ModSσM define an adjoint pair of functors.
The proof of Theorem 3.13 can be found in Section 8. As part of the data of the adjunction, we obtain maps η:LR(E,v)→(E,v) and ϵ:R→RL(R), these can be described as follows. For (E,v)∈Vectσ, the space LR(E,v) has underlying vector space limFm(v)⊂E and prevaluation defined by the spaces Grρ(R(E,v))=∑⟨ρ,m⟩≥rFm(v)⊂Grρ(v). This gives the map η. On the other hand, the object RL(R) is the M-graded module ⨁m∈MFm(vR), where Fm(vR)=⋂ρ∈σ∩NG⟨ρ,m⟩ρ(R). This gives the map ϵ.
Definition 3.14**.**
We say (E,v) (respectively R) is eversive if the map η (respectively ϵ) is an isomorphism. We let Vectσev⊂Vectσ and ModSσM,ev⊂ModSσM denote the full subcategories on eversive objects.
Proposition 3.15**.**
For any R∈ModSσM and (E,v)∈Vectσ, both R(E,v) and L(R) are eversive. Furthermore, ϵ and η are isomorphisms if and only if the corresponding objects are eversive. The functors R and L define an equivalence of categories Vectσev≅ModSσM,ev.
Example 3.16** (Eversive toric ideals).**
We give a description of the M-graded ideals which are eversive. Let I⊂Sσ be a M-graded ideal, and let Ω(I)⊂M∩σ∨ be the set of characters in I. For a subset X⊂M we define the σ−hull ∣X∣σ to be the set of m with the property that ⟨ρ,m⟩≤r whenever ⟨ρ,v⟩≤r for all v∈X. We claim that I is eversive if and only if Ω(I)=∣Ω(I)∣σ.
First, EI≅k, and for any ρ∈σ, Grρ(I)⊂EI is non-zero if and only if there is some u∈Ω(I) with ⟨ρ,u⟩≥r. Now, for any m∈M, the m-isotypical space of RL(I) is Fm(vI)=⋂⟨ρ,m⟩=rGrρ(I). The latter is nonzero if and only if for all ρ∈σ, r∈Q such that ⟨ρ,m⟩=r there exists a u∈Ω(I) such that ⟨ρ,u⟩≥r. As a consequence, Fm(vI) is non-zero if and only if for all ρ∈σ, ⟨ρ,m⟩ is less than the largest value ρ obtains on Ω(I). This second condition is equivalent to m∈∣Ω(I)∣σ.
Example 3.17** (Reflexive sheaves vs eversive sheaves).**
Recall that a sheaf F is said to be reflexive if it is isomorphic to its own double-dual: F≅(F∨)∨. In [Per04, Section 5.5], Perling characterizes T−linearized reflexive sheaves as those sheaves which are determined by the filtrations defined by the ray generators. In particular, over an affine toric variety Y(σ), R=⨁m∈MFm is reflexive if and only if Fm=⋂ρi∈σ(1)G⟨ρi,m⟩ρi. Observe that we always have the inclusions Fm⊂Fm(vR)⊂⋂ρi∈σ(1)G⟨ρi,m⟩ρi, hence if R is reflexive these inclusions must be equalities, and R must also be eversive by Definition 3.14.
On the other hand, Example 3.16 provides a way to find eversive sheaves which are not reflexive. We let I=⟨xy4,x2y2,x4y⟩⊂k[x,y]; this ideal defines an eversive sheaf by Example 3.16. However, in this case F(1,1)=0=G−1(−1,0)∩G−1(0,−1)=k, so I cannot be reflexive.
We let AlgSσM be the category of commutative algebras in ModSσM. If R∈AlgSσM then R/mR≅ER is a k-algebra, and vR is a quasivaluation on ER. Similarly, R(A,v) is an Sσ-algebra if (A,v)∈Algσ. Let Algσev and AlgSσM,ev be the categories of algebras whose underlying objects are eversive, then it is straightforward to show that Algσev≅AlgSσM,ev.
Finally, we construct the functor L for TN-equivariant quasicoherent sheaves ShY(Σ)M.
Definition 3.18**.**
Let ShY(Σ)M,ev be the full subcategory of ShY(Σ)M consisting of sheaves F such that Γ(Y(σ),F)∈ModSσM,ev for each σ∈Σ.
Recall that a functor F:(C,⊗)→(D,⊠) between symmetric monoidal categories is said to be strictly monoidal if F(A⊗B) is naturally isomorphic to F(A)⊠F(B).
Theorem 3.19**.**
There is a monoidal functor L:ShY(Σ)M→VectΣ. This functor gives an equivalence of ShY(Σ)M,ev and a full subcategory of VectΣ.
The functor L allows us to describe morphisms between equivariant sheaves using prevaluations. In the next corollary we give such a description of the space of maps when the domain is the invertible sheaf corresponding to a TN-Cartier divisor. We say that a function ψ:Σ∩N→Z is piecewise linear on the fan Σ if the restriction ψ∣σ is linear for each cone σ∈Σ. It is well-known (see [Ful93, Chapter 3]) that piecewise linear functions on Σ correspond to TN-Cartier divisors on Y(Σ). For such a ψ, we let O(ψ) denote the associated invertible sheaf on Y(Σ). In particular, for m∈M, O(m) denotes the linearization of the structure sheaf of Y(Σ) corresponding to the character χm. Also, recall that for any (E,v)∈VectΣ, Fψ(v)⊂E denotes the space of those f∈E such that v(f)≥ψ.
Corollary 3.20**.**
Let ψ be a piecewise linear function on Σ, and let F∈ShY(Σ)M,ev with L(F)=(E,v). Then HomShY(Σ)M(O(ψ),F) is isomorphic to Fψ(v).
3.3. Connection to Perling’s Σ-families
In [Per04], Perling proves that the category ShY(Σ)M is equivalent to the category of Σ-families. In the affine case, the σ-family of a module R∈ModSσM is the directed system formed by the isotypical spaces Rm for m∈M and the linear maps χu:Rm→Rm+u defined by the characters in Sσ.
In the case that R is eversive, the functor L:ModSσM→Vectσ recovers the data of the σ-family as the spaces Fm(vR)⊂ER, along with the inclusions Fm(vR)⊂Fm+u(vR). However, L can forget some of the data of the σ-family when R is not eversive. For example, if R is torsion, ER and all spaces Fm(vR) are [math]. Example 3.16 also gives a way to produce M-graded ideals I⊂Sσ where L(I) forgets some of the data of the associated σ-family.
3.4. Free, projective, and flat modules
In this section we see how the functor L behaves on free, projective and flat modules.
It is critical for the proof of our classification theorem (Theorem 1.2). In particular, we show that projective modules are eversive.
We start by observing that a direct sum is eversive if and only if the components are.
We leave the proof to the reader.
Proposition 3.21**.**
In ModSσM and Vectσ, a direct sum is eversive if and only if its components are eversive.
Let P be a free M-graded Sσ-module, then we can write P=⨁bi∈BSσbi for B an M-homogeneous basis. We let deg(bi)=λi∈M. For any m∈M we have a direct sum decomposition Pm=⨁ui+λi=mkχuibi.
Proposition 3.22**.**
The functors L and R give an equivalence of categories between M-homogeneous free modules and prevalued vector spaces with linear adapted bases (Definition 3.6). Furthermore, if P=⨁bi∈BSσ,bi is free with deg(bi)=λi, then P is eversive, L(P)≅⨁bi∈B(k,λi), and Fm(P) is isomorphic to the subspace of EP with basis Bm={bi∣m−λi∈σ∨∩M}.
Proof.
By Proposition 3.21 and Theorem 3.19 it suffices to note that if deg(b)=λ then L(Sσb)=(k,λ) and ϵ:RL(Sσb)≅Sσb.
∎
Propositions 3.22 and 3.21 imply that projective modules in ModSσM are eversive and correspond under L to summands of spaces with a linear adapted basis. From now on we identify the category of TN-vector bundles on Y(Σ) with the full subcategory of locally-free TN-sheaves.
Corollary 3.23**.**
Let (E,v)∈VectN with E a finite dimensional vector space. The following are equivalent.
(1)
The image v(E) is a finite subset of ON.
2. (2)
There is a finite complete fan Σ with (E,v)=L(E) for a locally free E∈ShY(Σ)M.
3. (3)
There is a finite complete fan Σ with (E,v)=L(F) for a coherent F∈ShY(Σ)M.
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 λi. We let grσ(E) be the associated graded vector space, and we choose a basis B^σ⊂grσ(E). Any lift Bσ⊂E of this basis is adapted to v∣σ, this proves (1)→(2). The case (2)→(3) is immediate. For (3)→(1), consider an Sσ-module R generated by f1,…,fN∈R with deg(fi)=λi. If p=∑i=1ℓχuifi, then ϕR(p)=q∈∑Fλi(R), so vR(q)≥⨁i=1ℓλi. Furthermore, Rm=0 only if m∈⋃i=1N(λi+(σ∨∩M)). It follows that vR(q) is the maximum over a finite number of expressions of the form ⨁i=1ℓλi, and only a finite number of such expressions are possible.
∎
Example 3.24**.**
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⊗kF gives a linear adapted basis as follows. For any ρ∈Σ∩N, Gsρ(v)∩B1⊂Gsρ(v) and Gtρ(w)∩B2⊂Gtρ(w) are bases, so Gs+tρ(v⋆w)∩(B1×B2) is a basis of Gs+tρ(v⋆w). This shows that B1×B2 is adapted to v⋆w. Moreover, (v⋆w)(b1⊗b2)=v(b1)+w(b2) for any b1×b2∈B1×B2, so B1×B2 is linear.
Example 3.25**.**
Example 3.24 allows us to find adapted bases of Sλ(E,v) for any Schur functor Sλ and a linear adapted basis B for (E,v). For any partition λ we obtain a basis Bλ of Sλ(E) by applying the symmetrizers sτ, corresponding to semi-standard fillings τ of λ, to B. We let bτ∈Bλ⊂Sλ(E) be the tensor obtained by applying sτ to the basis B. It is straightforward to check that if v is adapted to B⊂E, then every simple tensor appearing in bτ has the same value. The simple tensors with entries in B are an adapted basis of E⊗∣λ∣, so we conclude that sλ(v)(bτ) is linear if v(b) is linear for all b∈B. For any ρ∈σ∩N, we have sλ(v)ρ(∑Cτbτ)≥⨁sλ(v)ρ(bτ). Let ∑minCτbτ denote the sum of terms in ∑Cτbτ
which are minimum with respect to sλ(v)ρ. If ∑minCτbτ is non-zero, then some simple tensor in the elements of B in this sum achieves the top value, so this is an equality. If ∑minCτbτ=0, then Cτ=0 for all τ, as the Bλ={bτ} is a basis. It follows that Bλ is a linear adapted basis of Sλ(E,v).
Example 3.26**.**
For a TN-vector bundle E, the dual E∗ corresponds to the locally free sheaf with Γ(E∗,Y(σ))=⨁i=1rSσbi∗, where deg(bi∗)=−deg(bi)=−λi. We have L(E∗)=(E∗,v∗), where locally (E∗,v∗∣σ)=⨁i=1r(k,−λi).
For a module R∈ModSσM there are two k-vector spaces associated to a point ρ∈σ∩N. First, the fiber over a point in the orbit corresponding to the face τ containing ρ in its relative interior by taking the quotient R/mτR, where mτ=⟨{χu−1,χv∣u∈τ⊥,⟨ρ,v⟩<0∀ρ∈τ}⟩. Alternatively, we have the associated graded space grρ(ER)=⨁r∈QGrρ(R)/G>rρ(R). These two spaces are naturally isomorphic if R is flat. In this way we connect the geometric notion of fiber with the algebraic notion of associated graded.
Proposition 3.27**.**
The map ϕ^ρ:R→grρ(ER) sending Rm to G⟨ρ,m⟩ρ(R)/G>⟨ρ,m⟩ρ(R) is surjective and factors through the surjection R→R/mτR, giving a map ϕρ:R/mτR→grρ(ER). If R is flat, then ϕρ is also injective. If R is an algebra, ϕρ is a map of algebras.
Finally, with the following proposition we recover the fact that a vector bundle E is captured by its Klyachko spaces. For each ray ϱi∈Σ(1) let ρi∈ϱi denote the first integral point.
Proposition 3.28**.**
Let F∈ShY(Σ)M,ev be flat and let ψ be a piecewise linear function on Σ, then:
[TABLE]
3.5. Classification of flat sheaves of graded algebras
As an important application of the machinery set up in Section 3, we obtain a theorem classifying TN-equivariant degenerations of k-algebras using quasivaluations into O^Σ (Theorem 3.30). In the next section (Section 4) we prove a more refined classification theorem (Theorem 1.2). It can be regarded as Theorem 3.30 in the presence of an additional finiteness hypothesis.
Fix a k-algebra A, and a direct sum decomposition A=⨁i∈IAi (not necessarily a grading) into finite dimensional k-vector spaces.
A homogeneous Y(Σ)-degeneration of A is defined to be a flat sheaf of algebras A∈AlgSΣM which can be written A=⨁i∈IAi where each Ai is coherent, such that EA=⨁i∈IEAi=⨁i∈IAi=A. A quasivaluation v:A→O^Σ is said to be homogeneous if (A,v)≅⨁i∈I(Ai,v∣Ai) in VectΣ.
Theorem 3.30**.**
Any homogeneous degeneration is eversive, L(A)=(A,v), where v:A→O^Σ is homogeneous, and each v∣σ has a linear adapted basis. In particular, v takes values in OΣ. Moreover, all homogeneous quasivaluations such that (A,v∣σ) has an adapted basis for all σ∈Σ arise as L(A) for a homogeneous degeneration A.
Proof.
A degeneration A=⨁i∈IAi is flat if and only if each Ai is flat, and therefore locally projective, and therefore locally 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 3.19 to a unique homogeneous degeneration.
∎
4. Khovanskii bases
In this section we define the notion of a Khovanskii basis for a valuation v:A→O^Σ, and we prove two of our main theorems, namely, Theorems 1.2 and 1.4.
We begin by recalling elements of the theory of Khovanskii bases for quasivaluations with values in Q. Let B={b1,…,bd}⊂A be a generating set. We let π:k[x]→A be the surjection with π(xi)=bi∈B and I=ker(π). Any w∈QB gives a valuation v^w:k[x]→Q where v^w(xα)=⟨w,α⟩. We let vw=π∗(v^w) and refer to vw as the weight quasivaluation on A associated to w∈QB (see [KM19, Definition 3.1]). By construction, the quasivaluation vw takes values in Z when w∈ZB. We let grw(A) denote the associated graded algebra of the quasivaluation vw (see Equation (2)). By [KM19, Lemma 3.4] we have grw(A)≅k[x]/inw(I) where inw(I)⊂k[x] is the initial ideal of I corresponding to w. Moreover, π(x)=B⊂A is a Khovanskii basis for vw, that is, the image of B is a generating set for the associated graded algebra grw(A) (Definition 2.8).
Let GF(I)⊂QB denote the Gröbner fan of I (see e.g. [Stu96, Proposition 2.4]). Recall that if w,w′∈QB lie in the relative interior of the same face of GF(I) then inw(I)=inw′(I). For a maximal cone τ∈GF(I), we let Bτ⊂A be the associated standard monomial basis (see [Stu96, Proposition 1.1]). We will need the following proposition from [KM19].
Proposition 4.1**.**
[KM19, Proposition 4.9]**
Let B={b1,…,bd}⊂A be a Khovanskii basis for v:A→Q and put w=(v(b1),…,v(bd))∈GF(I). Then v=vw. Moreover, if Bτ⊂A is a standard monomial basis with w∈τ, then Bτ is adapted to v and v(bα)=⟨w,α⟩ for any bα∈Bτ.
It can happen that vw=vw′ for distinct w,w′∈QB. To account for this, the map ι:QB→QB defined by 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 an integral piecewise linear map on each face of GF(I), and that the image of ι is the support of a subfan K(I)⊂GF(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
In this section we extend the above definitions to quasivaluations into OΣ.
It is more convenient to begin by considering valuations into O^Σ. Recall that O^Σ is the semialgebra of functions ψ:∣Σ∣→Q with the property that ψ(ℓρ)=ℓψ(ρ) for all ℓ∈Q≥0, and that AlgΣ is the category whose objects are pairs (A,v), where A is an algebra and v:A→O^Σ is a quasivaluation (see Section 3).
Definition 4.2**.**
A generating set B⊂A is said to be a Khovanskii basis of (A,v)∈AlgΣ if it is a Khovanskii basis of vρ:A→Q, for every ρ∈∣Σ∣, in the sense of Definition 2.8.
For (A,v)∈AlgΣ and a subset B⊂A we define ΦB:∣Σ∣→QB to be the function
[TABLE]
We can think of ΦB as a tuple (v(b1),…,v(bd))∈O^Σd. When it is clear from context we may drop the subscript B.
Recall that the tropical variety TropOΣ(I) over the semialgebra OΣ can be viewed as the collection of tuples (ϕ1,…,ϕd)∈OΣd which satisfy the tropical relations coming from the ideal I (see Section 2). We note that a tuple (ϕ1,…,ϕd)∈OΣd will satisfy the tropical relations coming from I if and only if this is so for (ϕ1(ρ),…,ϕd(ρ)) for all ρ∈∣Σ∣. Hence, we can also view TropOΣ(I) as the set of integral piecewise linear maps Φ:∣Σ∣→Trop(I)⊂Qd.
We now define the analogues of the fans K(I) and GF(I) over the semialgebra OΣ.
Definition 4.3**.**
Let KOΣ(I) and GFOΣ(I) be the sets of integral piecewise linear maps on ∣Σ∣ with values in the supports of fans K(I) and GF(I) respectively. The sets KO^Σ(I), and GFO^Σ(I) are defined similarly.
For any Φ=(ϕ1,…,ϕd)∈O^Σd there is a canonical valuation v^Φ:k[x]→O^Σ defined by:
[TABLE]
In other words, we require that {xα∣α∈Z≥0d} is an adapted basis for v^Φ and we set v^Φ(xα)=∑iαiϕi, for all α=(α1,…,αd)∈Z≥0d. The pushforward π∗v^Φ:A→O^Σ is then a quasivaluation on A. We use this construction to give an alternative characterization of Khovanskii bases.
Proposition 4.4**.**
Let (A,v)∈AlgΣ, {b1,…,bd}=B⊂A, and suppose that Φ=(v(b1),…,v(bd)) lies in GFO^Σ(I). Then B is a Khovanskii basis for v if and only if v=π∗v^Φ. In this case Φ∈KO^Σ(I).
Proof.
By definition of pushforward, (π∗v^Φ)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 2.8. 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∗v^Φ. Conversely, if v=π∗v^Φ, then vρ=vΦ(ρ). So each vρ has B as a Khovanskii basis.
∎
Recall that if I is homogeneous with respect to a positive grading on k[x] then the support of GF(I) is all of QB, so the condition Φ∈GFO^Σ(I) is satisfied for any map Φ:∣Σ∣→QB whose components are in O^Σ.
Now we deal with quasivaluations with values in OΣ.
Proposition 4.5**.**
Let A be a positively graded k-algebra with a Khovanskii basis B={b1,…,bd}. For any quasivaluation v:A→OΣ let Φ=(v(b1),…,v(bd)). We have Φ∈KOΣ(I), and moreover, if v is a valuation, Φ∈TropOΣ(I). The set KOΣ(I) is in bijection with the set of quasivaluations with Khovanskii basis B.
Proof.
For any quasivaluation v:A→OΣ with Φ as above and ρ∈∣Σ∣∩N we have Φ(ρ)∈K(I). Now let v,v′ be quasivaluations with Khovanskii basis B with tuples Φ and Φ′, and let ρ∈Σ∩N. If Φ=Φ′, then vρ(b)=vρ′(b) for all b∈B. By Proposition 4.1 we must have vρ=vρ′ for all ρ∈∣Σ∣∩N, so v=v′. Moreover, the fibers of A coincide with the associated graded algebras of the vρ by Proposition 3.27. The fibers of the flat family are domains if and only if the vρ are valuations, so we may use Proposition 2.6.
∎
By abuse of notation we let ι:GFOΣ(I)→GFOΣ(I) denote the map Φ→ι∘Φ. It is straightforward to check that ι2=ι and that the image of ι is KOΣ(I).
4.2. Classification of toric flat families: proof of Theorem 1.2
In this section we complete the proof of Theorem 1.2. We start with some preparation.
Let A be a flat sheaf of algebras over a toric variety Y(Σ) with general fiber A. As usual A gives us a toric flat family π:X→Y(Σ) constructed as the relative spectrum X=SpecY(Σ)(A).
First we show that finite Khovanskii bases for A appear when X is of finite type. We start with the case where the base is an affine toric variety. Recall that Sσ=k[σ∨∩M] denotes the affine semigroup algebra associated to a cone σ.
Proposition 4.6**.**
Let R be a finitely generated Sσ-algebra giving a sheaf of algebras A over Y(σ). Let A be the fiber of A over the distinguished point x0 in Y(σ) and let L(R)=(A,v)∈Algσ (see Definition 3.11). Then (A,v) has a finite Khovanskii basis B. Moreover, if R is flat and positively graded then R is a finitely generated Sσ-algebra if and only if the special fiber R/m0R is a finitely generated k-algebra.
In this latter case, B⊂A can be chosen so that ΦB is linear on σ, ΦB(σ) is contained in a face of K(I), and v has a linear adapted basis given by the standard monomials Bτ for any maximal face τ∈GF(I) which contains the image ΦB(σ).
Proof.
There is a presentation π^:Sσ[x]→R, let b^i=π^(xi)∈Fλi(R), and bi=ϕR(b^i)∈ϕR(Fλi(R))⊂A. By assumption, for each face η⊂σ, the images of the b^i in R/mηR generate as a k-algebra. Each map ϕρ:R/mηR→grρ(A) is a surjection of algebras (Proposition 3.27), hence the images of the b^i generate grρ(A) as well. By definition, the image ϕρ(b^i) is in Fλi(R)⊂G⟨ρ,λi⟩ρ(R)/G>⟨ρ,λi⟩ρ(R). As a consequence, for each bi∈A, ϕρ(b^i) coincides with the equivalence class b~i∈grρ(A).
Let R be a flat, positively graded Sσ-algebra, R=⨁n≥0Rn, where each Rn is a coherent Sσ module. Then each Rn is free as an Sσ-module, so it follows that L(R) has a linear adapted basis B, and we may write R=⨁b∈BSσ⊗kb. Let B⊂B be such that the equivalence classes B~⊂R/m0R generate as a k-algebra, then a graded version of Nakayama’s lemma implies that B generates R.
For any ρ∈σ∘, the relative interior of σ, the set 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 η⊂σ, so it follows that B⊂A is a Khovanskii basis of L(R). From above, the presentation of R/mηR by the image of the Khovanskii basis can be obtained by composing the inclusion k[x]⊂Sσ[x] with π^ and the quotient map. The kernel of this presentation is inρ(I). It follows that if ρ,ρ′∈η∘ then inρ(I)=inρ′(I), so Φ(ρ) and Φ(ρ) must lie in the same face of the Gröbner fan GF(I). This implies that ρ and ρ′ are in the same face of the subfan K(I)⊂GF(I). By construction, the set B~⊂A is k-linearly independent, and each v(b) defines a linear function on σ. Let τ∈GF(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.
∎
In the next Proposition we give criteria for v:A→OΣ to come from a flat sheaf of algebras on Y(Σ).
Proposition 4.7**.**
Suppose for each σ∈Σ the restriction (A,v∣σ) has a finite Khovanskii basis Bσ, and that the image ΦBσ(σ∩N)⊂ZBσ lies in a face of the Gröbner fan GF(IBσ). Then (A,v∣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).
Proof.
By Proposition 4.1, vρ=vΦBσ(ρ). It follows that Bτ is an adapted basis of vρ for any ρ∈σ∩N, where ΦBσ(σ)⊂τ⊂GF(IBσ). If v(bi)∣σ∈Mσ for bi∈Bσ, then v(bα)∣σ∈Mσ for any bα∈Bτ by Proposition 4.1, so Bτ⊂A is a linear adapted basis for (A,v∣σ)∈Vectσ.
Suppose π:X→Y(Σ) is flat and of finite-type, where X=SpecY(Σ)(A), and A is positively graded. Let Rσ be the coordinate ring of the induced family over the affine chart 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∈ΣBσ gives a finite Khovanskii basis of L(A). Let I be the ideal of relations among the elements of B. For any ρ∈σ∩N, the point Φ(ρ)∈K(I) is computed by finding a standard monomial expression for each b∈B with respect to monomials in Bσ: b=∑Cαbα, then taking the minimum min{⟨ΦBσ(ρ),α⟩∣Cα=0}. Let ≺σ be any monomial 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 GF(I).
By Proposition 4.7, if (A,v)∈AlgΣ, each (A,v∣σ) has a finite Khovanskii basis, ΦBσ is linear, and ΦBσ(σ) is contained in a face of K(IBσ), then (A,v)=L(A), where A is a locally free sheaf of algebras of finite type. Finally, Theorem 3.30 implies that A and L(A) can be recovered from one another. If Σ is finite, then B=⋃σ∈ΣBσ is a finite Khovanskii basis of L(A). Moreover, v is a valuation if and only if X has reduced, irreducible fibers; in this case Φ∈TropOΣ(I).
∎
4.3. The diagram of a toric flat family
Proposition 4.5 implies that any quasivaluation v:A→OΣ with Khovanskii basis B is determined by its values on B. We represented this set of values by an element Φ∈KOΣ(I). For a cone σ∈Σ, Proposition 4.6 then shows that the points Φ(ρi)∈Qd for the ray generators ρi of σ, must all lie in a common maximal face τ∈GF(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 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 “compatibility conditions” with respect to certain subcomplexes of K(I). This is a far extension of the Klyachko data of a toric vector bundle.
Definition 4.8**.**
An apartmentAτ⊂K(I) is a subcomplex of the form K(I)∩τ, where τ is a maximal face of the Gröbner fan GF(I).
Let Bτ⊂A be the standard monomial basis associated to a facet τ∈GF(I), and let πτ:K(I)→QB∩Bτ be the projection along the components associated to B∩Bτ. For any monomial xα∈Bτ, xi∣xα only if xi corresponds to an element of B∩Bτ. It follows that w∈Aτ is determined by the projection πτ(w). Let ϕτ:QB∩Bτ→QB be the map which takes the value min{⟨w,α⟩∣b=∑Cαbα,Cα=0} on the component corresponding to b, where b=∑Cαbα is the expression of b in the basis Bτ.
Definition 4.9** (Σ-adapted map).**
We say Φ:∣Σ∣→K(I) is adapted to Σ if for every σ∈Σ there is an apartment Aτ such that Φ(σ)⊂Aτ and πτ∘Φ:σ→QB∩Bτ is a linear map.
Definition 4.10** (Diagram of a piecewise linear map).**
Let Φ:∣Σ∣→K(I) be as above, then the diagram D(Φ) is defined to be the matrix whose (j,i)-th entry is vΦ(ρj)(bi), where ρj is the ray generator of ϱi∈Σ(1) and bi∈B.
We can now give a proof Corollary 1.3 from the introduction.
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 toric flat 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′∈σ and b∈B we have vΦ(ρ+ρ′)(b)=min{⟨Φ(ρ+ρ′),α⟩∣Cα=0}, where b=∑Cαbα is the standard monomial expression for b with respect to τ. We have vΦ(ρ+ρ′)(bα)=vΦ(ρ)+Φ(ρ′)(bα) 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∈ϱi∈σ(1). Now, if Φ:NQ→K(I) is as above, then for any σ∈Σ, the quasivaluations vΦ(ρ) have a common linear adapted basis given by the standard monomials associated to τ, so we may use Theorem 1.2.
∎
Remark 4.12**.**
Given a simplicial fan Σ⊂NQ and a generating set B⊂A, we can produce toric flat families with general fiber Spec(A) by choosing wi∈K(I) in bijection with the rays of Σ such that for each σ∈Σ, the wi corresponding to ϱi∈σ(1) lie in a common apartment Aτ. Proposition 4.11 can then be used 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 4.13**.**
The term “apartment” is taken from the theory of buildings (see [KM22]). 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 ρ∈τB is entirely determined by its values on B, so τB∩Trop(L)≅Qm, where m=∣B∣. Under the natural identification of the spherical building of GLm(k) with the valuations on k[x]/L framed by bases, ϕB(Qm) is mapped onto the apartment of the spherical building corresponding to B.
Example 4.14**.**
Let Σ be any fan which refines GF(I), then the map ι:GF(I)→K(I)
defines a canonical piecewise-linear map ι:∣Σ∣→K(I). Corollary 1.3 then implies that there is an associated toric flat family over Y(Σ). The diagram of this family can be computed by applying ι to the integral generators of rays of Σ.
4.4. Toric vector bundles as tropical points: proof of Theorem 1.4
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 3.30, this is the degeneration corresponding to the algebra Sym(E∗,v∗)=(Sym(E∗),s(v∗)). In this case, s(v∗):Sym(E∗)→OΣ⊂O^Σ is a valuation.
Given a toric vector bundle E, we consider the prevalued vector space (E∗,v∗)=L(E∗) associated to the dual bundle (Example 3.26). Theorem 3.19 implies that the restriction of v∗ to each face σ∈Σ has a linear adapted basis. In turn, we obtain a linear adapted basis of monomials in each (Symn(E∗),sn(v∗)∣σ) (Example 3.25). 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, Definition 2.6.3] if Trop(I)=⋂i=1mTrop(⟨fi⟩).
Proposition 4.15**.**
If {f1,…,fm}⊂I is a tropical basis for Trop(I) then it is also a tropical basis over ON.
Proof.
A function Φ∈ONd 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 of a tropical basis of I. The latter equivalence implies that TropON(I)=⋂i=1mTropON(⟨fi⟩) for {f1,…,fm}⊂I a tropical basis.
∎
Lemma 4.16**.**
For any linear ideal L, K(L)=Trop(L) and KON(L)=TropON(L).
Proof.
It is always the case that Trop(L)⊂K(L). Let w∈K(L) and ∑Cixi∈L be a circuit. It is known that the circuits of a linear ideal form a tropical basis. Suppose for some j, wj<wk for all k=j, Ck=0. We obtain a contradiction: vw(xj)≥max{wk∣k=j,Ck=0}>wj=vw(xj). As a consequence w∈Trop(L).
∎
If (A,v)∈AlgΣ has a finite Khovanskii basis B and v is a valuation, the image of Φ:∣Σ∣→Trop(I) must lie in the locus Tropp(I) of points w∈GF(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 [IM19]) if Tropp(I)=Trop(I).
Proposition 4.17**.**
Let B⊂E∗ and let L⊂k[x] be the corresponding linear ideal. Valuations v:Sym(E∗)→OΣ with Khovanskii basis B are in bijection with points Φ∈TropOΣ(L). Moreover, for any Φ∈TropOΣ(L) there is a refinement π:Σ^→Σ such that the corresponding valuation corresponds to a vector bundle E over Y(Σ^).
Proof.
Valuations v:Sym(E∗)→OΣ are classified by their values Φ∈KOΣ(L)=TropOΣ(I). Moreover, any point Φ∈TropOΣ(L) determines a quasi-valuation vΦ with Khovanskii basis B which must be a valuation since L is well-poised. By construction, the restriction of v to E∗ is computed on f∈E∗ by taking max of expressions of the form ⨁i=1ℓϕi where Φ=(ϕ1,…,ϕd). There are only a finite number of functions of this type, so we conclude that v takes a finite number of values on E∗. By Corollary 3.23, there is a fan Σ such that B contains a linear adapted basis for each restriction v∣σ, σ∈Σ.
∎
Remark 4.18**.**
The direct sum and tensor product operations on toric vector bundles are straightforward to compute with diagrams. If E is defined by the ideal L⊂k[y1,…,yr] and diagram D1, and F is defined by K⊂k[w1,…,ws] and daigram D2, then E⊕F is defined by the ideal ⟨L,K⟩⊂k[y1,…,yr,w1,…,ws] and the diagram [D1,D2]. The tensor product E⊗F is defined by the ideal in k[…,yi⊗wj,…] generated by forms ∑i=1rCi[yi⊗wj], ∑j=1sKjyi⊗wj, where ∑Ciyi∈L and ∑Kjwj∈K. The the column of the diagram D1⊙D2 for E⊗F corresponding to yj⊗wj is obtained by adding the i-th column of D1 to the j-th column of D2. In particular, if F is the TN-linearized line bundle corresponding to integers r1,…,rn, then D1⊙D2 is computed by adding ri to every entry in the the i-th row of D1.
4.5. Connection with the work of Di Rocco, Jabbusch, and Smith
In [DRJS18], Di Rocco, Jabbusch, and Smith define a matroid M(E) which captures many properties of the toric vector bundle E. We finish this section by relating M(E) to the notion of Khovanskii basis.
Proposition 4.19**.**
With notation as above, let B⊂E be a representation of the matroid M(E) as constructed in [DRJS18, Algorithm 3.2], then B is a Khovanskii basis of v.
Proof.
In [DRJS18, Section 3] it is observed that the Klyachko spaces Grρi(v) for Z≥0ρi=ϱi∩N,ϱi∈σ(1), generate a distributive lattice under intersection, so there is a “compatible basis” Bσ⊂B. This means that Bσ∩Grρi(v) is a basis for each Klyachko space. By Proposition 3.28, Bσ∩Fm(v∣σ) is also a basis for m∈M and σ∈Σ, and by Proposition 3.22, Bσ∩Grρ(v) is a basis for all ρ∈σ∩N. Take b∈Bσ and write it as a linear combination ∑i=1ℓCibi′ of members of a linear adapted basis B′ of v∣σ with deg(bi′)=λi. It follows that b∈∑i=1ℓFλi(v). The set Bi=Bσ∩Fλi(v) is a basis, and b∈⋃i=1ℓBi, so for some j, b∈Fλj(v). As B′ is adapted, it follows that each bi′∈Fλj(v) as well, so λi≤λj for each i. As a consequence, v(b)=⨁i=1ℓλi=λj. It follows that Bσ is a linear adapted basis.
∎
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 cone σ∈Σ is linear and has integer values on lattice points. In turn, such functions are determined by their values ψ(ρi) for ρi∈ϱi∈Σ(1). Thus, letting ∣Σ(1)∣=n, we identify the TN-linearized line bundles on Y(Σ) with the free group Zn.
Let A be a toric flat sheaf of algebras on Y(Σ) corresponding to a valuation 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-equivariant maps from the TN-linearized line bundles whose underlying line bundle is L:
[TABLE]
By Corollary 3.20, each space HomShY(Σ)M(O(ψ),A) can be identified with the Klyachko space Fψ(v)⊂A. The total section ring R(A,v^) of A is defined as the following direct sum:
[TABLE]
Let SΣ be the Cox ring of Y(Σ); recall that SΣ is isomorphic to the polynomial ring k[Xϱ∣ϱ∈Σ(1)]. The monomial in SΣ corresponding to ϕ∈Z≤0n acts on R(A,v^) by shifting a graded component Fψ(v) into Fψ+ϕ(v). By Proposition 3.13, the total section ring defines a quasivaluation v^:A→OCΣ, where CΣ=Q≤0n; whence the notation R(A,v^).
When A=Sym(E) for a toric vector bundle E, the total section ring (denoted R(E)) is isomorphic to the Cox ring of the projectivized vector bundle PE of rank 1 quotients of E, see [GHPS12]. The main result of this section is Theorem 1.5 which characterizes the finite generation of R(A,v^) in terms of the map Φv, or equivalently the diagram D(Φv).
5.1. Total section rings as Rees algebras
Corollary 1.3 implies that the total section ring R(A,v^) is determined by the valuations vρi: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 be quasivaluations with a common Khovanskii basis B. For r=(r1,…,rn)∈Zn, let Fr=⋂i=1nGri(vi). The multi-Rees algebra is the direct sum:
[TABLE]
For a toric flat algebra A, the total section ring R(A,v^) is the multi-Rees algebra defined by vi=vρi. Moreover, if we let Σ(n)⊂Qn 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 in Trop(IB). Total section rings for families over Y(Σ) are then those multi-Rees algebras whose valuations vi satisfy the apartment compatibility conditions of Remark 4.12. 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 description of a total section algebra. For v:A→OΣ, j≤n and r∈Zj, let Frj(v)=⋂1≤i≤jGri(vi). We define the partial multi-Rees algebras Rj=⨁r∈ZjFrj(v), and let Sj=k[X1,…,Xj], and Tj be the Laurent polynomial ring k[t1±,…tj±]. Let mj=⟨X1−1,…,Xj−1⟩⊂Sj.
Lemma 5.2**.**
For each j, Rj is an algebra over Sj, and is naturally identified with a subalgebra of A⊗kTj. For each j, vρj+1 defines a Zj-homogeneous quasivaluation on Rj, and the Rees algebra of this valuation is Rj+1. In particular, Rj≅Rj+1/⟨Xj+1−1⟩, grvρj+1(Rj)≅Rj+1/⟨Xj+1⟩, and R(A,v^)=Rn. Moreover, R(A,v^) is eversive, and if w∈Cσ⊂CΣ for σ∈Σ, then v^w=vw.
Proof.
The algebra R1 is clearly a Rees algebra of A, and a subalgebra of A⊗kk[t1±], where we view X1 as t1−1. If this statement holds for j, then we use Rj⊂A⊗kTj and the fact that vρj+1 defines a Zj-homogeneous quasivaluation on A⊗kTj. It is then straightforward to check that the rj+1−st filtration level of Fr1,…,rjj(v) with respect to this quasivaluation is Fr1,…,rj,rj+1j+1(v).
It is straightforward to check that the graded components of Rj/⟨Xj−1⟩ are the spaces ⋂i=1j−1Griρi. Consequently, Rj/⟨Xj−1⟩≅Rj−1, and the algebra R(A,v^)/mnR(A,v^) is isomorphic to R1/⟨X1−1⟩≅A. It follows that L(R(A,v^))=(A,v^) for some quasivaluation v^:A→OCΣ. Moreover, if w∈Cσ⊂CΣ, where w=∑ρi∈σ(1)wiρ^i for ray generators ρ^i∈Cσ, we have:
[TABLE]
In particular, this implies that v^ρ^i=vρi as quasivaluations on A. Now, by construction, the quasivaluation v^ is concave with respect to the rays of CΣ, so we have:
[TABLE]
The middle term Fψ(v) is a graded component of R(A,v^), and the inclusion into the space ⋂w∈CΣG∑wiψ(ρi)w(v^) is a consequence of the definition of the functor L. The equality on the left, and the inclusion on the right then shows that R(A,v^) is eversive.
∎
5.2. Strong Khovanskii bases and proof of Theorem 1.5
By Proposition 4.6 and Lemma 5.2 above, if R(A,v^) is finitely generated as a k-algebra, then the image B⊂A of this set of generators is a finite Khovanskii basis for both v^ 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,v^).
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 that B will be a Khovanskii basis for v, indeed, B monomials span each Grρ(v) 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 assumption, each πψ is surjective. The algebra R(k[x],w^)⊂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],w^) is a saturated affine semigroup algebra, and is therefore finitely generated.
∎
Unfortunately, it is not enough to show that v^:A→OCΣ has a finite Khovanskii basis. Recall that this means that the associated graded grρ(A) is finitely generated for all ρ∈CΣ∩Zn. Finite generation of the toric fixed point fiber R(A,v^)/mCΣR(A,v^) is the obstruction to finite generation of R(A,v^), however the maps ϕ^ρ:R(A,v^)/mCΣR(A,v^)→grρ(A) from Proposition 4.6 are only surjections in general. It could be the case that the associated graded algebras grρ(A) are finitely generated when R(A,v^) is not. In particular, R(A,v^) may not be flat over SΣ. Moreover, if R(A,v^) is flat as an SΣ module and A is positively graded then R(A,v^) 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 7.4. 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 by w(xα)=∑αiϕi, where ϕi=v(bi) for bi∈B. For each ψ∈Div(Σ) there is a map πψ:Fψ(w)→Fψ(v). The space Fψ(w) has as a basis those monomials xα 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 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 construct 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=⟨G⟩⊂k[x].
(1)
For an element g∈G, compute h=g(b1,…,bd)∈A and h∈grv(A).
2. (2)
Check if h can be written as a polynomial in B⊂grv(A).
3. (3)
If h cannot be written as a polynomial in B then add h to B.
4. (4)
Repeat steps (1)−(3) for each element of G.
5. (5)
If B is unchanged then JB=inv(IB) so return B, otherwise go back to step (1).
The Rees algebra R(A,v)=⨁r∈ZFr(v) is naturally a subalgebra of A⊗kk[t±]. A generating set B⊂grv(A) then lifts to a k[t−1]-algebra generating set B^⊂R(A,v), where b^i=bi⊗tv(bi). In particular R(A,v) is generated by B^∪{t−1} 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,v^).
Algorithm 5.6**.**
Let R0=A, and B0⊂R0 be a finite generating set.
(1)
Starting with a generating set Bj⊂Rj, use **[KM19, Algorithm 2.18]** to construct a Khovanskii basis Bj′ of vρj+1.
2. (2)
Set Bj+1=Bj′^∪{tj+1−1}⊂Rj+1⊂A⊗kTj+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 an algebraic criterion for Algorithm 5.6 to stop. This will be a key step in the proof of Theorem 1.5.
Proposition 5.8**.**
Let the pair (A,v) have Khovanskii basis B⊂A with v a valuation, and let IB be the kernel of the map k[X,Y]→A⊗kk[t1±,…,tn±], where Xi→1⊗ti−1 and Yj→bj⊗tv^(bj), then B is a strong Khovanskii basis if and only if the ideal ⟨IB,Xi⟩ is prime for all i∈{1,…,n}.
Proof.
If B is a strong Khovanskii basis, then we can reorder the variables X1,…,Xn so that k[X,Y]/⟨IB,Xi⟩≅grvρi(Rn−1) as in Algorithm 5.6. The latter algebra is a domain since vρi is a valuation, so Lemma 5.2 implies that ⟨I,Xi⟩ is prime for all i∈{1,…,n}.
We prove the converse by induction. We let Rm⊂A⊗kTm be as in Algorithm 5.6, and we show that Rm is finitely generated by the images of X1,…,Xm and Y for each m∈{1,…,n}. The base case R0=A is handled by the fact that B is a weak Khovanskii basis. Assume this holds for Rm, and 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 valuation vρm+1:A⊗kTm+1→Z, and let Jm⊂k[X1,…,Xm,Y] denote the kernel of the map ψm:k[X1,…,Xm,Y]→grvρm+1(Rm)⊂grvρm+1(A)⊗kTm. Observe that vρm+1 is homogeneous with respect to the Zm+1 grading on Rm. We have in(Im)⊂Jm as above, and moreover, the images of X1,…,Xm,Y are a Khovanskii basis for vρm+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, and the algebra obtained from this quotient by inverting the X1,…,Xm is grvρm+1(A)⊗kTm. Therefore, it suffices to show that the quotient algebra of in(Im) is also a domain of the same dimension.
We prove the required 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/⟨x−1⟩ is prime. To see this, let ϕ be an isomorphism between a polynomial rings k[…,xdeg(yi)y,…] and k[y] obtained by setting x to 1. Let (x1J)0⊂k[…,xdeg(yi)y,…] be the prime ideal of degree [math] forms. Each element of (x1J)0 is mapped into J0 by ϕ, and both ideals have the same height, so it follows that the image ϕ((x1J)0) is J0. Moreover, the dimension of S/⟨x−1⟩ is one less than the dimension of S. By repeatedly applying this observation to the prime ideal ⟨I,Xm+1⟩, we see that the kernel Im′ of the map k[X1,…,Xm,Y]→k[X,Y]/⟨Xm+1,Xm+2−1,…,Xn−1⟩ is prime, and the image algebra has the same dimension as grvρm+1(A)⊗kTm.
Next we consider the general case of an algebra S≅k[x,y]/J as above. If S=⨁m∈ZSm, then the images Fm of the spaces Sm in S/⟨x−1⟩ form an algebra filtration, and xFm⊂Fm−1. The image B of the set y in S/⟨x−1⟩ generates, and the same holds for the image B⊂S/⟨x⟩≅grF(S). It follows that B is a Khovanskii basis for the filtration F. If J0 denotes the kernel of the presentation of S/⟨x−1⟩ and J0′ denotes the kernel of the presentation of S/⟨x⟩, 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]/⟨I,Xm+2−1,…,Xn−1⟩ 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 to A⊗kTn. As a consequence, there is an isomorphism π~:Trop(IB)→Trop(IB)×Qn, where π~(w1,…,wd,m1,…,mn)=(…,wj+∑vρi(bj)mi,…,−m1,…,−mn). We let π:Trop(IB)→Trop(IB) be the composition of π~ 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).
The ideal IB is homogeneous with respect to the Zn grading on k[X,Y], so for any m=(0,…,0,m1,…,mn)∈Qd+n 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 addition by m∈{(0,…,0)}×Qn⊂Qd+n. 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 form C~≅C×Qn 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.
The set B⊂A is a strong Khovanskii basis if and only if w~i=s∘Ψ(ρi)∈C~i⊂Trop(IB) is a prime point for each ϱi∈Σ(1).
Proof.
For each ray ρi, the valuation vρi:A→Z defines a valuation on A⊗kTn, which in turn induces a valuation on R′. We let wρi:R′→Z denote the weight quasivaluation induced by the values of vρi on the generating set {ϕ(X),ϕ(Y)}⊂R′. In particular, we have wρi(ϕ(Xj))=0 and wρi(ϕ(Yj))=vρi(bj), so the weight of wρi is Ψ~(ρi)∈C~i.
There is an inclusion of algebras R′⊂R(A,v^). We let ordi:R′→Z be the quasivaluation obtained by the ⟨ϕ(Xi)⟩-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+degi=wρi 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,v^), and ⟨ϕ(Xi)⟩⊂R′ is a prime ideal. It follows that ordi is a valuation. Since degi is a homogeneous grading function, this implies that wρi is a valuation as well, so C~i is a prime cone. Similarly, if C~i is a prime cone, then wρi is a valuation, and ordi must also be a valuation; this implies that ⟨ϕ(Xi)⟩ is prime. Proposition 5.8 completes the proof.
∎
6. Cox rings of projectivized toric vector bundles
We apply our results from Section 5 to the Cox rings of 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′). Because of this observation and Remark 4.18, we assume that the minimum entry in each row of a diagram D is [math].
We start by classifying those diagrams D whose associated bundle E has R(E) presented by the “expected” relations. We call such a bundle complete intersection bundle. More precisely, suppose that the linear ideal LB⊂k[y] is generated by linear forms ℓ={ℓ1,…,ℓd}, where ℓi=∑j=1rci,jyj, and let M be the matrix with (i,j)-th entry equal to ci,j. We let αi be the entry-wise minimum of the values v^(bj) taken over the yj in the support of ℓi. Let pi=Xαi1∑j=1rci,jXv^(bj)Yj∈k[X,Y]. It is straightforward to verify that ⟨p1,…,pd⟩⊆IB.
Definition 6.1**.**
Let E be a toric vector bundle corresponding to a v:Sym(E)→OΣ with Khovanskii basis B⊂E. We say that E is a complete intersection bundle if B is a strong Khovanskii basis and IB=⟨p1,…,pd⟩, for pi1≤i≤d associated to ℓi∈LB as above.
By Theorem 1.5, the projectivization PE of any complete intersection bundle E is a Mori dream space. Moreover, in this case the Cox ring R(E) can be presented as a complete intersection. For a subset A⊂{1,…,n} let MA be the matrix whose j-th row is given by the nonzero entries of the j-th row of M where the rows wi in the diagram have a common minimal entry. We let mA=rank(MA).
Proposition 6.2**.**
The image of the map ΦB:k[X,Y]→Sym(E)⊗kTn is R(E) with kernel IB=⟨p1,…,pd⟩ if and only if for every A⊂{1,…,n} we have d<∣A∣+mA, and for every B⊂{1,…,n} and i∈B we have 1+m{i}<∣B∣+mB.
Proof.
Suppose the stated inequalities hold. We show that ⟨p1,…,pd⟩ is a prime complete intersection. For dimension reasons, this implies that the pi generate IB. Let V⊂An×kAr be the scheme defined by ⟨p1,…,pd⟩. After inverting X1,…,Xn, the pi present Sym(E)⊗kTn, which has dimension n+r−d. For A⊂{1,…,n}, let AA⊂An be the coordinate subspace defined by Xi∣i∈A, and let TA⊂AA be the torus orbit. The polynomials pi,A=pi∣xi=0,i∈A∈k[Xi∣i∈/A,Y] define V∩AA×kAr, and V∩TA×kAr is a trivial vector bundle over TA of rank r−mA with dimension n−∣A∣+r−mA. By assumption, n−∣A∣+r−mA<n+r−d, so the dimension of V is n+r−d, and V is a generically reduced complete intersection. The latter implies that V is unmixed and reduced, so ⟨p1,…,pd⟩ is prime. A similar argument using the inequality 1+m{i}<∣B∣+mB shows that ⟨IB,Xi⟩ is prime for each i. Conversely, we suppose IB=⟨p1,…,pd⟩ and ⟨IB,Xi⟩ are prime. It follows that the height of IB is strictly smaller than the height of ⟨IB,Xi⟩. Moreover, ⟨IB,Xi⟩ must have height strictly smaller than that of ⟨IB,Xi,Xj⟩. These two observations imply the inequalities d<∣A∣+mA and 1+m{i}<∣B∣+mB. ∎
Definition 6.3**.**
We say a bundle E is uniform if we can find a Khovanskii basis B⊂E such that the associated matrix M has all nonzero minors.
Remark 6.4**.**
Note that our condition defining uniform bundles is stronger than requiring B be a representation of a uniform matroid. The B with M having all nonzero minors form a dense, open subset in the representations of the uniform matroid.
Corollary 6.5**.**
Let E be a uniform bundle with diagram D and Khovanskii basis B⊂E, then ΦB has image R(E) with kernel IB=⟨p1,…,pd⟩ if and only if for any A⊂{1,…,n} the nonzero entries of the corresponding rows of D are contained in r+∣A∣−2 columns.
Proof.
We have assumed that M has all nonzero minors, so mA is equal to the minimum of d and the number sA of columns of D where the rows from A have a common zero entry. Now we use Proposition 6.2. Observe that m−s{i}<r=m−d, so s{i}>d for any i∈{1,…,n}, and m{i}=d. Now the presentation statement is equivalent to 1+d<∣A∣+min{sA,d} for all A, which is in turn equivalent to the requirement that all nonzero entries of the A-rows of D be contained in r+∣A∣−2 columns.
∎
In the case that d=1 ( the hypersurface case), this condition states that any two rows of such a diagram D must have a common zero entry. Moreover, if we work with bundles over Pn−1, then any collection of n−1 rows in D must share a common apartment. It follows that these rows share at least d+1 zero entries.
Corollary 6.6**.**
Let E be a uniform bundle over Pn−1 as above with d<n−1, then PE is a Mori dream space.
We say a bundle E is sparse if we can find a Khovanskii basis B⊂E such that D has at most one non-zero entry in each row. Without loss of generality, we assume that the minimal circuits in LB have size at least 3. In this case m{i}=d.
Corollary 6.7**.**
Let E be a sparse bundle with diagram D and Khovanskii basis B, then ΦB has image R(E) with kernel IB=⟨p1,…,pd⟩.
Proof.
Let M=[TA∣MA], then ∣A∣ is at least equal to the number of columns in TA. Without loss of generality we assume that ∣A∣ equals the number of columns in TA. We must show that d+1<∣A∣+mA. We consider three cases. First, assume that the rank of TA is less than or equal to ∣A∣−2, then mA≥d−∣A∣+2. Next, assume that the rank of TA is ∣A∣−1. We select an independent set I1 from the columns of TA and complete it to a basis with columns I2 from MA. We rearrange M so that I1∪I2 come first, and row-reduce. As the minimal circuits of LB are of size at least 3, the first ∣I1∣ rows of the resulting matrix must contain a non-zero entry in the columns past the first d×d block. The row containing such an entry along with the last I2 rows form a linearly independent set of size d−∣A∣+2, so mA≥d−∣A∣+2. Similarly, if TA has rank ∣A∣, then ∣I2∣=d−∣A∣. We consider the vectors formed by the first I1 entries in the remaining columns in MA. Two of these vectors must be linearly independent, else L contains a linear binomial in the members of B corresponding to the columns of TA. Taking the columns of MA in the positions of the two linearly independent vectors, plus the columns in I2 then gives a linearly independent set of size d−∣A∣+2. In all cases, we conclude that d+1<∣A∣+mA.
∎
We can apply Corollary 6.7 to rank 2 vector bundles, which were considered in [Gon12], 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 the diagram of a sparse bundle. The bundles considered in [GHPS12, Section 6.2] and [Nod, Proposition 5.2] are sparse by a similar argument.
Now we have several examples illustrating Theorem 1.5 and Algorithm 5.6.
We consider the hypersurface bundle E over P2 defined by the ideal L=⟨∑i=16yi⟩⊂k[y] and the following diagram:
[TABLE]
Observe that E is uniform, but not sparse. Let B={b1,…,b6}⊂E be the corresponding arrangement. If ψi is the i-th column, then Fψi(v)=⟨bi⟩⊂E. The rays ϱ0,ϱ1,ϱ2 of the fan of P2 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{ϱ1,ϱ2}, B∖{b2} is adapted over σ1=R≥0{ϱ0,ϱ2}, and B∖{b3} is adapted over σ2=R≥0{ϱ0,ϱ1}. This diagram satisfies the condition in Corollary 6.5, so R(E) is presented by the following polynomial:
[TABLE]
Next we see a hypersurface bundle over P1×kP1 whose Khovanskii basis is not strong.
Example 6.9** (An application of Algorithm 5.6).**
Let E be the rank 3 bundle over P1×kP1 defined by the ideal L=⟨∑i=14y1⟩⊂k[y] and the following diagram.
[TABLE]
Let B⊂E be the corresponding arrangement. The ideal IB is generated by the polynomial Y1X12X4+Y2X1X22+Y3X2X32+Y4X3X42. The ideal ⟨IB,X1⟩ is not prime, so B is not a strong Khovanskii basis by Theorem 1.5. 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 obtained by applying the weight valuations represented by the rows of D to c1 and c2:
[TABLE]
The ideal IB′⊂k[X1,X2,X3,X4,Y1,Y2,Y3,Y4,Z1,Z2] is generated by 5 polynomials:
[TABLE]
[TABLE]
[TABLE]
We conclude that R(E)≅k[X,Y,Z]/IB′.
Example 6.10** (A non-example).**
We take E to be the rank 3 bundle defined by the diagram:
[TABLE]
Let R2⊂Sym(E)⊗kk[t1±,t2±] be the 2nd Rees algebra in Algorithm 5.6. This algebra is presented as a quotient of k[Y1…Y5,X1,X2] by ⟨Y5X1−Y2X2−Y3,Y4X12−Y1X22−Y2⟩. This ideal is the kernel of the map determined by sending Y1→ω1t22, Y2→ω2t2, Y3→ω3, Y4→(ω1+ω2)t12t2, 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 valuation vρ3 defined by the third row of D is not finitely generated. Let F0,n;n⊂R2 be the subspace Fn,0∩Symn(E), and let Γ=⨁n≥0F0,n;n. As gr3(Γ) 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:
[TABLE]
[TABLE]
It can be shown by induction that the 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 included, as is ω23n (resp. ω23n−1,ω23n−2). Also included are the polynomials (2ω22ω3+ω2ω32)kω2p, where p+3k=3n (resp. 3n−1, 3n−2). This basis can be shown to be adapted to the valuation vρ3. The components of the associated graded algebra gr3(Γ)⊂gr3(R2) are similarly described, except the polynomials (2ω22ω3+ω2ω32)kω2p are replaced with the monomials (2ω22ω3)kω2p. This is the source of the infinite generation. In the component F0,3n;3n, the corner element ω1ω2n−1ω32n is only reachable as a difference of (2ω22ω3+ω2ω32)n−1ω1ω22∈F0,3(n−1);3(n−1)F0,3;3 and monomials obtained from the product 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 every n, ω1ω2n−1ω32n∈F0,3n;3n is an essential generator of gr3(Γ).
6.1. Diagram alterations and the fan Δ(Σ,Trop⋆(IB))
We finish this section by showing that the language of prime cones also provides a mechanism to make new Mori dream space bundles from old. In principle, these results can be formulated for any multi-Rees algebra, but we focus on the projectivized toric vector 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 turn defines the point (vw(b1),…,vw(br))∈Trop⋆(IB).
Definition 6.11**.**
Let Δ(Σ,Trop∗(IB)) be the subset of Trop∗(IB)n⊂Qn×r consisting those matrices D which are adapted to Σ.
By Proposition 4.11, any D∈Δ(Σ,Trop⋆(IB)) defines a toric vector bundle over Y(Σ).
Proposition 6.12**.**
The set Δ(Σ,Trop⋆(IB))⊂Qn×r is the support of a polyhedral fan.
Proof.
Give Trop⋆(IB) the Gröbner fan structure. If (w1,…,wn)∈Δ(Σ,Trop⋆(IB)), and wi∈Ci, where Ci is the face of Trop⋆(IB) containing wi, then C1×…×Cn⊂Δ(Σ,Trop⋆(IB)) as well.
∎
Next we describe alterations to the rows of a diagram D∈Δ(Σ,Trop⋆(IB)) which preserve the finite generation of the corresponding total section ring.
Proposition 6.13**.**
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′∈Δ(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′′∈Δ(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).
∎
Proposition 6.13 can be used to decide when the pullback of a Mori dream space bundle along a single toric blow-up is a Mori dream space bundle.
Corollary 6.14**.**
Let E be a toric vector bundle over a toric variety Y(Σ) with diagram D(Φ)∈Δ(Σ,Trop⋆(IB)), let β:Y(Σ′)→Y(Σ) be a blow-up of a toric orbit of Y(Σ) corresponding to a new ray ρ=Q≥0p⊂∣Σ∣, and let β∗E be the pullback bundle. If Φ(p) is contained in the image of the union of the set of prime cones of Trop(IB) under π:Trop(IB)→Trop(IB), then R(β∗E) is finitely generated.
By Corollary 6.14, if IB is well-poised, then R(β∗E) is finitely generated for any such blow-up β:Y(Σ′)→Y(Σ). In particular, this holds for any sparse bundle.
By Theorem 1.5, each row wi of D is in the image π(C~i)⊂Trop(IB) of a prime cone C~i⊂Trop(IB). Let Ki be the polyhedral complex π(C~i)∩Trop⋆(IB). By Proposition 6.13, 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∈Δ(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∈Δ(Σ,Trop⋆(IB)). We show the existence of a generating set G⊂IB with well-behaved initial forms.
Lemma 6.15**.**
There exists a generating set {f1,…,fℓ}=G⊂IB with the property that for each wi, the initial forms of G 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 grading on IB. We let G=⋃i=1nGi. It remains to show that ins(wi)(f)=f∣Xj=0 for each f∈F. This is a consequence of the homogeneity and irreducibility of f.
∎
We define CG⊂Δ(Σ,Trop⋆(IB)) to be the set of those diagrams D′ with the property that ins(wi′)(f)=ins(wi)(f) for each f∈F, where wi′ is the i-th row of of D′.
Proposition 6.16**.**
Let E′ be the bundle defined by D′∈CG, then R(E′) is finitely generated.
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 1.5. By construction, ins(w1′)(I0)=⟨f1∣X1=0,…,fℓ∣X1=0⟩ is prime, it follows that the Rees algebra of vs(w1′) and R(E) can be presented as a quotient of k[X,x1′,Y] by the ideal I1=⟨f~1,…,f~ℓ⟩, where the f~i are obtained as follows. The form f~ is constructed from f=∑Cα,βXαYβ by appending a power (x1′)p(α,β), where p(α,β) is the difference between vw1′(bβ) and the minimum such value that appears among the monomials of f. By construction, f~ is in the kernel of the presentation of the Rees algebra, and the initial forms ins(w1′)(f~)=ins(w1)(f) generate a prime ideal of height equal to the height of I1. It follows that the lifts f~ generate I1, and constitute a set G~⊂I1 satisfying the properties of Lemma 6.15. We may now repeat this procedure with w2′, and so on.
∎
Proposition 6.16 shows that the locus of diagrams in Δ(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 Δ(Σ,Trop⋆(IB)) (Conjecture 1.6). The next proposition is a weakened version of such a result.
Proposition 6.17**.**
There is a finite fan F supported on Δ(Σ,LB) such that the set of diagrams D∈Δ(Σ,LB) which satisfy Proposition 6.2 for some minimal generating set {ℓ1,…,ℓd}⊂LB is the union of faces of F.
Proof.
Fix a minimal generating set ℓ={ℓ1,…,ℓd}⊂LB. We define Fℓ to be the fan supported on Trop(LB) by first taking the Gröbner fan of LB and then refinining by the domains of linearity for the piecewise linear functions obtained by tropicalizing the ℓi. In particular there is a specific choice of initial forms of the ℓi corresponding to each face of Fℓ. We let Fℓ be the fan obtained by taking the common refinement of Δ(Σ,LB) with Fℓ×⋯×Fℓ⊂Trop(LB)n.
Any face σ∈Fℓ determines initial forms of ℓi for each of n choices of points from Trop(LB), and this information in turn determines each matrix MA. It follows that for a given choice of ranks M={mA∣A⊂Σ(1)}, there is a union of faces FM,ℓ⊂Fℓ such that each diagram D∈σ with σ⊂FM,ℓ realizes M. As a consequence, the set of diagrams which satisfy the inequalities in Proposition 6.2 are the points contained in union of the appropriate FM,ℓ. Now we observe that Fℓ only depends on the supports of the ℓi∈ℓ, and that there is a finite set of possible selections of supports. We take the common refinement of the possible Fℓ⊂Trop(LB); this defines a fan structure F on Δ(Σ,LB).
∎
Example 6.18**.**
We apply Propositions 6.16 and 6.13 to the bundle from Example 6.9. To find an appropriate set G⊂IB we must add two polynomials to the generating set:
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
In this case B′ is composed of linear forms, so IB′=LB′. The cone CG⊂Δ(Σ,Trop(LB′)) is composed of diagrams of the form:
[TABLE]
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 6.9. 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-up BLp(P1×kP1), where p is the toric fixed point corresponding to the cone spanned by {(1,0),(0,−1)} in the fan of P1×kP1. It can be shown that the following diagram is adapted to Σ′, where the bottom row is associated to the new ray in Σ′.
[TABLE]
By Proposition 6.13, the algebra R(E′′) corresponding to the bundle E′′ defined by this diagram is finitely generated.
7. Examples from flag varieties and cluster algebras
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.
7.1. Vector bundles from tropicalization
By Theorem 1.4, 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 7.1**.**
Let Ψ=ψ be an n-tuple of piecewise linear functions, and suppose that for each circuit ∑i∈Jaixi∈L, ⨁i∈Jψi=⨁i∈J∖{j}ψi for each j∈J, then ψ∈TropON(L). Moreover, suppose P is a collection of n 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 define a point in TropON(L).
Proposition 7.2**.**
Let λ(t)=(λ1(t),…,λn(t)) with λi(t)∈k(TN) be in the kernel of the matrix MI whose rows are given by the coefficients of the minimal circuits of L, then (wN(λ1),…,wN(λn))∈TropON(L). Moreover, if λi(t)∈S0⊂k(TN) for each 1≤i≤n, then each wn(λi) is the support function of a polytope as in Lemma 7.1.
Proof.
Proposition 2.6 immediately implies that (wN(λ1),…,wN(λn))∈TropON(L).
∎
7.2. ON tropical points on Grassmannian varieties
Let I2,n be the Plücker ideal which cuts out the affine cone X2,n⊂⋀2(kn) over the Grassmannian variety Gr2(kn) of 2-planes in n-space. In general, the ideal I2,n is well-poised. It follows that, similar to tropicalized linear spaces, flat TN families π:G→Y(Σ) with reduced, irreducible fibers, general fiber X2,n, and a TN-equivariant embedding into the trivial bundle ⋀2(kn)×Y(Σ) correspond to piecewise linear maps Ψ:Σ→Trop(I2,n), or equivalently a point ψ∈TropON(I2,n). Moreover, the Plücker generators pijpkℓ−pikpjℓ+piℓ4pjk for 1≤i<j<k<ℓ≤n form a tropical basis of I2,n.
Let us consider the Plücker ideal I2,4=⟨p12p34−p13p24+p14p23⟩. We choose four points m={m1,m2,m3,m4}⊂M and we let Δ(m) be the convex hull of the set m. Finally, we let Pij=conv{mi,mj}. The following is straightforward to verify by hand.
Lemma 7.3**.**
The support functions ψij of the edges Pij define a point on TropON(I2,4).
Proposition 7.4**.**
Let m={m1,…,mn}⊂M, with Δ(m) and Pij defined as above, then the support functions ψij of Pij define a point on TropON(I2,n).
Proof.
Plücker relations are a tropical basis, and by Lemma 7.3 we have ψ=(…ψij,…)∈TropON(⟨pijpkℓ−pikpjℓ+piℓ4pjk⟩) for each 1≤i<j<k<ℓ≤n.
∎
From Proposition 7.4 we conclude that the 1-skeleton of an n-simplex is a point on the ON tropical (2,n) Grassmannian. The corresponding statement for m-skeleton of a simplex and the (m,n) Grassmannian also holds. Let Am,n denote the (m,n) Plücker algebra.
Theorem 7.5**.**
Let m and Δ(m) be as above, and let PI=conv{mi∣i∈I} for I⊂{1,…,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 7.2. Let L(m,t) be the following matrix with entries in k(TN):
[TABLE]
where the aij are chosen generically in k. By sending the Plücker generator pI to the I-minor δI of L(m,t), we obtain a ring map ϕm:Am,n→k(TN), and a valuation wN∘ϕm:Am,n→ON. One checks that δI(L(m,t))=∑i∈Iditmi, where di is a non-zero m−1×m−1 determinant in the aij. It follows that wN∘ϕm(pI) is the support function of PI. The theorem is then a consequence of Proposition 2.6.
∎
Example 7.6**.**
We return to the case X2,4⊂⋀2(k4), and its corresponding Plücker ideal ⟨p12p34−p13p24+p14p23⟩. Let m1=(0,0),m2=(1,0),m3=(0,1), and m4=(0,0). We view the convex hull of the mi as a degenerate tetrahedron in Q2. Following Theorem 7.5, there is an associated ring map A2,4→Q(Z2)∗ given by the minors of a matrix:
[TABLE]
with generic aij. We obtain an associated valuation w:A2,4→OZ2. Letting x:Z2→Z and y:Z2→Z be projections onto the x and y axes, respectively, we have:
[TABLE]
[TABLE]
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.
[TABLE]
[TABLE]
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 ⟨p12p34−p13p24⟩. The family G(Φ)→Y(Σ) has fibers X2,4, except over the divisors D(−1,−1),D(1,1)⊂Y(Σ), where the fiber is the toric variety V(⟨p12p34−p13p24⟩)⊂⋀2(k4).
Example 7.7**.**
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 consider a map Ψ:∣Σ∣→Trop(I2,4) which folds Σ into all three non-trivial faces of Trop(I2,4):
[TABLE]
[TABLE]
[TABLE]
On the ray generators of Σ we have:
[TABLE]
[TABLE]
We let w:A2,4→OΣ denote the valuation associated to Ψ.
By Theorem 1.5, we compute the map ψ:k[X1,…,X6,Y12,…,Y34]→R(A2,4,w^), where ψ(Xi)=ti−1∈F−ei(w), and:
[TABLE]
[TABLE]
[TABLE]
Here pij∈Fr1,…,r6(w) 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 by the image of the k-th ray is −rk. We have ker(ψ)=⟨X12Y12Y34−X52Y13Y24+X32Y14Y23⟩. The pij are a Khovanskii basis for w, and the ideal ⟨ker(ψ),Xi⟩ is prime for all i∈[6], so
[TABLE]
In particular, the Plücker generators are a strong Khovanskii basis of w.
7.3. 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 coneC⊂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].
7.4. 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 that A has no frozen variables, and that Θ=Aprin(ZT) (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 a monomial zn∈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⇢TN and a piecewise linear map μk:N→N. There is also a corresponding dual piecewise linear map Tk:M→M. Let Hk={m∣⟨ek,m⟩=0}⊂M, with Hk+ and Hk− the two corresponding half-spaces. On Hk−, Tk is the identity map, and on Hk+, Tk(m)=m+⟨ek,m⟩P∗(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. 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)=⟨{θ∣m−gs(θ)∈P∗(σs∨)}⟩. We let Ss=⨁m∈MFm(s) be the Rees space of this system of filtrations.
Proposition 7.8**.**
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.
Proof.
Once again, the M-grading on Ss is given by assigning znθ 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 3.22.
∎
Let σ^s⊂NQ be the cone that maps to σs⊂MQ under P∗. For ρ∈σs∩M with ρ=P∗(ρ′) we have vs(θ)(ρ)=⟨ρ′,gs(θ)⟩.
Once again, we fix a seed s. The g-fan Δs⊂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 Δs to Δs′, and each restriction Tk:σs,s′′→σs′,s′′ is linear. For σs,s′∈Δs, we use the map Tk:σs,s′→σs to define a valuation vs,s′:can(A)→Oσs,s′, where vs,s′(f)(ρ)=vs′(f)(Tk(ρ)).
Proposition 7.9**.**
The valuations vs,s′:can(A)→Oσs,s′ fit together to define a valuation ws:can(A)→O^Δs. Furthermore, for each θ∈Θ, there is a line bundle Ds(θ) on the toric variety YΔs, and L(⨁θ∈ΘDs(θ))=(can(A),ws).
Proof.
We show that for any σs,s′ and σs,s′′ meeting at a facet in Δs, 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)=⟨P∗(ρ′),gs′(θ)⟩, for ρ′∈NQ in the hyperplane which maps to Hk under P∗. This is equal to ⟨ρ′,Tk(gs(θ))⟩=⟨ρ′,gs(θ)+⟨ek,gs(θ)⟩P∗(ek)⟩. But by assumption ⟨ρ′,P∗(ek)⟩={ρ′,ek}=−{ek,ρ′}=−⟨ek,P∗(ρ′)⟩=0, since P∗(ρ′)∈Hk. This implies that the vs,s′ fit together into a valuation ws:can(A)→O^Δs with adapted basis Θ. For each θ∈Θ, the restriction of ws(θ) to a face of Δs is linear; it follows that there is a toric line bundle Ds(θ) on YΔs corresponding to the piecewise linear function ws(θ). Theorem 3.19 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 Δs and the valuations ws are all related by piecewise linear maps.
First we show that L:ModSσM→Vectσ and R:Vectσ→ModSσM are functors. For R∈ModSσM, we check that vR(f)∈O^σ, it is then immediate that vR is a prevaluation. Fix ℓ∈Z>0, and suppose vR(f)(ρ)=r, so f∈Grρ(R)∖Gr−1ρ(R). By definition, f∈Gℓrℓρ(R), and if f∈Gℓr−1ℓρ(R), then for some n1,…,nd we have f∈Fn1(R)+⋯+Fnd(R) where ⟨ℓρ,ni⟩>ℓr. But then ⟨ρ,ni⟩>r, which contradicts vR(f)(ρ)=r.
For an ModSσM-morphism ψ:R1→R2 there is an induced map L(ψ):ER1→ER2, and for any m∈M the m-component ψm satisfies ϕR∘ψm=L(ψ)∘ϕR. It follows that L(ψ) maps Grρ(R1) into Grρ(R2), so L(ψ) is a morphism in Vectσ.
For (E,v)∈Vectσ, it is clear that R(E,v)∈ModSσM, so 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 the action of χm commutes with the inclusions ψ, so R(ψ) is a morphism in ModSσM. It is straightforward to show R(ψ∘ψ′)=R(ψ)∘R(ψ′).
Now we prove the adjunction statement. Recall the universal property of colimit: if we are given a system of maps ψm:Rm→E which commute with χu for u∈σ∨∩M, then there is a unique map ℓ(ψ)=limψm:ER→E such that ψm=(ℓ(ψ)∘ϕR)m. Let ϕ:L(R)→(E,v) be a morphism in Vectσ. The image 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:Rm→Fm(v). The following diagram commutes:
[TABLE]
as the right two vertical arrows are inclusions, with the middle occuring in the colimit ER. As a consequence, r(ϕ)∈HomModSσM(R,R(A,v)).
Given ψ:R→R(E,v), we have a family of maps ψm:Rm→Fm(v) which commute with each χu. We get ℓ(ψ):ER→E, where ψm=ℓ(ψ)∘ϕR on Rm. Furthermore, Grρ(R) is mapped into ∑⟨ρ,m⟩≥rFm(v)⊂Grρ(v) so ℓ(ψ)∈HomVectσ(L(R),(E,v)).
Now we show that ℓ and r are inverses of each other. Given ϕ:L(R)→(E,v) and f∈ER, pick f^∈Rm so that ϕR(f^)=f. Then r(ϕ)(f^)=ϕ∘ϕR(f^)=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.
Theorem 3.13 implies that the fiber EF over id∈TN⊂Y(Σ) carries a prevaluation vF,σ:EF→O^σ for each face σ∈Σ. We must check that for any f∈EF, and τ=σ1∩σ2, the restrictions vF,σ1(f)∣τ and vF,σ2(f)∣τ coincide. More generally, 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 −⊗SσSτ:ModSσM→ModSτM′. We also have a functor ι†:Vectσ→Vectτ obtained by composing v:E→O^σ with the map on semialgebras ι♯:O^σ→O^τ given by precomposition with ι. We show that these two functors coincide under L. As a consequence, the restriction of L(R)=(ER,vR) to a facet τ⊂σ is L(R⊗SσSτ).
Proposition 8.1**.**
The following diagram commutes.
[TABLE]
Proof.
Let R′=R⊗SσSτ and R0=R⊗Sσk[M]. We have R′⊗Sτk[M′]≅R0⊗k[M]k[M′]. By setting χm′=1 for each m′∈M′ we obtain ER≅ER′, so ϕR′∘(ι∗⊗1)=ϕR. Now let ρ∈τ∩N′, and consider Grι(ρ)(R)=∑⟨ι(ρ),m⟩≥rFm(R). This space is the image of ⨁⟨ι(ρ),m⟩≥rRm⊂R under ϕR. Similarly, Grρ(R′) is the image of ⨁⟨ρ,m⟩≥rRm′ under ϕR′ where
[TABLE]
We have ⟨ι(ρ),m⟩≥r if and only if ⟨ρ,ι∗(m)⟩≥r, so Fm(R)⊂Fι∗(m)(R′) and Grι(ρ)(R)⊂Grρ(R′) in ER≅ER′. Similarly, if f∈Grρ(R′) then we can write f=∑ϕR(gi), for gi∈Fni(R)=ϕR′((ι∗⊗1)(Rni⊗kkχu)), where ι∗(ni)⊙u=mi with ⟨ρ,mi⟩≥r. But then ⟨ρ,ι∗(ni)⟩≥r, so that ⟨ι(ρ),ni⟩≥r. It follows that f∈Grι(ρ)(R) and Grι(ρ)(R)=Grρ(R′). Now fix ρ∈τ∩N′ and f∈AR, then ι♯vR(f)(ρ)=vR(f)(ι(ρ)), which is equal to vR′(f)(ρ) by the above calculation. This shows that ι†∘Lσ(R)=(ER,ι♯vR) is isomorphic to (ER′,vR′)=Lτ∘(R⊗SσSτ) by the map which identifies ER with ER′.
∎
Proposition 8.1 implies that L(F)=(EF,vF)∈VectΣ. By Proposition 3.13 a morphism ϕ:F→G gives a map L(ϕ)∣σ:(EF,vF∣σ)→(EG,vG∣σ), so we have a functor L:ShY(Σ)M→VectΣ. We observe ([Sta19, Lemma 17.15.4]) that for any σ∈Σ, and F,E∈ShY(Σ)M, 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)⊗SσΓ(Y(σ),F)) since E,F are quasi-coherent 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).
Now let F,G∈ShY(Σ)M,ev, and consider a map ψ:(EF,vF)→(EG,vG). 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 quasi-coherent Y(σ) sheaves: R~(ψ∣σ):R~(EF,vF∣σ)→R~(EG,vG∣σ). The sheaves F,G were chosen in ShY(Σ)M,ev, so R~(EF,vF∣σ) and R~(EG,vG∣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} (see e.g. [Sta19, Lemma 68.3.4]). This map induces a unique map between F and G. It’s straightforward to check that this construction is inverse to L on ShY(Σ)M,ev, in particular L(ϕ):(EF,vF)→(EG,vG) is an isomorphism if and only if ϕ:F→G is an isomorphism.
We make use of Lazard’s Theorem ([Sta19, Theorem 10.80.4]). Any f∈Grρ(R) is a sum of elements fi∈Fmi(R) with ⟨ρ,mi⟩≥r, and all Fm(R) with ⟨ρ,m⟩=r are in the image of ϕ^ρ. If g∈mτR then we can write g=∑χvjgj+∑(χui−1)hi for ⟨ρ,vj⟩<0 and ⟨ρ,ui⟩=0, where gj and hi are homogeneous. Let deg(g)=m−v so that χvg∈Rm, then ϕ^ρ(χvg)∈Fm−v(R)⊂G⟨ρ,m⟩ρ(R). It follows that ϕ^ρ(χvg)=0. Similarly, since ⟨ρ,u⟩=0, ϕ^ρ(χuh)=ϕ^ρ(h). If R∈AlgSσM 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 ⟨ρ,mi⟩=r, and suppose that ϕρ(∑i=1npi)=0. Let B⊂EP be a linear adapted basis with deg(bj)=λj for bj∈B. Then pi=∑j=1kcijχvijbj with vij+λj=mi. If ⟨ρ,vij⟩<0 then cijχvijbj∈mτP, so without loss of generality we assume that ⟨ρ,vij⟩=0. This implies that ⟨ρ,λj⟩=r for each bj. Since the bj with ⟨ρ,λj⟩=r form a basis of Grρ(P)/G>rρ(P), we conclude that ∑i=1ncij=0 for each j. Then ∑i=1npi=∑j=1k(∑i=1ncijχvij)bj=∑j=1k(∑i=1ncij(χvij−1))bj∈mτP. Finally, observe that grρ:Vectσ→Vectk is a functor which 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 limPi=R for Pi free. It follows that R/mτR≅limPi/mτPi≅limgrρ(L(Pi))≅grρ(limL(Pi))≅grρ(L(R))=grρ(ER).
By definition, Fψ(v)⊂⋂ϱi∈Σ(1)Gψ(ρi)ρi(v). We show that if v(f)(ρi)≥⟨ρi,mσ⟩ for each ϱi∈σ(1) then v(f)∣σ≥mσ, so Fmσ(v∣σ)=⋂ϱi∈σ(1)Gψ(ρi)ρi(v). A function ϕ∈O^σ is said to be concave if for any ρ1,…,ρℓ∈σ we have ϕ(∑i=1ℓρi)≥∑i=1ℓϕ(ρi). We show that vR:ER→O^σ takes concave values when R is flat.
If (E,v)=L(R), where R∈ModSσM, let ρ,ρ′∈σ∩N, and consider f∈Grρ+ρ′(R) for some r∈Z. We can write f=∑i=1ℓfi for fi∈Fmi(R) with ⟨ρ+ρ′,mi⟩≥r. We let si=⟨ρ,mi⟩ and ti=⟨ρ′,mi⟩ with si+ti=r, so that fi∈Gsiρ(R)∩Gtiρ′(R). It follows that f∈∑s+t≥rGsρ(R)∩Gtρ′(R), and Grρ+ρ′(R)⊂∑s+t≥rGsρ(R)∩Gtρ′(R). We show that containment holds in the other direction when R is flat. Let f∈Gsρ(R)∩Gtρ′(R) for s+t≤r, then f=∑gi=∑hj for gi∈Fmi(R) and hj∈Fnj(R) with ⟨ρ,mi⟩≥s and ⟨ρ′,nj⟩≥t. Since R is flat it is torsion-free; we choose an appropriate m and identify f∈Rm with f=∑χuigi=∑χvjhj for ui+mi=vj+nj=m. This is always possible as σ∨ is a full dimensional cone in MQ. Now we choose a presentation π^:P→R with pi∈Pmi and qj∈Pnj such that π^(pi)=gi, π^(hj)=qj. We can find ϕ:P→P′, π:P′→R such that π∘ϕ=π^ with P′ free and π(∑χuipi)=π(∑χvjqj)=p∈Gsρ(P′)∩Gtρ′(P′)⊂Grρ+ρ′(P′). We have L(π)(Grρ+ρ′(P′))⊂Grρ+ρ′(R), so L(π)(p)=f∈Grρ+ρ′(R). In conclusion, if R is flat then:
[TABLE]
If v(f)(ρi)=ti, it follows that f∈Gt1ρ1(R)∩⋯∩Gtℓρℓ(R)⊂Gr∑i=1ℓρi(R), where r=∑i=1ℓti, so that v(f)(∑i=1ℓρi)≥r=∑i=1ℓv(f)(ρi). This proves that v(f) is concave.
Now we consider Fm(v)=⋂ρ∈σ∩NG⟨ρ,m⟩ρ(R)⊂⋂ϱi∈σ(1)G⟨ρi,m⟩ρi(R). Pick ρ∈σ∩N and d∈Z≥0 so that dρ=∑ϱi∈σ(1)diρi for some di∈Z≥0. Now suppose that v(f)(ρi)≥⟨ρi,m⟩ for m∈M. It follows that v(f)(dρ)=v(f)(∑ϱi∈σ(1)diρi)≥∑ϱi∈σ(1)v(f)(diρi). We have v(f)∈O^σ, so dv(f)(ρ)=v(f)(dρ)≥∑ϱi∈σ(1)v(f)(diρi)≥∑ϱi∈σ(1)⟨diρi,m⟩=d⟨ρ,m⟩, and v(f)(ρ)≥⟨ρ,m⟩. As ρ was arbitrary, it follows that f∈Fm(v).
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