Birational geometry of cluster algebras
aa r X i v : . [ m a t h . AG ] A p r BIRATIONAL GEOMETRY OF CLUSTER ALGEBRAS
MARK GROSS, PAUL HACKING, AND SEAN KEEL
Abstract.
We give a geometric interpretation of cluster varieties in terms of blowupsof toric varieties. This enables us to provide, among other results, an elementary geo-metric proof of the Laurent phenomenon for cluster algebras (of geometric type),extend Speyer’s example [Sp13] of upper cluster algebras which are not finitely gen-erated, and show that the Fock-Goncharov dual basis conjecture is usually false.
Contents
Introduction 11. Log Calabi-Yau varieties and a geometric motivation for cluster varieties. 42. Review of the X and A cluster varieties 133. The geometry of cluster varieties 213.1. Elementary transformations. 223.2. The X - and A prin -cluster varieties up to codimension two. 284. The A t and A prin cluster varieties as torsors 325. The X variety in the rank ǫ = 2 case 386. Examples of non-finitely generated upper cluster algebras 447. Counterexamples to the Fock-Goncharov dual bases conjecture 46References 48 Introduction
Cluster algebras were introduced by Fomin and Zelevinsky in [FZ02a]. Fock andGoncharov introduced a more geometric point of view in [FG09], introducing the A and X cluster varieties constructed by gluing together “seed tori” via birational mapsknown as cluster transformations.In this note, motivated by our study of log Calabi-Yau varieties initiated in thetwo-dimensional case in [GHK11], we give a simple alternate explanation of basic con-structions in the theory of cluster algebras in terms of blowups of toric varieties. Each seed roughly gives a description of the A or X cluster variety as a blowup of a toricvariety, and a mutation of the seed corresponds to changing the blowup descriptionby an elementary transformation of a P -bundle. Certain global features of the clus-ter variety not obvious from the expression as a union of tori are easily seen from thisconstruction. For example, it gives a simple geometric explanation for the Laurent phe-nomenon (originally proved in [FZ02b]), see Corollary 3.11. From the blowup pictureit is clear that the Fock-Goncharov dual basis conjecture, particularly the statementthat tropical points of the Langlands dual A parameterize a natural basis of regularfunctions on X , can fail frequently, see § §
1, we explain the basic philosophical point of view demonstratinghow a study of log Calabi-Yau varieties can naturally lead to the basic notions ofcluster algebras. This section can be read as an extended introduction; its role inthe paper is purely motivational. In §
2, we review the definitions of cluster varieties,following [FG09]. We pay special attention to the precise procedure for gluing tori viacluster transformations, as this has not been discussed to the precision we need in theliterature. § P -bundles in algebraic geometry. This procedure blows up acodimension two center in a P -bundle meeting any P fibre in at most one point, andblows down the proper transform of the union of P fibres meeting the center.This is a very general construction, covered in § A and X cluster varieties in § A cluster variety, but does work for the A variety with principalcoefficients . This variety A prin fibres over an algebraic torus with A being the fibreover the identity element of the torus. Properties such as the Laurent phenomenon for A can be deduced from that for A prin . Many of the phenomena discussed here alsowork for a very general fibre A t of the map from A prin ; we call such a cluster varietyan A cluster variety with general coefficients . The algebra of regular functions of sucha cluster variety are of the kind considered by Speyer in [Sp13].The key result is Theorem 3.9, which gives the precise description of the X , principal A cluster varieties and A cluster varieties with general coefficients up to codimensiontwo in terms of a blowup of a toric variety. The toric variety and the center of the More precisely, the A prin , A t (defined in §
2) or X cluster variety. IRATIONAL GEOMETRY OF CLUSTER ALGEBRAS 3 blowup is specified very directly by the seed data determining the cluster variety. Animmediate consequence is the Laurent phenomenon, Corollary 3.11.In §
4, we give another description of the principal A cluster variety and A clustervariety with general coefficients in terms of line bundles on the X cluster variety. Thereis in fact an algebraic torus which acts on A prin , and the quotient of this action is X ,making A prin a torsor of X . We give a precise description of this family in terms ofline bundles on X . Furthermore, there are tori T K ∗ and T K ◦ such that there is a map X → T K ∗ and an action of T K ◦ on any A cluster variety with general coefficientsdetermined by the seed data. We show that for any such sufficiently general A clustervariety A t , there is a φ = φ ( t ) ∈ T K ∗ such that up to codimension two, A t is the universal torsor of X φ , essentially obtained as Spec L L∈ Pic( X φ ) L . In particular, thisallows us to identify the corresponding upper cluster algebra with the Cox ring of X φ .This is a slight simplification of the discussion: see the main text for precise statements.The Cox ring of any variety with finitely generated torsion free Picard group is factorial,see [Ar08] and [BH03]. This explains the ubiquity of factorial cluster algebras remarkedon, e.g., in [K12], § X → T K ∗ is a family of surfaces. In fact, the fibresare essentially the interiors of Looijenga pairs . A Looijenga pair is a pair (
Y, D ) where Y is a rational surface and D ∈ | − K Y | is a cycle of rational curves, U := Y \ D is theinterior. We study moduli of such pairs in [GHK12]. Here, we show (Theorem 5.5) thatessentially X → T K ∗ coincides with a type of universal family constructed in [GHK12].Our construction implies that in many cases, the kernel of the skew-symmetric ma-trix carries a canonical symmetric form, invariant under mutations, see Theorem 5.6.Though not (as far as we know) previously observed, this symmetric form controls thegross geometry of X , in particular the generic fibre of X → T K ∗ . Indeed, the fibre isaffine if and only if the form is negative definite; when the form is indefinite the fibresare the complement of a single point in a compact complex analytic space, and thushave no non-constant global functions. Thus in this indefinite case (which from theblowup point of view is the generic situation) the only global functions on X are pulledback from T K ∗ , contradicting the dual basis conjecture of [FG09], see § §
6, we give a general procedure for constructing upper cluster algebras with generalor principal coefficients which are not finitely generated. These examples generalizethat given by Speyer in [Sp13], and suggest that “most” upper cluster algebras arenot finitely generated. These examples arise because Cox rings tend not to be finitely
MARK GROSS, PAUL HACKING, AND SEAN KEEL generated. Indeed, finite generation of the Cox ring of a projective variety is a verystrong (Mori Dream Space) condition, see [HK00].In this paper, we will always work over a field k of characteristic zero. Acknowledgments : The genesis of our results on cluster varieties was a conversationwith M. Kontsevich. He pointed out to us that (in the skew-symmetric case) a seed isthe same thing as a collection of vectors in a symplectic lattice and the piecewise linearcluster mutation is just like moving worms in the integral affine manifolds centralto mirror symmetry for open Calabi-Yau varieties, see e.g., [GHK11]. Before thisconversation we had been incorrectly assuming the cluster picture was a very specialcase of the mirror construction in [GHK11]. However, Kontsevich’s remarks led us tothe correct view that in dimension two, the scope of the two theories are exactly thesame. This in turn led to the simple blowup description of cluster varieties we describehere.We first learned of the connection between mirror symmetry and cluster varietiesfrom conversations with A. Neitzke. We received considerable inspiration from conver-sations with V. Fock, S. Fomin, A. Goncharov, B. Keller, B. Leclerc, G. Musiker, M.Shapiro, Y. Soibelman, and D. Speyer. Special thanks go to Greg Muller, who pointedout a crucial mistake in a draft version of this paper, see Remark 3.13.The first author was partially supported by NSF grant DMS-1105871 and DMS-1262531, the second by NSF grants DMS-0968824 and DMS-1201439, and the third byNSF grant DMS-0854747.1.
Log Calabi-Yau varieties and a geometric motivation for clustervarieties.
To a geometer, at least to the three of us, the definition of a cluster algebra is ratherbizarre and overwhelming. Here we explain the geometric motivation in terms of logCalabi-Yau varieties. There are two elementary constructions of log Calabi-Yau (CY)varieties. The first method is to glue together tori in such a way that the volume formspatch. The second method is to blow up a toric variety along a codimension two centerwhich is a smooth divisor in a boundary divisor, and then remove the strict transformof the toric boundary. As we will see, the simplest instances of either construction areclosely related, and either leads to cluster varieties. The first approach extends theviewpoint of [FG09], the second was inspired by [L81].
Definition 1.1.
Let (
Y, D ) be a smooth projective variety with a normal crossingdivisor, and let U = Y \ D . By [I77], the vector subspace H ( Y, ω Y ( D ) ⊗ m ) ⊂ H ( U, ω ⊗ mU ) IRATIONAL GEOMETRY OF CLUSTER ALGEBRAS 5 (where the inclusion is induced by restriction) depends only on U , i.e., is independentof the choice of normal crossing compactification. We say U is log Calabi-Yau if forall m this subspace is one-dimensional, generated by Ω ⊗ m for a volume (i.e., nowherevanishing) form Ω ∈ H ( U, ω U ). Note that by definition Ω is unique up to scaling.In practice, log Calabi-Yau varieties are often recognized using the following: Lemma 1.2.
Let ( Y, D ) be a dlt pair with K Y + D trivial (in particular Cartier), and Y projective. Let U ⊂ Y \ D be a smooth open subset, with ( Y \ D ) \ U of codimensionat least . Then U is log CY. For the definition of dlt (divisorial log terminal), see [KM98], Def. 2.37. As thissection should be viewed as purely motivational, the reader who wishes to avoid thetechnicalities of the minimal model program should feel free to assume that the pair(
Y, D ) is in fact normal crossings.
Proof.
When (
Y, D ) has normal crossings this is immediate from Definition 1.1. Thedefinition of dlt is such that the vector space of Definition 1.1 can be computed usinga dlt (instead of normal crossing) compactification. (cid:3)
Remark . The data (
Y, D ) with U ⊆ Y \ D as in the lemma is called a minimalmodel for U . One example of a minimal model is a pair ( Y, D ) with D ∈ | − K Y | areduced normal crossings divisor. This is a minimal model for U = Y \ D . The mainconjectures of the minimal model program would imply every log CY has a minimalmodel, see [BCHM]. Lemma 1.4. (1)
Let U ⊂ V be an open subset, with ( U, Ω) log CY. Then V is logCY if and only if Ω extends to a volume form on V , and in this case Ω is ascalar multiple of the volume form of V . (2) Let µ : U V be a birational map between smooth varieties which is anisomorphism outside codimension two subsets of the domain and range. Then U is log CY if and only if V is.Proof. For (1), if V is log CY, then clearly its volume form restricts to a scalar multipleof the volume form on U . Now suppose U is log CY, and its volume form Ω extendsto a volume form on V . We have U ⊆ V ⊆ Y where Y is a compactification of both U and V . Thus Ω (and its powers) obviously has at worst simple poles on any divisorcontained in Y \ V , and it is unique in this respect, since we have the same propertiesfor Ω as a volume form on U . Next (2) follows from (1), passing to the open subsetswhere the map is an isomorphism, noting that in (1), when the complement of U hascodimension at least two, the extension condition is automatic. (cid:3) MARK GROSS, PAUL HACKING, AND SEAN KEEL
Definition 1.5.
We say a log CY U has maximal boundary if it has a minimal model( Y, D ) with a zero-dimensional log canonical center. For example, this is the case if(
Y, D ) is a minimal model for U such that D is simple normal crossings and containsa zero-dimensional stratum, i.e., a point which is the intersection of dim( Y ) distinctirreducible components of D . Example 1.6.
Consider the group G = PGL n . There are the 2 n − n × n matrix given by the square submatrices in the upper right corner or the lowerleft corner. For example, for n = 3 these are the 4 minors a , , a , , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a , a , a , a , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a , a , a , a , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) and the determinant of the 3 × D ⊂ Y = P (Mat n × n ) = P n − bethe union of the 2 n − D is 1 + 2 + · · · + ( n −
1) + 1 + 2 + · · · + ( n −
1) + n = n , so D ∈ | − K Y | . With some non-trivial effort, one can check ( Y, D ) is dlt, with a zero-dimensional log canonical center, and thus (
Y, D ) is a minimal model for the smoothaffine log CY with maximal boundary U ⊂ G , the non-vanishing locus of this collectionof minors. U is by definition the open double Bruhat cell in G .A log CY U with maximal boundary will (in dimension at least two) always haveinfinitely many minimal models. The set of possibilities leads to a fundamental invari-ant: Definition 1.7.
Let ( U, Ω) be a log CY. Define(1.1) U trop ( Z ) := { divisorial discrete valuations v : k ( U ) \ { } → Z | v (Ω) < } ∪ { } := { ( E, m ) | m ∈ Z + , E ⊂ ( Y \ U ) , Ω has a pole along E } ∪ { } . Here k ( U ) is the field of rational functions of U , a discrete valuation is called divisorialif it is given by the order of vanishing of a divisor on some variety birational to U .Furthermore, we define v ( gdz ∧ · · · ∧ dz n ) := v ( g )for z , . . . , z n local coordinates in a neighborhood of the generic point of the divisorcorresponding to v ; this is independent of the choice of coordinates as a change ofcoordinates only changes g by a unit. In the second expression E is a divisorial irre-ducible component of the boundary in some partial compactification U ⊂ Y , and two IRATIONAL GEOMETRY OF CLUSTER ALGEBRAS 7 divisors on two possibly different birational varieties are identified if they give the samevaluation on their common field of fractions.The simplest example of a log CY with maximal boundary is an algebraic torus T N := N ⊗ G m , for N = Z n . Note H ( T N , O T N ) = k [ M ], where M = Hom( N, Z ) is the characterlattice of T N . Lemma 1.8.
Restriction of valuations to the character lattice M induces a canonicalisomorphism T trop N ( Z ) = N. A minimal model for T N is the same as a complete T N -equivariant toric compactifica-tion.Proof. This is an easy log discrepancy computation, using e.g., [KM98], Lemmas 2.29and 2.45. (cid:3)
Thus U trop ( Z ) gives an analog for any log CY of the cocharacter lattice of a torus.Note however that in general U trop ( Z ) is not a group as addition does not make sense.We conjecture there is also an analog of the character lattice, or equivalently, the dualtorus: Conjecture 1.9. [GHK11]
Let ( Y, D ) be a simple normal crossings minimal model fora log CY with maximal boundary U = Y \ D , and assume D supports an ample divisor(note this implies U is affine). Let R = k [Pic( Y ) ∗ ] . The free R -module V with basis U trop ( Z ) has a natural finitely generated R -algebra structure whose structure constantsare non-negative integers determined by counts of rational curves on U . The associatedfibration p : Spec( V ) → Spec( R ) = T Pic( Y ) is a flat family of affine log CYs withmaximal boundary. Letting K be the kernel of the natural surjection Pic( Y ) ։ Pic( U ) , p is T K -equivariant. The quotient family Spec( V ) /T K → T Pic( U ) depends only on U (isindependent of the choice of minimal model), and is the mirror family to U .Remark . An analog of Conjecture 1.9 is expected for compact Calabi-Yaus, butperhaps only with formal (e.g., Novikov) parameters, and for Calabi-Yaus near theso-called large complex structure limit. This will be discussed in forthcoming work.The maximal boundary condition means the boundary is highly degenerate — we arethus already in some sense in the large complex structure limit, and so one can hopethat no formal power series or further limits are required. This is one reason to focuson this case. The other is the wealth of fundamental examples.
MARK GROSS, PAUL HACKING, AND SEAN KEEL
The conjecture is of interest independently of mirror symmetry: in many instancesthe variety U and its prospective mirror are known varieties of compelling interest.The conjecture then gives a new construction of a variety we already care about,a construction which in particular endows the mirror (and each fibre of the family)with a canonical basis of functions. In any case mirror symmetry is conjecturally aninvolution, the mirror of the mirror being a family of deformations of the original U .Thus the conjecture says in particular that any affine log CY with maximal boundaryis a fibre of the output of such a construction, and thus in particular has a canonicalbasis of functions, B U . One then expects B U to be the tropical set of the conjecturalmirror.We call a partial compactification U ⊂ Y a partial minimal model if the volumeform Ω has a pole on every irreducible divisorial component of Y \ U . One checks usingLemma 1.8 that a partial minimal model for an algebraic torus is the same thing as atoric variety. We further conjecture that for any partial minimal model (not necessarilyaffine) of an affine log CY U with maximal boundary, B U ∩ H ( Y, O Y ) ⊂ H ( Y, O Y )is a basis of regular functions on Y . For example we conjecture that the open doubleBruhat cell U ⊂ G has a canonical basis of functions, and that the subset of basiselements which extend regularly to G give a basis of functions on G .After tori, the next simplest example of a log CY with maximal boundary is obtainedby gluing together algebraic tori in such a way that the volume forms patch. Moreprecisely, suppose that A = [ s ∈S T N, s is a variety covered by open copies of the torus T N indexed by the set S . This givescanonical birational maps µ s , s ′ : T N, s T N, s ′ for each pair of seeds s , s ′ ∈ S . Then A will be log CY if and only if each birational map is a mutation , i.e., preserves thevolume form: µ ∗ (Ω) = Ω. In this case each choice of seed torus T N, s ⊂ A gives acanonical identification A trop ( Z ) = T trop N, s ( Z ) = N .We can reverse the procedure. Beginning with a collection of such mutations satisfy-ing the cocycle condition, we can canonically glue together the tori along the maximalopen sets where the maps are isomorphisms to form a log CY A . See Proposition 2.4for details. The simplest example of a mutation comes from a pair ( n, m ) ∈ N × M with h n, m i := m ( n ) = 0. It is defined by(1.2) µ ∗ ( n,m ) ( z m ′ ) = z m ′ · (1 + z m ) h m ′ ,n i IRATIONAL GEOMETRY OF CLUSTER ALGEBRAS 9 where z m ′ , z m ∈ k [ M ] are the corresponding characters of T N . Cluster varieties are logCYs formed by gluing tori by mutations of this simple sort (and compositions of such)for a particular parameterizing set S . See § m, − n ) ∈ Hom( N × M, Z ) = M × N, so that µ ( m, − n ) defines a birational automorphism of T M . Thus for each A := S s ∈S T N, s built from such maps, there is a canonical dual X := S s ∈S T M, s , just obtained byreplacing each torus (and each mutation) by its dual. For the particular parameterizingset S used in cluster varieties, Fock and Goncharov made the following remarkableconjecture: Conjecture 1.11. A trop ( Z ) parameterizes a canonical vector space basis for H ( X , O X ) .The structure constants for the algebra H ( X , O X ) expressed in this basis are non-negative integers. (Here we are treating the notationally simpler case of skew-symmetric cluster vari-eties, the general case involving a Langlands dual seed.)Note as stated A and X are on completely equal footing, so the conjecture includesthe analogous statement with the two reversed. Fock and Goncharov have a differentdefinition of e.g., A trop ( Z ), which they denote A ( Z t ), as points of A valued in thetropical semi-field. But it is easy to check this agrees with our definition, which hasthe advantage that it makes sense for any log CY, while theirs is restricted to varietieswith a so-called positive atlas of tori.In somewhat more detail, a skew-symmetric cluster variety is defined using initialdata of a lattice N with a skew-symmetric form {· , ·} : N × N → Z , and each mutationis given by the pair ( n, { n, ·} ) for some n ∈ N . When {· , ·} fails to be unimodular,the dual M does not have a skew-symmetric form, and in this case A and X are onunequal footing. In this case A , by the Laurent phenomenon, always has lots of globalfunctions, but X may have very few.Conjecture 1.11 was inspired by the case A := U ⊂ G of Example 1.6, which hasa celebrated canonical basis of global functions constructed by G. Lusztig. See [L90].Conjecture 1.9 suggests the existence of this basis may have nothing a priori to do with representation theory, or cluster varieties, but is rather a general feature of affine logCYs with maximal boundary.In § X is affine, it becomes a very special case of Conjecture 1.9, and for that reasonwe refer to X , A as Fock-Goncharov mirrors . In view of the highly involved existingproposals for synthetic constructions of mirror varieties, [KS06], [GS11], [GHK11], thissimple alternative — replace each torus in the open cover by its dual — is an attractivesurprise. We will prove many instances of Conjecture 1.11 in [GHKK].We now turn to the main idea in this paper, which connects the above traditionaldescription of cluster varieties via gluing tori to the description we will develop in thispaper, involving blowups of toric varieties. Here is some cluster motivation for theblowup approach. Each seed s gives a torus open subset T N, s ⊂ A , together with n cluster variables, a basis of characters. These give a priori rational functions on A andthus a birational map b : A A n , whose inverse restricts to an isomorphism of thestructure torus G nm ⊂ A n with T N, s ⊂ A . The Laurent Phenomenon is equivalent to thestatement that b is regular, and thus in particular suggests that each seed determines aconstruction of A as (an open subset of) a blowup of a toric variety (in fact A n ) alonga locus in the toric boundary. Stated this way, it is natural to wonder if it holds for X as well. We’ll show this indeed holds for X , and while it fails for general A , a slightlyweaker version is true which is still good enough for the Laurent Phenomenon.Log CYs with maximal boundary are closed under blowup in the following sense: Lemma 1.12.
Let ¯ U ⊂ ¯ Y be a log CY open subset of a smooth (not necessarilycomplete) variety, ¯ D := ¯ Y \ ¯ U , and H ⊂ ¯ D \ Sing( ¯ D ) be a smooth codimension two(not necessarily irreducible) subvariety. Let b : Y → ¯ Y be the blowup along H , D ⊂ Y the strict transform of ¯ D and U := Y \ D . Then U is log CY, with unique volume formthe pullback under b of the volume form on ¯ U . In addition, U has maximal boundaryif ¯ U does.Proof. If E is the exceptional divisor of b , then it is standard that K Y = b ∗ K ¯ Y + E (using H codimension two) and that D = b ∗ ¯ D − E . Thus K Y + D = 0. (cid:3) Now starting with the simplest example, an algebraic torus, we get lots of examplesvia:
Definition 1.13.
Continuing with notation as in the lemma, we say that U = Y \ D is a cluster log CY and b : ( Y, D ) → ( ¯ Y , ¯ D ) a toric model for U if(1) ( ¯ Y , ¯ D ) is toric and the fan for ¯ Y consists only of one-dimensional cones R ≥ v i for v i ∈ N primitive, with T N the structure torus of ¯ Y . (Equivalently, theboundary D is a disjoint union of codimension one tori). IRATIONAL GEOMETRY OF CLUSTER ALGEBRAS 11 (2) The connected components of H are the subtori z w i + 1 = 0 ⊂ T N/ Z · v i for some w i ∈ ( N/ Z · v i ) ∗ = v ⊥ i ⊂ M .As the name suggests, the log CYs obtained by this simple blowup constructionand those obtained in the previous discussion as tori glued in the simplest way arefrequently the same. Note the toric model determines a canonical torus open subset T N = ¯ Y \ ¯ D ⊂ Y \ D = U. Remarkably there are (usually infinitely) many other torus open sets. Given a toricmodel for U , and a choice of a center, i.e., a connected component of H , or equivalently,a choice of one of the primitive lattice points v = v k , there is a natural mutation whichproduces a new log CY U ′ , with a birational map U U ′ . Under certain conditions,this map will be an isomorphism outside of codimension two subset (of domain andrange). In these nice situations, this produces, up to codimension two, a second copyof T N living in U . Iterating the procedure produces an atlas of torus open sets. Hereis a sketch; full details are given in § µ : U V between log CY varieties canonically induces an isomorphism of tropicalsets µ t : U trop ( Z ) → V trop ( Z ) , v → v ◦ µ ∗ . For the mutation µ ( n,m ) : T N T N of Equation (1.2), one computes(1.4) µ t ( n,m ) : N = T trop N ( Z ) → T trop N ( Z ) = N, µ t ( n ′ ) = n ′ + [ h m, n ′ i ] − n where for a real number r , [ r ] − := min( r, µ t is piecewise linear but not linear (unless µ is an isomorphism). This explains thegeometric origin of piecewise linear maps in the cluster theory (and tropical geometry,see [HKT09], § U trop ( Z ) as a collection of valuations. If we thinkof elements of U trop ( Z ) as boundary divisors with integer weight, as in the secondformula in equation (1.1), µ t is simply strict transform (also called pushforward) forthe birational map µ .Now we explain how to mutate from one toric model of a cluster log CY to another.Continuing with the situation of Definition 1.13, we choose one index, k , and let v = v k ,with corresponding divisor D k . The center H k = H ∩ D k determines what is known asan elementary transformation in algebraic geometry. We explain this in a simplified,but key, situation.Let Σ v be the fan, with two rays, with support R v , so that the corresponding toricvariety X Σ v ∼ = T N/ Z v × P , with π : X Σ v → T N/ Z v the projection. Write D ± for thetwo toric divisors corresponding to the rays generated by ± v . Viewing X Σ v \ D − as an open subset of ¯ Y , the center H k is identified with a codimension two subscheme H + ⊂ D + ⊂ X Σ v . Let H − = π − ( π ( H + )) ∩ D − .There is then a birational map µ : X Σ v X Σ v obtained by blowing up H + andthen blowing down the strict transform of π − ( π ( H + )). One checks that µ is describedby Equation (1.2). Clearly by construction µ is resolved by the blowup b : Y ′ → X Σ v along H + , and one can check that µ ◦ b : Y ′ → X Σ v is regular as well, being the blowupalong H − , see Lemma 3.2.This description of the elementary transformation extends to give birational mapsbetween closely related toric models. For simplicity assume − v = v i for any i (in § + be the fan consisting of rays R ≥ v i togetherwith − R ≥ v . The toric model gives us a blowup b : Y → X Σ + . (This is a slight abuseof notation, because we added one ray, − R ≥ v . But note we do not blow up along thenew boundary divisor D − ⊂ X Σ + , and in forming U we throw away the strict transformof boundary divisors, so adding this ray does not change U at all). Let Σ − be the fanwith rays R ≥ µ t ( v i ) together with − R ≥ v = − R ≥ µ t ( v ). In § b ′ := µ ◦ b : Y → X Σ − is regular off a codimension two subset and give formulae for the centers, which againare of the cluster log CY sort. Thus the elementary transformation induces a new toricmodel for U (up to changes in codimension two), and in particular a second torus opensubset of U . This recovers the standard definition of mutations for cluster algebras[FZ02a]. From this perspective, each seed is interpreted as the data for a toric modelof the same (up to codimension two) cluster log CY. Note in the mutated toric model b ′ : Y → X Σ − there is now a center in the boundary divisor D − , but no center in D + . In the original model b : Y → X Σ + there is a center in D + ⊂ X Σ + (this divisoris the strict transform of D + ⊂ X Σ − ) but no center in D − ⊂ X Σ + . For all the otherboundary divisors there is a center in either model. This difference between the chosenindex k and the other indices accounts for the peculiar sign change in the formula forseed mutation, see Equation (2.3).Unfortunately, this procedure does not always give a precise identification betweenthe picture of cluster varieties as obtained from gluing of tori and the picture given byblowups of toric varieties. The reason is that b ′ above need not always be regular off acodimension two subset. It turns out that this works in certain cases, including all X cluster varieties and principal A cluster varieties. See § § Remark . There is no need to restrict to the special centers of Definition 1.13, (2):one can consider the blowup of an arbitrary hypersurface in each boundary divisor.
IRATIONAL GEOMETRY OF CLUSTER ALGEBRAS 13
An elementary transform gives a mutation of a toric model in the same way, but theformulae for how the centers change are more complicated. For a general center, wechecked one obtains the mutation formulae of [LP12]. In this note we restrict ourtreatment to the cluster variety case, as it is simpler and sufficient for our applications.There are lots of formulae in the Fomin-Zelevinsky, Fock-Goncharov definitions ofcluster algebras, which we reproduce in the next section. But we note that only one,Equation (1.2), is essential. This is the birational mutation, µ , between tori in the A -atlas. Its canonical dual, arising from Equation (1.3), gives the mutation for the Fock-Goncharov mirror, see Equations (2.6) and (2.5) below. The formula for the changeof seed, Equation (2.3), comes from the tropicalisation, µ t , of the birational mutation,Equation (1.4). Note in Equation (2.3), e ′ i = µ t ( e i ) for i = k , e ′ k = µ t ( − e k ) = − e k .This is the peculiar sign change explained above.2. Review of the X and A cluster varieties We follow [FG09], with minor modifications. We will fix once and for all in thediscussion the following data, which we will refer to as fixed data : • A lattice N with a skew-symmetric bilinear form {· , ·} : N × N → Q . • An unfrozen sublattice N uf ⊆ N , a saturated sublattice of N . If N uf = N , wesay the fixed data has no frozen variables. • An index set I with | I | = rank N and a subset I uf ⊆ I with | I uf | = rank N uf . • Positive integers d i for i ∈ I with greatest common divisor 1. • A sublattice N ◦ ⊆ N of finite index such that { N uf , N ◦ } ⊆ Z , { N, N uf ∩ N ◦ } ⊆ Z . • M = Hom( N, Z ), M ◦ = Hom( N ◦ , Z ).Given this fixed data, seed data for this fixed data is a labelled collection of elementsof N s := ( e i | i ∈ I )such that { e i | i ∈ I } is a basis of N , { e i | i ∈ I uf } a basis for N uf , and { d i e i | i ∈ I } is abasis for N ◦ . This terminology is not standard in the cluster literature. Rather, what we call fixed data alongwith seed data is referred to as seed data in the literature. We prefer to distinguish the data whichremains unchanged under mutation from the data which changes.
A choice of seed data s defines a new (non-skew-symmetric) bilinear form on N by[ · , · ] s : N × N → Q [ e i , e j ] s = ǫ ij := { e i , e j } d j Note that ǫ ij ∈ Z as long as we don’t have i, j ∈ I \ I uf . We note this bilinear formdepends on the seed. We drop the subscript s if it is obvious from context. Remark . Suppose we specify a basis e i , i ∈ I for a lattice N , I uf ⊆ I , positiveintegers d i , and a matrix ǫ ij satisfying d i ǫ ij = − d j ǫ ji and ǫ ij ∈ Z provided we don’t have i, j ∈ I \ I uf . This data determines the data N , N uf , N ◦ , {· , ·} , etc. It will turn out that ǫ ij for i, j ∈ I \ I uf does not affect the schemeswe construct, and it is standard in the literature to just consider rectangular matrices( ǫ ij ) i ∈ I uf ,j ∈ I . We wish however to emphasize that the fixed data does not depend onthe particular choice of seed.Given a seed s , we obtain a dual basis { e ∗ i } for M , and a basis { f i } of M ◦ given by f i = d − i e ∗ i . We use the notation h· , ·i : N × M ◦ → Q for the canonical pairing given by evaluation. We also write for i ∈ I uf v i := { e i , ·} ∈ M ◦ . We have two natural maps defined by {· , ·} : p ∗ : N uf → M ◦ p ∗ : N → M ◦ /N ⊥ uf N uf ∋ n ( N ◦ ∋ n ′
7→ { n, n ′ } ) N ∋ n ( N uf ∩ N ◦ ∋ n ′
7→ { n, n ′ } )For the future, let us choose a map(2.1) p ∗ : N → M ◦ such that, (a) p ∗ | N uf = p ∗ and (b) the composed map N → M ◦ /N ⊥ uf agrees with p ∗ .Different choices of p ∗ differ by a choice of map N/N uf → N ⊥ uf .Given seed data s , we can associate two tori X s = T M = Spec k [ N ] and A s = T N ◦ = Spec k [ M ◦ ]. We note that [FG09] gives an incorrect definition when N uf = N , as the formula p ∗ ( n ) = { n, ·} may not give a result in M ◦ . IRATIONAL GEOMETRY OF CLUSTER ALGEBRAS 15
We write X , . . . , X n as coordinates on X s corresponding to the basis vectors e , . . . , e n ,i.e., X i = z e i , and similarly coordinates A , . . . , A n corresponding to the basis vectors f , . . . , f n , i.e., A i = z f i . The coordinates X i , A i are called cluster variables . Thesecoordinates give identifications(2.2) X s → G nm , A s → G nm . We write these two split tori as ( G nm ) X and ( G nm ) A in the X and A cases respectively. Remark . These tori come with the following structures:(1) Let K = ker p ∗ . Then the inclusion K ⊆ N induces a map X s → T K ∗ =Spec k [ K ]. Furthermore, the torus T ( N/N uf ) ∗ = Spec k [ N/N uf ] is a subtorus of X s and hence acts on X s .(2) Let K ◦ = K ∩ N ◦ . Then the inclusion K ◦ → N ◦ induces a map of tori T K ◦ → A s . This gives an action of T K ◦ on A s . Furthermore, there is a naturalinclusion N ⊥ uf = { m ∈ M ◦ | h m, n i = 0 ∀ n ∈ N uf } ⊆ M ◦ . This induces a map A s → T N ◦ /N uf ∩ N ◦ = Spec k [ N ⊥ uf ].(3) The chosen map p ∗ : N → M ◦ defines a map p : A s → X s . Furthermore, p ∗ induces maps p ∗ : K → N ⊥ uf ⊆ M ◦ and p ∗ : N/N uf → ( K ◦ ) ∗ ,giving maps p : T N ◦ /N uf ∩ N ◦ → T K ∗ , p : T K ◦ → T ( N/N uf ) ∗ , respectively. We then obtain commutative diagrams A s p / / (cid:15) (cid:15) X s (cid:15) (cid:15) T N ◦ /N uf ∩ N ◦ p / / T K ∗ T K ◦ p / / (cid:15) (cid:15) T ( N/N uf ) ∗ (cid:15) (cid:15) A s p / / X s We next define a mutation of seed data.For r ∈ Q define [ r ] + = max(0 , r ). Given seed data s and k ∈ I uf , we have a mutation µ k ( s ) of s given by a new basis(2.3) e ′ i := e i + [ ǫ ik ] + e k i = k − e k i = k. Note that { e ′ i | i ∈ I uf } still form a basis for N uf and the d i e ′ i still form a basis for N ◦ .Dually, one checks that the basis { f i } for M ◦ changes as f ′ i := − f k + P j [ − ǫ kj ] + f j i = kf i i = k. One also checks that the matrix ǫ ij changes via the formula(2.4) ǫ ′ ij := { e ′ i , e ′ j } d j = − ǫ ij k ∈ { i, j } ǫ ij ǫ ik ǫ kj ≤ , k
6∈ { i, j } ,ǫ ij + | ǫ ik | ǫ kj ǫ ik ǫ kj > , k
6∈ { i, j } . We also define birational maps µ k : X s X µ k ( s ) µ k : A s A µ k ( s ) defined via pull-back of functions µ ∗ k z n = z n (1 + z e k ) − [ n,e k ] , n ∈ N (2.5) µ ∗ k z m = z m (1 + z v k ) −h d k e k ,m i , m ∈ M ◦ . (2.6)These maps are more often seen in the cluster literature as described via pull-backs ofcluster variables:(2.7) µ ∗ k X ′ i = X − k i = kX i (1 + X − sgn( ǫ ik ) k ) − ǫ ik i = k and(2.8) A k · µ ∗ k A ′ k = Y j : ǫ kj > A ǫ kj j + Y j : ǫ kj < A − ǫ kj j , µ ∗ k A ′ i = A i , i = k. The correspondence between these two descriptions can be seen using X i = z e i , X ′ i = z e ′ i and A i = z f i , A ′ i = z f ′ i . Remark . Note in the notation of Equation (1.2), the mutation (2.6) is µ ( − d k e k ,v k ) : T N ◦ T N ◦ . By Equation (1.4) its tropicalisation is µ tk ( n ) = n + [ h v k , n i ] − ( − d k e k ) = n + [ { n, d k e k } ] + e k and thus the seed mutation (2.3) is also given by(2.9) e ′ i = µ tk ( e i ) i = k − e k = − µ tk ( e k ) i = k. IRATIONAL GEOMETRY OF CLUSTER ALGEBRAS 17
On the other hand, the mutation (2.5) is µ ( d k v k ,e k ) : T M T M . This tropicalizes to µ tk ( m ) = m + [ h d k e k , m i ] − v k . Noting that as p ∗ is a linear function, the v i transform under the mutation in the sameway the e i do, i.e., v ′ k = − v k , v ′ i = v i + [ ǫ ik ] + v k for i = k . But µ tk ( v i ) = v i + [ ǫ ik ] − v k = v ′ i , so we do not obtain an equation analogous to (2.9). Rather, one checks that(2.10) − v ′ i = µ tk ( − v i ) i = k − µ tk ( − v k ) i = k (cid:3) One checks easily the commutativity of the diagrams(2.11) T K ◦ / / = (cid:15) (cid:15) A s p / / µ k (cid:15) (cid:15) ✤✤✤ X s / / µ k (cid:15) (cid:15) ✤✤✤ T K ∗ = (cid:15) (cid:15) T K ◦ / / A µ k ( s ) p / / X µ k ( s ) / / T K ∗ (2.12) T ( N/N uf ) ∗ / / = (cid:15) (cid:15) X s µ k (cid:15) (cid:15) ✤✤✤ T ( N/N uf ) ∗ / / X µ k ( s ) A s / / µ k (cid:15) (cid:15) ✤✤✤ T N ◦ /N uf ∩ N ◦ = (cid:15) (cid:15) A µ k ( s ) / / T N ◦ /N uf ∩ N ◦ We can now define the X and A cluster varieties associated to the seed s . We willfirst need the following general gluing construction: Proposition 2.4.
Let { X i } be a collection of integral, separated schemes of finite typeover a field k , with birational maps f ij : X i X j for all i, j , with f ii the identityand f jk ◦ f ij = f ik as rational maps. Let U ij ⊆ X i be the largest open subset such that f ij : U ij → f ij ( U ij ) is an isomorphism. Then there is a scheme X obtained by gluingthe X i along the open sets U ij via the maps f ij .Proof. First, the sets U ij exist: take U ij to consist of all points x in the domain of f ij at which f ij is a local isomorphism. By [Gr60], 6.5.4, these are precisely the points x such that f ∗ ij : O X j ,f ij ( x ) → O X i ,x is an isomorphism. By [Gr60], 8.2.8, f ij | U ij is an openimmersion. By [H77], Ex. II 2.12, it is now sufficient to check that f ij ( U ij ∩ U ik ) = U ji ∩ U jk .Clearly f ij ( U ij ∩ U ik ) ⊆ U ji . If x ∈ U ij ∩ U ik , then f jk can be defined at f ij ( x ) ∈ U ji via f ik ◦ f − ij . Then clearly f jk is a local isomorphism at f ij ( x ), so f ij ( x ) ∈ U jk . Conversely,if y ∈ U ji ∩ U jk , then y = f ij ( x ) for some x ∈ U ij . Clearly f ik = f jk ◦ f ij is a localisomorphism at x , so x ∈ U ik also and y ∈ f ij ( U ij ∩ U ik ). (cid:3) Let T be the oriented rooted tree with | I uf | outgoing edges from each vertex, labelledby the elements of I uf . Let v be the root of the tree. Attach the seed s to the vertex v . Now each simple path starting at v determines a sequence of seed mutations, justmutating at the label attached to the edge. In this way we attach a seed to eachvertex of T . We write the seed attached to a vertex w as s w . We further attach copies X s w , A s w to w .If T has a directed edge from w to w ′ labelled with k ∈ I uf , with associated seeds s w and µ k ( s w ) = s w ′ , we obtain mutations µ k : X s w X s w ′ , µ k : A s w A s w ′ . We canview these maps as arising from traversing the edge in the direction from w to w ′ ; weuse the inverse maps µ − k if we traverse the edge from w ′ to w .Now for any two vertices w, w ′ of T there is a unique simple path γ from one to theother. We obtain birational maps µ w,w ′ : A s w A s w ′ , µ w,w ′ : X s w X s w ′ , between the associated tori. These are obtained by taking the composition of mutationsor their inverses associated to each edge traversed by γ in the order traversed, usinga mutation µ k associated to the edge if the edge is traversed in the direction of itsorientation, and using µ − k if traversed in the opposite direction.These birational maps clearly satisfy µ w ′ ,w ′′ ◦ µ w,w ′ = µ w,w ′′ as birational maps, andhence by Proposition 2.4, we obtain schemes X or A by gluing these tori using thesebirational maps. Remark . Note that µ k ◦ µ k : A s A µ k ( µ k ( s )) is not the identity when expressedas a map Spec k [ M ◦ ] Spec k [ M ◦ ]; rather, it is the isomorphism given by the linearmap M ◦ → M ◦ , m m − h d k e k , m i v k . This map takes the basis { f i } for the seed µ k ( µ k ( s )) to the basis { f i } for the seed s . This is why µ k ◦ µ k is only the identity whenviewed as an automorphism of Spec k [ A ± , . . . , A ± n ]. Remark . As we shall see in Theorem 3.14, the A variety is always separated, butthe X variety usually is not. It is not clear, however, whether either of these schemesis Noetherian. This will sometimes cause problems in what follows, but these problemsare purely technical. In particular, given any finite connected regular subtree T ′ of T ,we can use the seed tori corresponding to vertices in T ′ to define open subschemes of X IRATIONAL GEOMETRY OF CLUSTER ALGEBRAS 19 and A . We shall write these subschemes as X ft and A ft respectively. We will not needto be particularly concerned about which subtree T ′ we use, only that it be sufficientlybig for the purpose at hand. However, we shall always assume T ′ contains the rootvertex v and all its adjacent vertices. Remark . The structures (1)-(4) of Remark 2.2 described on individual seed tori,being compatible with mutations as seen in Equations (2.11) and (2.12), induce corre-sponding structure on X and A . In particular, (1) there is a canonical map λ : X → T K ∗ and a canonical action of T ( N/N uf ) ∗ on X ; (2) there is a canonical action of T K ◦ on A and a canonical map A → T N ◦ /N uf ∩ N ◦ ;(3) there is a map p : A → X . This map is compatible with the maps and actions of (1) and (2) as indicated in Remark2.2, (3).
Definition 2.8.
The X -cluster algebra ( A -cluster algebra) associated to a seed s isΓ( X , O X ) (or Γ( A , O A )). Remark . The A -cluster algebra is usually called the upper cluster algebra in theliterature, see [BFZ05]. This can be viewed as the algebra of Laurent polynomialsin k [ M ◦ ] which remain Laurent polynomials under any sequence of mutations. Sucha Laurent polynomial is called a universal Laurent polynomial . The algebra whichis usually just called the cluster algebra is the sub-algebra of the field of fractions k ( A s ) = k ( A , . . . , A n ) of A s generated by all functions { µ ∗ v,w ( A ′ i ) | A ′ i is a coordinate on A s w , w a vertex of T } . We note that the cluster algebras arising via this construction are still a special caseof the general definition given in [FZ02a], and are called cluster algebras of geometrictype in the literature. These include most of the important examples.We end this section with several variants of the above constructions.
Construction 2.10.
When there are frozen variables (i.e., N uf = N ) one frequentlymight want to allow the frozen variables X i , i I uf or A i , i I uf to take the value 0.Thus one replaces X s , A s with X s := Spec k [ { X ± i | i ∈ I uf } ∪ { X i | i I uf } ] , A s := Spec k [ { A ± i | i ∈ I uf } ∪ { A i | i I uf } ] . These varieties can be defined somewhat more abstractly as toric varieties, with fansthe set of faces of the cone generated by { e ∗ i | i I uf } and { d i e i | i I uf } respectively.One sees from (2.7) and (2.8) that no X i or A i for i I uf is inverted by mutations.Thus cluster varieties X , A can be defined via gluing these modified spaces as before.In particular, we obtain a map A →
Spec k [ { A i | i I uf } ].In any event, Fock and Goncharov [FG11] define the special completion of the X variety, written as b X , by replacing each X s with the affine space Spec k [ X , . . . , X n ],and using the same definition for the birational maps between the X s as usual. Construction 2.11.
We define the notion of cluster algebra with principal coefficients .In general, given fixed data N, {· , ·} as usual along with seed data s , we construct the double of the lattice via e N = N ⊕ M ◦ , { ( n , m ) , ( n , m ) } = { n , n } + h n , m i − h n , m i . We take e N uf = N uf ⊆ e N , and e N ◦ the sublattice N ◦ ⊕ M . The lattice e N with its pairing {· , ·} and sublattices e N uf , e N ◦ can now play the role of fixed data. Given a seed s forthe original fixed data, we obtain a seed ˜ s for e N with basis { ( e i , , (0 , f α ) } . We usethe convention that indices i, j, k ∈ I are used to index the first set of basis elementsand α, β, γ ∈ I are used to index the second set of basis elements. The integer d i associated with ( e i ,
0) or d α associated to (0 , f α ) is then taken to agree with d i or d α of the original seed. Then the matrix ˜ ǫ determined by this seed is given by˜ ǫ ij = ǫ ij , ˜ ǫ iβ = δ iβ , ˜ ǫ αj = − δ αj , ˜ ǫ αβ = 0 . One notes that f M = Hom( e N , Z ) = M ⊕ N ◦ and f M ◦ = M ◦ ⊕ N . Furthermore, givena choice of p ∗ : N → M ◦ , we can take the map p ∗ : e N → f M ◦ to be given by p ∗ ( e i ,
0) = ( p ∗ ( e i ) , e i ) , p ∗ (0 , f α ) = ( − f α , , so that p ∗ is an isomorphism.With this choice of fixed and seed data, the corresponding A cluster variety will bewritten as A prin . The ring of global functions on A prin is the upper cluster algebra withprincipal coefficients at the seed s of [FZ07], Def. 3.1. A prin has an additional relationship with X . There are two natural inclusions˜ p ∗ : N → f M ◦ , π ∗ : N → f M ◦ n ( p ∗ ( n ) , n ) , n (0 , n )The first inclusion induces for any seed s an exact sequence of tori1 −→ T N ◦ −→A prin , s ˜ p −→X s −→ . IRATIONAL GEOMETRY OF CLUSTER ALGEBRAS 21
One checks that ˜ p commutes with the mutations µ k on A prin , s and X s . Thus we obtaina morphism ˜ p : A prin → X . The T N ◦ action on A prin , s gives a T N ◦ action on A prin ,making ˜ p the quotient map for this action and A prin is a T N ◦ -torsor over X . On theother hand, π ∗ induces a projection(2.13) π : A prin → T M . We note that if e ∈ T M denotes the identity element, then π − ( e ) = A . To see this,note the fibre of π : A prin , s → T M over e is canonically A s , and a mutation µ k on A prin , s specializes to the corresponding mutation on A s . The open subset on whicha mutation µ w,w ′ : A prin , s w → A prin , s w ′ is an isomorphism onto its image restrictsto the corresponding open subset of A s w ; otherwise, A prin would not be separated,contradicting Theorem 3.14. Definition 2.12.
Let t ∈ T M . We write A t for the fibre π − ( t ). We call this an A cluster variety with general coefficients . Construction 2.13.
In case there are no frozen variables, i.e., N = N uf , we have p ∗ = p ∗ and K = ker p ∗ . We then have a commutative diagram N ˜ p ∗ / / f M ◦ K λ ∗ O O i ∗ / / N π ∗ O O where both i ∗ and λ ∗ are the inclusion. This induces a commutative diagram(2.14) X λ (cid:15) (cid:15) A prin˜ p o o π (cid:15) (cid:15) T K ∗ T Mi o o Note that for t ∈ T M , ˜ p restricts to a map p t : A t → λ − ( i ( t )) = X i ( t ) . The geometry of cluster varieties
We now give our description of cluster varieties as blowups of toric varieties andmutations as elementary transformations of P -bundles. This gives rise to most of theresults in this paper, including a simple explanation for the Laurent phenomenon andcounterexamples to some basic conjectures about cluster algebras. Elementary transformations.
The basic point is that the gluing of adjacentseed tori can be easily described in terms of blow-ups of toric varieties, and thatmutations have a simple interpretation as a well-known operation in algebraic geometryknown as an elementary transformation. To describe this in general, we fix a lattice N with no additional data, and a primitive vector v ∈ N . The projection N → N/ Z v gives a G m -bundle π : T N → T N/ Z v . A non-zero regular function f on T N/ Z v can be viewed as a map f : T N/ Z v \ V ( f ) → T Z v ⊆ T N = N ⊗ Z G m t v ⊗ f ( t )to obtain a birational map µ f : T N T N t f ( π ( t )) − · t. Note that on the level of pull-back of functions, this is defined, for m ∈ M =Hom( N, Z ), by z m z m ( f ◦ π ) −h m,v i . Indeed, this is easily checked by choosing a basis f , . . . , f n of M with h f , v i = 1, h f i , v i = 0 for i >
1. This gives coordinates x i = z f i , 1 ≤ i ≤ n , on T N so that theprojection π is given by ( x , . . . , x n ) ( x , . . . , x n ), and the map µ f is given by( x , . . . , x n ) ( f ( x , . . . , x n ) − x , x , . . . , x n ) . Now consider the fan Σ v, + = { R ≥ v, } in N . This defines a toric variety TV(Σ v, + )isomorphic to A × T N/ Z v , and contains a toric divisor D + . It has a canonical projection π : TV(Σ v, + ) → T N/ Z v , which induces an isomorphism D + ∼ = T N/ Z v . Set Z + = π − ( V ( f )) ∩ D + . This hypersurface may be non-reduced. Define g TV(Σ v, + ) → TV(Σ v, + ) the blowup of Z + ,˜ D + the proper transform of D + , U v, + = g TV(Σ v, + ) \ ˜ D + . Note that Γ( U v, + , O U v, + ) = Γ(TV(Σ v, + ) , O TV(Σ v, + ) )[ f /x ].We can also use µ f to define a variety X f obtained by gluing together two copies of T N using µ f along the open subsets T N \ V ( f ◦ π ) ⊆ T N . IRATIONAL GEOMETRY OF CLUSTER ALGEBRAS 23
We then obtain the following basic model for describing gluings of tori as blowupsof toric varieties:
Lemma 3.1.
There is an open immersion X f ֒ → U v, + such that U v, + \ X f is codi-mension two in U v, + . Furthermore, the projection π : U v, + → T N/ Z v is a G m -bundleover T N/ Z v \ V ( f ) , while the fibres of π over V ( f ) are each a union of two copies of A meeting at a point. The locus where π is not smooth is precisely U v, + \ X f .Proof. Using coordinates ( x , . . . , x n ) for TV(Σ v, + ) as before, with D + given by x = 0,note the ideal of Z + is ( x , f ). Thus the blow-up of Z + is given by the equation ux = vf in P × TV(Σ v, + ). We define two embeddings of T N , ι : ( x , . . . , x n ) (cid:0) ( f, x ) , ( x , . . . , x n ) (cid:1) ι : ( x , . . . , x n ) (cid:0) (1 , x ) , ( x f, x , . . . , x n ) (cid:1) Noting that µ f = ι − ◦ ι , it is clear that these maps give an embedding of X f . Thedivisor ˜ D + is given by the equation v = x = 0, so the only points of U v, + missedby the open immersion X f ֒ → U v, + are the points where u = x = 0, i.e., points ofthe form (cid:0) (0 , , (0 , x , . . . , x n ) (cid:1) with f ( x , . . . , x n ) = 0. The remaining statements areclear. (cid:3) Next we examine how this gives a basic model for a mutation. Consider the fanΣ v := { R ≥ v, R ≤ v, } . This defines a toric variety we write as P , and it comes withdivisors D + , D − corresponding to the two rays and a map π : P → T N/ Z v , identifying D + and D − with T N/ Z v . Let Z + = D + ∩ V ( f ◦ π ) ,Z − = D − ∩ V ( f ◦ π ) . We have two blow-ups b ± : ˜ P ± → P being the blow-ups of Z + and Z − . Lemma 3.2.
The rational map µ f : T N T N extends to a regular isomorphism µ f : ˜ P + → ˜ P − .Proof. Working in coordinates ( x , . . . , x n ) as before, we can describe P as P × T N/ Z v with coordinates ( x : y ) on P and coordinates x , . . . , x n on T N/ Z v . Here D + is givenby x = 0 and D − by y = 0. Then µ f is given as (cid:0) ( x : y ) , ( x , . . . , x n ) (cid:1) (cid:0) ( x , f ( x , . . . , x n ) y ) , ( x , . . . , x n ) (cid:1) . This fails to be defined precisely where x = f = 0, i.e., along Z + , and blowing up Z + clearly resolves this indeterminacy. Thus µ f : P P lifts to a morphism µ f : ˜ P + → P .On the other hand, since the ideal sheaf of Z − in P (locally generated by y and f )pulls back via µ f to an invertible sheaf on ˜ P + , this morphism factors as a morphism µ f : ˜ P + → ˜ P − by the universal property of blowing up.To see that µ f as viewed in this way is a regular isomorphism, note the inverserational map µ − f can be written as t f ( π ( t )) · t , and thus as a map P P iswritten as (cid:0) ( x : y ) , ( x , . . . , x n ) (cid:1) (cid:0) ( f ( x , . . . , x n ) x , y ) , ( x , . . . , x n ) (cid:1) . This lifts to a well-defined morphism µ − f : ˜ P − → ˜ P + as before. Thus µ f is an isomor-phism between ˜ P + and ˜ P − . (cid:3) Remark . This lemma should be interpreted as saying that µ f : P P can beviewed as the birational map described as the blow-up of Z + followed by the contractionof the proper transform of π − ( V ( f )) ⊆ P in ˜ P + to Z − ⊆ P . This is a birationaloperation called an elementary transformation in algebraic geometry.Furthermore, let ˜ D ± be the proper transform of D ± in either ˜ P + or ˜ P − . Thencombining Lemmas 3.1 and 3.2, this tells us that there are open immersions of X f in˜ P ± \ ( ˜ D + ∪ ˜ D − ), missing a codimension two subset. The roles the two coordinate toriof X f play are reversed under these two immersions; one of the tori of X f is the inverseimage of the big torus orbit under the blow-up ˜ P − → P , and the other torus in X f isthe inverse image of the big torus orbit under the blow-up ˜ P + → P .We need an extended version of the above setup: Construction 3.4.
Suppose we have the data of a fan Σ = { R ≥ v i | ≤ i ≤ ℓ } ∪ { } where v , . . . , v ℓ ∈ N are primitive, w , . . . , w ℓ ∈ M with h v i , w i i = 0. We allowsome of the v i ’s to coincide. Let a , . . . , a ℓ be positive integers, c , . . . , c ℓ ∈ k × , and µ i : T N T N be defined as before by the data f i = (1 + c i z w i ) a i and v i , where c i ∈ k × . Let TV(Σ) be the toric variety defined by Σ, and let D i be the toric divisorcorresponding to R ≥ v i .In what follows, we use the notation ¯ V ( f i ) for the closure of V ( f i ) ⊆ T N in TV(Σ).Define Z j = D j ∩ ¯ V ( f j ) ,π : g TV(Σ) → TV(Σ) the blow-up along S ℓi =1 Z i ,˜ D j the proper transform of D j . IRATIONAL GEOMETRY OF CLUSTER ALGEBRAS 25
On the other hand, define a scheme X as follows. Let T , . . . , T ℓ be ℓ + 1 copies ofthe torus T N . The map µ i is viewed as an isomorphism between open sets ϕ i := µ i : U i → U i of T and T i respectively, with U i taken as the largest possible such open subset.Indeed, we can take U i = T \ V ( f i ) and U i = T i \ V ( f i ). In addition, for 1 ≤ i, j ≤ ℓ ,define ϕ ij := µ j ◦ µ − i , and define U ij to be the largest subset of T i on which ϕ ij definesan open immersion. The identifications ϕ ij then provide gluing data to obtain a scheme X , in general not separated, by Proposition 2.4. Lemma 3.5.
There is a natural morphism ψ : X → ˜ U Σ := g TV(Σ) \ [ i ˜ D i , which in special cases satisfies the following properties: (1) If dim Z i ∩ Z j < dim Z i for all i = j , then ψ is an isomorphism off a set ofcodimension ≥ . (2) If Z i ∩ Z j = ∅ for all i = j , then ψ is an open immersion. In particular, in thiscase, X is separated.Proof. This is just a slightly more involved version of the argument of Lemma 3.1. Wefirst describe maps of the tori T i , 0 ≤ i ≤ ℓ into ˜ U Σ . We have a canonical identificationof T with the big torus orbit T N of TV(Σ), isomorphic to π − ( T N ) ⊆ ˜ U Σ . On the otherhand, for a given i , let J be the set of indices such that v j = v i if and only if j ∈ J .Note TV(Σ v i , + ) is an open subset of TV(Σ). Using coordinates x , . . . , x n on TV(Σ v i , + )as in the proof of Lemma 3.1, we obtain an open subset of g TV(Σ) described as a subsetof TV(Σ v i , + ) × P given by the equation ux = v Q j ∈ J f j . With this description, wedefine ι i : T i → ˜ U Σ by ι i : ( x , . . . , x n ) (cid:0) ( Y j ∈ J \{ i } f j , x ) , ( f i x , x , . . . , x n ) (cid:1) . Note that ι i contracts the locus f i = Q j ∈ J \{ i } f j = 0 in T i so this is not an embeddingunless the Z j are disjoint. In this coordinate chart, ι is given by ι : ( x , . . . , x n ) (cid:0) ( Y j ∈ J f j , x ) , ( x , . . . , x n ) (cid:1) . From this one sees that ι i ◦ µ i = ι on U i . In particular the maps ι i , 0 ≤ i ≤ n arecompatible with the gluings ϕ ij , and hence we obtain the desired map ψ .In the case (1), each ι i , i ≥
1, is an open immersion off of a codimension ≥ ≥ ι i is an open immersion. Thus ψ is a local isomorphism, and it is enough to show ψ is injective to see that it is an open immersion. Certainly ψ is injective on each T i . If x ∈ T i , y ∈ T j have ψ ( x ) = ψ ( y ), then ι i ( x ) = ι j ( y ). Noting that ϕ ij = ι − j ◦ ι i as rational maps, we see that ϕ ij is a local isomorphism at x and ϕ ij ( x ) = y . Thus x ∈ U ij and x and y are identified by the gluing maps so they give the same point in X . (cid:3) Next we understand the general setup for a mutation.Given elements v ∈ N , w ∈ M with h w, v i = 0, define the piecewise linear transfor-mation T v,w : N R → N R , n n + [ h n, w i ] − v Note this coincides with the tropicalization of µ ( v,w ) in (1.2) as given in (1.4).Now in the situation of this construction, let us impose one additional restriction onthe starting data v i , w i , namely,(3.1) h w i , v j i = 0 ⇔ h w j , v i i = 0 . Pick some index k and let Σ + = Σ ∪ { R ≤ v k } , and define Σ − by applying T − v k ,a k w k to each ray of Σ + . Let D k, + ⊆ TV(Σ + ) be thedivisor corresponding to R ≥ v k in Σ + and D k, − ⊆ TV(Σ − ) be the divisor correspondingto R ≤ v k in Σ − . For j = k , write D j, ± for the divisor corresponding to R ≥ v j in Σ + or R ≥ T − v k ,a k w k ( v j ) in Σ − . Finally, we can set Z j, + = ¯ V ( f j ) ∩ D j, + Z j, − = ¯ V ( f j ) ∩ D j, − if h w k , v j i ≥ V (cid:0) (1 + c j c a k h w j ,v k i k z w j + a k h w j ,v k i w k ) a j (cid:1) ∩ D j, − if h w k , v j i ≤ . Let g TV(Σ ± ) be the blowups of TV(Σ ± ) at this collection of subschemes. Lemma 3.6. µ k = µ f k : T N T N defines a birational map µ k : g TV(Σ + ) g TV(Σ − ) . If dim ¯ V ( f k ) ∩ Z j, + < dim Z j, + whenever h w k , v j i = 0 , then this extension is an iso-morphism off of sets of codimension ≥ .Proof. We first analyze the map µ k before blowing up the hypersurfaces Z j, + , j = k .So abusing notation, assume g TV(Σ ± ) is just obtained by blowing up Z k, ± . Off of aclosed subset of codimension two, we can cover g TV(Σ + ) with open sets, one isomorphicto ˜ P + with v = v k , and the remaining ones of the form U ρ \ ¯ V ( f k ). Here ρ ranges over IRATIONAL GEOMETRY OF CLUSTER ALGEBRAS 27 dimension one cones of Σ + not equal to R ≥ v k or R ≤ v k , and U ρ denotes the standardaffine toric open subset of TV(Σ + ) corresponding to ρ . Denoting D ρ ⊆ U ρ the toricdivisor, note that D ρ ∩ ¯ V ( f k ) = ∅ if w k is non-zero on ρ , as then either z w k or z − w k vanishes on D ρ and ¯ V (1 + z w k ) = ¯ V (1 + z − w k ). Thus we only fail to cover codimensiontwo subsets of the form D ρ ∩ ¯ V ( f k ) such that w k is zero on ρ . So for the purposes ofdescribing the extension of µ k up to codimension two, it will be sufficient to restrict tothe open subset U of g TV(Σ + ) covered by these open sets.By Lemma 3.2, µ k gives a well-defined morphism on the open subset isomorphicto ˜ P + , so we need to check µ k defines a morphism on each of the remaining sets. If h w k , ρ i ≥
0, then for any m ∈ ρ ∨ ∩ M = ( T − v k ,a k w k ( ρ )) ∨ ∩ M , µ ∗ k acts by z m z m f −h m,v k i k , taking a regular function to a regular function on U ρ \ ¯ V ( f k ). If h w k , ρ i <
0, then if m ∈ ( T − v k ,a k w k ( ρ )) ∨ ∩ M we see µ k acts by z m z m (1 + c k z w k ) − a k h m,v k i = z m − a k h m,v k i w k ( c k + z − w k ) − a k h m,v k i . But m − a k h m, v k i w k ∈ ρ ∨ by definition of T − v k ,a k w k , so this is again a regular functionon U ρ \ ¯ V ( f k ). This shows µ k is a morphism on U ; to show it is an isomorphism ontoits image, we repeat the same process for µ − k .To prove the result after blowing up the hypersurfaces Z j, ± , first note that if h w k , v j i 6 =0, then Z j, + ⊆ U , and we need to show that µ k ( Z j, + ) = Z j, − . This can be checked incases. If h w k , v j i ≥
0, then Z j, − is defined by the equation f j on D j . Now if h w k , v j i > f k | D j = 1, so that µ ∗ k ( f j ) | D j = f j | D j . If h w k , v j i = 0, then h w j , v k i = 0 byAssumption (3.1), so that µ ∗ k z w j = z w j , so again µ ∗ k ( f j ) = f j . If h w k , v j i <
0, thennoting the definition of Z j, − in this case, µ ∗ k ((1 + c j c a k h w j ,v k i k z w j + a k h w j ,v k i w k ) a j )= (1 + c j c a k h w j ,v k i k z w j + a k h w j ,v k i w k (1 + c k z w k ) − a k h w j ,v k i ) a j = (1 + c j c a k h w j ,v k i k z w j ( c k + z − w k ) − a k h w j ,v k i ) a j . However, z − w k vanishes identically on D j in this case, so restricting to D j this coincideswith f j . This shows µ k extends to a regular map after blowing up U along the Z j, ± forthose j with h w k , v j i 6 = 0.Finally, if h w k , v j i = 0, then we do not necessarily have Z j, + ⊆ U , and if ¯ V ( f k )contains an irreducible component of Z j, + , the map µ k need not extend as an isomor-phism across the exceptional divisor of the blowup of Z j, + . Hence we need to use thestated hypothesis, which implies that Z j, + \ U is codimension ≥
3. Since µ ∗ k ( f j ) = f j when h w k , v j i = 0, it then follows that µ k extends to an isomorphism off of a set ofcodimension ≥ g TV(Σ + ). (cid:3) The X - and A prin -cluster varieties up to codimension two. Since the ringof functions on a non-singular variety is determined off a set of codimension two, wecan study the X - and A prin -cluster algebras by describing the corresponding varietiesup to codimension two.Suppose given fixed data as in §
2. Let s be a seed. Consider the fansΣ s , A := { } ∪ { R ≥ d i e i | i ∈ I uf } Σ s , X := { } ∪ {− R ≥ d i v i | i ∈ I uf } in N ◦ and M respectively. These define toric varieties TV s , A and TV s , X respectively.We remark that the minus signs in the definition of Σ s , X are forced on us by (2.10).Each one-dimensional ray in one of these fans corresponds to a toric divisor, whichwe write as D i for i ∈ I uf (not distinguishing the X and A cases). For i ∈ I uf , we candefine closed subschemes Z A ,i := D i ∩ ¯ V (1 + z v i ) ⊆ TV s , A ,Z X ,i := D i ∩ ¯ V ((1 + z e i ) ind d i v i ) ⊆ TV s , X , (3.2)where ind d i v i denotes the greatest degree of divisibility of d i v i in M . Let ( g TV s , A , D )and ( g TV s , X , D ) be the pairs consisting of the blow-ups of TV s , A and TV s , X along theclosed subschemes Z A ,i and Z X ,i respectively, with D the proper transform of the toricboundaries.We note that in the A case the divisors D i are distinct and hence the centers of theblow-ups are disjoint. In the X case, however, we might have v i and v i ′ being positivelyproportional to each other, so that D i = D i ′ . Then the two centers Z X ,i , Z X ,i ′ mayintersect. However, it is easy to see this intersection occurs in higher codimension, i.e.,dim Z X ,i ∩ Z X ,i ′ < dim Z X ,i . Thus in the X case we are in the situation of Lemma 3.5,(1) and in the A case we are in the situation of Lemma 3.5, (2).Finally we define U s , A := g TV s , A \ D, U s , X := g TV s , X \ D. Clearly these varieties contain the seed tori A s and X s , and hence given vertices w, w ′ ∈ T , we obtain a birational map µ w,w ′ of seed tori inducing birational maps µ w,w ′ : U s w , A U s w ′ , A , µ w,w ′ : U s w , X U s w ′ , X . IRATIONAL GEOMETRY OF CLUSTER ALGEBRAS 29
Since A prin is defined to be a special case of the construction of the A cluster variety,we also obtain in the same way birational maps µ w,w ′ : U s w , A prin U s w ′ , A prin . In this case the projection e N ◦ → M projects all rays of Σ s , A prin to 0, so we obtain amorphism TV s , A prin → T M . The fibres of this map are (non-canonically) isomorphicto TV s , A . After blowing up the centers Z A prin ,i , we get morphisms π : U s , A prin → T M which commute with the mutations µ w,w ′ . Write a fibre of π over t ∈ T M as U s , A t . Wethen obtain birational maps on fibres of π over t : µ w,w ′ : U s w , A t U s w ′ , A t . We recall from [BFZ05]:
Definition 3.7.
A seed s is coprime if, writing (2.8) as A k · µ ∗ k A ′ k = P k , the P k , k ∈ I uf , are pairwise coprime. We say a seed s is totally coprime if all seeds obtainedby repeated mutations of s are coprime.We then have Lemma 3.8.
Let U ′ s , A ⊂ A (resp. U ′ s , X ⊂ X ) be the union of the tori A s (resp. X s )and A µ i ( s ) (resp. X µ i ( s ) ), i ∈ I uf . (1) For k ∈ I uf , with w ′ = µ k ( w ) , the maps µ w,w ′ : U s w , X U s w ′ , X , µ w,w ′ : U s w , A prin U s w ′ , A prin are isomorphisms outside codimension two. (2) µ w,w ′ : U s w , A U s w ′ , A is an isomorphism outside codimension two if the seed s w is coprime. (3) µ w,w ′ : U s w , A t U s w ′ , A t is an isomorphism outside of codimension two for t ∈ T M general (i.e., t contained in some non-empty Zariski open subset). (4) U ′ s , A U s , A is an open immersion with image an open subset whose comple-ment has codimension at least two. (5) U ′ s , X U s , X is an isomorphism outside of codimension two.Proof. These are all special cases of Construction 3.4. For (1) and (2), in the X (resp. A prin , A ) case, we take the vectors v i to be − d i v i / ind( d i v i ) ∈ M (resp. ( d i e i , ∈ e N ◦ , d i e i ∈ N ◦ ) for i ∈ I uf , the vectors w i to be e i ∈ N (resp. ( v i , e i ) ∈ f M ◦ , v i ∈ M ◦ ).In all these cases, the constants c i are taken to be 1. The integers a i are taken to be a i = ind( d i v i ) (resp. a i = 1). In all three cases, the cluster mutation µ k coincides withthe µ k as defined in Construction 3.4. In the notation of Lemma 3.6, taking Σ + = Σ s w , X (resp. Σ s w , A prin , Σ s w , A ), we observe that T d k v k ,e k (resp. T ( − d k e k , , ( v k ,e k ) , T − d k e k ,v k ) applied to the rays of Σ + gives Σ − := Σ s w ′ , X , (resp. Σ s w ′ , A prin , Σ s w ′ , A ) as follows immediatelyfrom (2.3) and Remark 2.3.We now only need to check the hypothesis of Lemma 3.6 to see that µ w,w ′ is anisomorphism off codimension two subsets. In the X case, f k = 1 + z e k , and from thisthe condition is easily checked. In the A case, f k = 1 + z v k , which coincides with P k upto a monomial factor. The hypothesis then follows from the coprime condition, and theprincipal coefficient case is automatically coprime as the ( v k , e k ), k ∈ I uf are linearlyindependent.The A t case (3) is similar to the A case, except that now f k = 1 + z e k ( t ) · z v k , so wetake c k = z e k ( t ). If t is chosen generally, then the hypothesis of Lemma 3.6 continuesto hold.(5) follows from part (1) of Lemma 3.5. (4) follows from part (2) of Lemma 3.5. (cid:3) The main result in this section is then:
Theorem 3.9.
Let w, w ′ be vertices in T . (1) The induced birational maps U s w , X µ w,w ′ U s w ′ , X X ft U s w , A prin µ w,w ′ U s w ′ , A prin A ftprin are isomorphisms outside of codimension two. (See Remark 2.6 for X ft , A ft .We use a finite subtree of T containing both w and w ′ .) (2) If the initial seed is totally coprime, then U s w , A µ w,w ′ U s w ′ , A A ft is an isomorphism outside a codimension two set. (3) If t ∈ T M is very general (outside a countable union of proper closed subsets),then U s w , A t µ w,w ′ U s w ′ , A t A ft t is an isomorphism outside a codimension two set.In particular, as all schemes involved are S , these maps induce isomorphisms on ringsof regular functions.Proof. That the maps µ w,w ′ are isomorphisms outside of codimension two follows fromLemma 3.8. For the remaining statements in (1-3) consider just the X case, as theother cases are identical. By Lemma 3.5, each of the U s , X is isomorphic, outside ofcodimension two, to the gluing of the seed torus X s to its adjacent seed tori X µ k ( s ) , k ∈ I uf . This gives a birational map U s , X X ft ⊆ X . (Here we use any choice ofregular subtree of T containing the vertex corresponding to s and its adjacent vertices. IRATIONAL GEOMETRY OF CLUSTER ALGEBRAS 31
The subtree is taken to be finite but as large as we would like.) Since X ft is covered,up to codimension two subsets, by some finite collection { U s w , X } we see each U s w , X isisomorphic to X ft off a codimension two subset. We need to use X ft rather than X , forif X is not Noetherian, the subset of X we fail to cover need not be closed. (cid:3) Remark . More generally than the principal coefficient case, the totally coprime hy-pothesis also holds if the matrix ( ǫ ij ) i ∈ I uf , ≤ j ≤ n has full rank. See [BFZ05], Proposition1.8. Of course, this holds in particular for the principal coefficient case.We immediately obtain from this a geometric explanation for the well-known Laurentphenomenon: Corollary 3.11 (The Laurent phenomenon) . For a seed s , let q ∈ M ◦ (resp. q ∈ N )have non-negative pairing with each e i (resp. each − v i ) for i ∈ I uf . Equivalently, z q is a monomial which is a regular function on the toric variety TV s , A (resp. TV s , X ).Then z q is a Laurent polynomial on every seed torus, i.e., z q ∈ H ( A , O A ) (resp. z q ∈ H ( X , O X ) ).Proof. By assumption z q is a regular function on TV s , A (or TV s , X ), and hence pullsback and restricts to a regular function on U s , A (resp. U s , X ). In the X case, the resultthen follows from Theorem 3.9, since then z q also defines a regular function on X ft forany choice of subtree of T , and hence also defines a regular function on X .The A case then follows from the A prin case, since the mutation formula (2.6) for A is obtained from that for A prin by setting z (0 ,e i ) = 1 for ( e , . . . , e n ) the initial seed. (cid:3) Remark . Note that in the A case, with no frozen variables, (i.e., I uf = I ) thecondition on q is exactly that q is in the non-negative span of the e ∗ i , i.e., that z q is a monomial, with non-negative exponents, in the cluster variables of the seed. Inparticular, this applies to any cluster variable, in which case the statement gives theusual Laurent phenomenon. From this point of view the difference between A and X is that the fan Σ s , A always looks the same (it is the union of coordinate rays), andin particular TV s , A has lots of global functions (this is a toric open subset of A n ),while Σ s , X can be any arbitrary collection of rays, and TV s , X has non-constant globalfunctions if and only if all these rays lie in a common half space. Remark . By [BFZ05], Def. 1.1, the algebra H ( U ′ s , A , O U ′ s , A ) = H ( U s , A , O U s , A ) (seepart (4) of Lemma 3.8) is the upper bound . In an earlier version of this paper we claimedTheorem 3.9 for A (without any coprimality assumption), which would in particularimply the upper bound is equal to the upper cluster algebra. But Greg Muller set us straight, by giving us an example where the upper cluster algebra is strictly smallerthan the upper bound.We learned the following theorem, and its proof, from M. Shapiro: Theorem 3.14.
The canonical map ι : A →
Spec A up is an open immersion, where A up = Γ( A , O A ) is the upper cluster algebra. In particular, A is separated.Proof. A is covered by open sets of the form A s , for various seeds s . First note that theinduced map ι s : A s → Spec A up is an open immersion. Indeed, this map is inducedby the inclusion ι ∗ s : A up ⊆ k [ A ± , . . . , A ± n ] =: B , where A , . . . , A n are the clustercoordinates on A s . One checks this is a local isomorphism: given ( a , . . . , a n ) ∈ A s , a , . . . , a n = 0, the corresponding maximal ideal is m = h A − a , . . . , A n − a n i ⊆ B .By the Laurent phenomenon, A , . . . , A n ∈ A up , and thus A , . . . , A n are invertible inthe localization A up( ι ∗ s ) − ( m ) . Thus A up( ι ∗ s ) − ( m ) ∼ = B m , and ι s is a local isomorphism. Thusby [Gr60], I, 8.2.8, ι s is an open immersion.To show ι itself is now an open immersion, it is sufficient to show it is one-to-onesince it is a local isomorphism. Let x ∈ A s , y ∈ A s ′ be such that ι ( x ) = ι ( y ). Let A , . . . , A n be the cluster coordinates on A s . Again by the Laurent phenomenon, thereis an inclusion k [ A , . . . , A n ] ⊆ A up , hence a map ψ : Spec A up → A n . The composition ψ ◦ ι s is the obvious inclusion and ( ψ ◦ ι s ) − ◦ ( ψ ◦ ι s ′ ) agrees, as a rational map,with µ w ′ ,w , where w ′ , w are the vertices of T corresponding to the seeds s ′ , s . Thusthe map µ w ′ ,w is defined at y , since ψ ◦ ι s ′ is defined at y and ( ψ ◦ ι s ) − is defined at ψ ( ι s ′ ( y )) = ψ ( ι s ( x )). Furthermore, µ w ′ ,w is then a local isomorphism at y as it agreeswith ι − s ◦ ι s ′ at y , and ι s and ι s ′ are local isomorphisms at x and y respectively. So thegluing map defining A identifies x and y , and ι is injective. (cid:3) The A t and A prin cluster varieties as torsors Fix in this section fixed data and a seed s as usual. We shall assume that there areno frozen variables, i.e., I uf = I , N uf = N , and furthermore that the matrix ǫ has nozero row (or equivalently no zero column). Note that if ǫ does have a zero row the sameis true for all mutations, so this condition is mutation independent. We then obtainthe X , A , A prin and A t varieties.Denote by X the open subset of X obtained by gluing together the seed tori X s and X µ k ( s ) , 1 ≤ k ≤ n . This still comes with a map λ : X → T K ∗ as in Construction 2.13,and we write X φ for the fibre over φ ∈ T K ∗ .We first compute the Picard group of X and X φ : Theorem 4.1.
For φ ∈ T K ∗ , Pic( X ) ∼ = Pic( X φ ) ∼ = coker( p ∗ : N → M ◦ ) . IRATIONAL GEOMETRY OF CLUSTER ALGEBRAS 33
Proof.
We first need to describe precisely how X and X φ are glued together out of tori.Let U = X s , U i = X µ i ( s ) , 1 ≤ i ≤ n . We have birational gluing maps ϕ ij : U i U j given by ϕ j = µ j , ϕ ij = µ j ◦ µ − i . These glue over sets U ij as in Proposition 2.4. Notethat U j = X s \ V (1 + z e j ) , by (2.5) and the fact that no row or column of ǫ is zero. The same description appliesto U j . On the other hand, noting that( µ j ◦ µ − i ) ∗ ( z n ) = z n (1 + z e j (1 + z e i ) [ e j ,e i ] ) [ − n,e j ] (1 + z e i ) [ n,e i ] , one sees that if we set h ij = z e j (1 + z e i ) [ e j ,e i ] [ e j , e i ] ≥ z e i ) − [ e j ,e i ] + z e j [ e j , e i ] ≤ U ij = X µ i ( s ) \ ( V (1 + z e i ) ∪ V ( h ij )) . Now the U ij also map to T K ∗ , with fibres U ij,φ over φ , so that X φ is obtained by gluingthe sets U i,φ (the fibres of U i → T K ∗ over φ ) via the restriction of the ϕ ij to U ij,φ .Choose a splitting N = K ⊕ N ′ . A regular function on a fibre of X s → T K ∗ is a linearcombination of restrictions of monomials z n ′ , n ′ ∈ N ′ , to the fibre.In particular, we haveΓ( U i , O × U i ) = { cz n (1 + z e i ) − a | c ∈ k × , n ∈ N, a ∈ Z } , Γ( U ij , O × U ij ) = { cz n (1 + z e i ) − a h − bij | c ∈ k × , n ∈ N, a, b ∈ Z } , Γ( U i,φ , O × U i,φ ) = { cz n (1 + z e i ) − a | c ∈ k × , n ∈ N ′ , a ∈ Z } , Γ( U ij,φ , O × U ij,φ ) = { cz n (1 + z e i ) − a h − bij | c ∈ k × , n ∈ N ′ , a, b ∈ Z } , (4.1)noting that as e i K for any i by assumption on ǫ , 1 + z e i has some zeroes on U ij,φ .We will now compute Pic( X φ ), the argument for Pic( X ) being identical except that N ′ is replaced by N below. We compute Pic( X φ ) = H ( X φ , O × X φ ) using the ˇCech cover { U i,φ | ≤ i ≤ n } with U i,φ ∩ U j,φ identified with U ij,φ for i < j . Indeed, this covercalculates Pic( X φ ) because Pic( U i,φ ) = Pic( U ij,φ ) = 0 for all i and j . Thus a ˇCech1-cochain consists of elements g ij ∈ Γ( U ij,φ , O × U ij,φ ) for each i < j . In particular, if( g ij ) is a 1-cocycle, necessarily g ij = ( µ − i ) ∗ ( g − i g j ), and the g i ’s can then be chosenindependently. From (4.1), the group of 1-cocycles is then identified with Z := n M i =1 ( k × ⊕ N ′ ⊕ Z ) . On the other hand, Γ( U i,φ , O × U i,φ ) = k × ⊕ N ′ . A 0-cochain g = ( g i ) ≤ i ≤ n , g i ∈ k × ⊕ N ′ then satisfies ( ∂g ) i = g − µ ∗ i ( g i ) , where ∂ denotes ˇCech coboundary. Given Equation (2.5), we can then view ∂ as a map C := n M i =0 ( k × ⊕ N ′ ) → Z , ( c i , n i ) ≤ i ≤ n ( c i c − , n i − n , [ n i , e i ]) ≤ i ≤ n . Thus modulo ∂ ( C ), every element of Z is equivalent to some (1 , , a i ) ≤ i ≤ n . Thus Z /∂ ( C ) is isomorphic to Z n / ( ∂ ( C ) ∩ Z n ), where Z n ⊂ Z via the last component foreach i . But ∂ ( C ) ∩ Z n consists of the coboundaries of elements (1 , n ) ≤ i ≤ n , and thecoboundary of such an element is (1 , , [ n , e i ]) ≤ i ≤ n . If we identify Z n with M ◦ usingthe basis f i , then with n = e j , we obtain the element of M ◦ given by n X i =1 [ e j , e i ] f i = n X i =1 { e j , e i } e ∗ i = p ∗ ( e j ) . This proves the result. (cid:3)
Remark . We note the calculations in the above proof demonstrate easily how X e (and hence X and X ) can fail to be separated. Indeed, suppose that e i , e j agree afterprojection to N/K . In particular, [ n, e i ] = [ n, e j ] for any n ∈ N and [ e j , e i ] = 0. Thus µ j ◦ µ − i is the identity on k [ N/K ] ∼ = X µ i ( s ) ,e ∼ = X µ j ( s ) ,e , but U ij,e is a proper subset of X µ i ( s ) ,e . So the two tori are glued via the identity across a proper open subset of eachtorus, and we obtain a non-separated scheme. Construction 4.3.
We now recall the construction of the universal torsor over ascheme X with finitely generated Picard group. Ideally, we would like to define theuniversal torsor as the scheme affine over XU T X := Spec M L∈ Pic X L . However, the quasi-coherent sheaf of O X -modules appearing here doesn’t have a naturalalgebra structure, since elements of Pic X represent isomorphism classes of line bundles.If Pic X is in fact a free abelian group, we can proceed as in [HK00] and choose a set ofline bundles L , . . . , L n whose isomorphism classes form a basis for the Picard group,and write, for ν ∈ Z n , L ν := N ni =1 L ν i i . Then L ν ∈ Z n L ν does have a natural algebrastructure.If Pic X has torsion, then we need to make use of the definition given in [BH03], §
3. We can choose a sufficiently fine open cover U of X such that every isomorphismclass of line bundle on X is represented by a ˇCech 1-cocycle for O × X with respect to thiscover. Denoting the set of ˇCech 1-cocycles as Z ( U , O × X ), we choose a finitely generated IRATIONAL GEOMETRY OF CLUSTER ALGEBRAS 35 subgroup Λ ⊆ Z ( U , O × X ). If for λ ∈ Λ we denote by L λ the corresponding line bundle,we can choose Λ so that the natural map Λ → Pic X , λ [ L λ ], is surjective. Thenmultiplication gives a sheaf of O X -algebras structure to R := L λ ∈ Λ L λ .To obtain the universal torsor, we need to define an ideal I ⊆ R generated byrelations coming from isomorphisms L λ ∼ = L λ ′ . However, these isomorphisms must bechosen carefully, so [BH03] defines the notion of a shifting family . Let Λ = ker(Λ → Pic X ). A shifting family is a set of O X -module isomorphisms { ρ λ : R → R | λ ∈ Λ } such that(1) ρ λ maps L λ ′ to L λ ′ + λ , for every λ ∈ Λ , λ ′ ∈ Λ;(2) For every λ , λ ∈ Λ , ρ λ + λ = ρ λ ◦ ρ λ ;(3) If f , g are sections of L λ , L λ respectively, for λ , λ ∈ Λ, and λ ∈ Λ , we have ρ λ ( f g ) = f ρ λ ( g ).A shifting family defines a sheaf of ideals I ⊆ R such that I ( U ) is generated byelements of the form f − ρ λ ( f ) for f ∈ R ( U ), λ ∈ Λ .Given a shifting family, the universal torsor is then defined to be U T X := Spec R / I . A priori
U T X depends on the choice of shifting family (although [BH03] proves any twochoices are isomorphic provided that k is algebraically closed and Γ( X, O × X ) = k × , seeLemma 3.7 of [BH03]). If Pic X is torsion free, then this ambiguity disappears. Thus,in general, we will talk about a choice of universal torsor .Given a seed s , t ∈ T M , let A t (resp. A prin ) be the variety defined by gluing togetherthe seed tori A prin , s ,t , A prin ,µ i ( s ) ,t (resp. A prin , s , A prin ,µ i ( s ) ), 1 ≤ i ≤ n , analogously to X . Theorem 4.4. (1)
Let t ∈ T M , φ = i ( t ) ∈ T K ∗ (see (2.14) ). The torsor p t : A t → X φ of Construction 2.13 is a universal torsor for X φ . For very general t , p t : A ft t → X ft φ is a universal torsor for X φ . (2) For m ∈ M ◦ , let L m denote the line bundle on X associated to m under theidentification Pic( X ) ∼ = M ◦ /p ∗ ( N ) . Specifically, L m is the representative of theisomorphism class given by the ˇCech 1-cocycle represented by m ∈ M ◦ in theproof of Theorem 4.1. Then A prin = Spec M m ∈ M ◦ L m . Furthermore, the line bundle L m on X extends to a line bundle L m on X ft , andsimilarly A ftprin = Spec M m ∈ M ◦ L m (using the same finite subtrees of T to define both A ftprin and X ft ).Proof. We first prove the statements for A prin , X and A t , X φ . Continuing with thenotation of the proof of Theorem 4.1 and Construction 4.3, we take the open covers U = {X s } ∪ {X µ i ( s ) | ≤ i ≤ n } = { U i | ≤ i ≤ n } , U φ = {X s ,φ } ∪ {X µ i ( s ) ,φ | ≤ i ≤ n } = { U i,φ | ≤ i ≤ n } as usual. We saw in the proof of Theorem 4.1 that M ◦ isnaturally identified with a subgroup of both Z ( U , O × X ) and Z ( U φ , O × X φ ). Taking thesubgroup Λ of this cocycle group to be M ◦ , we obtainΛ = ker(Λ → Pic( X φ )) = ker( M ◦ → M ◦ /p ∗ ( N )) = p ∗ ( N ) . This then gives rise to a sheaf of O X -algebras R = L λ ∈ Λ L λ and a sheaf of O X φ -algebras R φ defined by the same formula. For the two cases, we have the maps ˜ p : A prin → X and p t : A t → X φ of Construction 2.13. Noting that U i,A prin := ˜ p − ( U i ) = A prin , s i = 0 A prin ,µ i ( s ) i > ,U i,A t := p − t ( U i,φ ) = A prin , s ,t i = 0 A prin ,µ i ( s ) ,t i > p, p t are affine. Thus to prove both parts of the theorem, it issufficient to construct morphisms of sheaves of O X -algebras or O X φ -algebras(4.2) ψ : R → ˜ p ∗ O A prin , ψ φ : R φ → p ∗ O A t such that ψ is an isomorphism and the kernel of ψ φ is an ideal I arising from a shiftingfamily.First, by construction, R φ | U i,φ ∼ = L m ∈ M ◦ O U i,φ e m , and the transition function on U i,φ for the generator e m is (1 + z e i ) −h d i e i ,m i . The same formulae hold for R . Let I φ ⊆ Γ( U i , O U i ) = k [ N ] be the ideal of the fibre U i,φ ⊆ U i , and let I t ⊆ Γ( U i,A prin , O U i,A prin ) = k [ f M ◦ ] be the ideal of the fibre U i,A t of π : U i,A prin → T M over t . Then R φ | U i,φ is thequasi-coherent sheaf associated to the free k [ N ] /I φ -module with basis { e m | m ∈ M ◦ } ,while R| U i is the quasi-coherent sheaf associated to the free k [ N ]-module with the samebasis.Second, note that ˜ p ∗ O U i,A prin (resp. p ∗ O U i,At ) is the quasi-coherent sheaf associated tothe k [ N ]-algebra k [ f M ◦ ] (resp. the k [ N ] /I φ -algebra k [ f M ◦ ] /I t ). The algebra structure isgiven by the map N → f M ◦ , n ( p ∗ ( n ) , n ). There are natural maps R| U i → ˜ p ∗ O U i,A prin and R φ | U i,φ → ( p t ) ∗ O U i,At induced by the maps of k [ N ]- or k [ N ] /I φ -modules given by e m z ( m, . We first check that these maps respect the transition maps. We do thisfor the case of A t → X φ , the case of A prin → X being identical. On U i,φ , e m is glued IRATIONAL GEOMETRY OF CLUSTER ALGEBRAS 37 to (1 + z e i ) −h d i e i ,m i e m as observed above, while z ( m, ∈ k [ f M ◦ ] /I t is transformed via the A mutation µ i . But using (2.6), µ ∗ i ( z ( m, ) z ( m, = (1 + z ( v i ,e i ) ) −h d i e i ,m i . This can be viewed via p ∗ t as the function on U i,φ given by (1 + z e i ) −h d i e i ,m i . This showsthat the transition maps match up, and we obtain the desired map (4.2).Note that ψ is easily seen to be an isomorphism. On the other hand, the kernel I of ψ φ is generated on U i,φ by elements of L e m k [ N ] /I φ of the form e m − e m + p ∗ ( n ) z − n for m ∈ M ◦ , n ∈ N . This arises from the family of identifications { ρ p ∗ ( n ) } defined by ρ p ∗ ( n ) ( e m ) = e m + p ∗ ( n ) z − n . This is easily checked to be a shifting family.This completes the proof for A prin , X and A t , X φ . To prove the result for A ftprin , X ft etc., one just notes that the corresponding spaces are equal to A prin , X etc. outside ofcodimension two. (cid:3) Definition 4.5.
Given a choice of shifting family for a scheme X over a field k withfinitely generated Picard group, the Cox ring
Cox( X ) of X is the k -algebra of globalsections of R / I . If Pic X is free, this coincides with the usual definitionCox( X ) = M ν Γ( X, L ν ) , after a choice of line bundles L , . . . , L n whose isomorphism classes give a basis ofPic X . Corollary 4.6. (1)
The upper cluster algebra with principal coefficients is isomor-phic to M m ∈ M ◦ Γ( X, L m ) . (2) If the initial seed is totally coprime, the upper cluster algebra is isomorphic tothe Cox ring of X e . (3) For t ∈ T M very general, Γ( A t , O A t ) is isomorphic to the Cox ring of X i ( t ) .Proof. This follows because with the hypotheses, the upper cluster algebra Γ( A , O A )coincides with Γ( A, O A ) by Theorem 3.9 and Lemma 3.5. The latter algebra has thedesired description by Theorem 4.4. The principal coefficient case is similar. (cid:3) Corollary 4.7. If Pic( X ) is torsion free (i.e., if M ◦ /p ∗ ( N ) is torsion free) then theupper cluster algebra with principal coefficients Γ( A prin , O A prin ) and for very general t ,the upper cluster algebra with general coefficients Γ( A t , O A t ) are factorial. If the initialseed is totally coprime, then the upper cluster algebra Γ( A , O A ) is factorial. Proof.
For the cases other than A prin , this follows from Theorem 1.1 of [Ar08], (see also[BH03], Prop. 8.4.)For the principal case, we note that the map ˜ p : A prin → X is a T N ◦ -torsor, andthat if Pic( X ) is torsion-free, then Pic( X ) ∗ ⊆ N ◦ and T Pic( X ) ∗ is a subtorus of T N ◦ .Write X ′ = A prin /T Pic( X ) ∗ . Note that X ′ = Spec L m ∈ p ∗ ( N ) L m . But L m ∼ = O X asa line bundle for m ∈ p ∗ ( N ), so X ′ → X is a trivial T ( p ∗ ( N )) ∗ -torsor. In particular,Pic( X ′ ) ∼ = Pic( X ) and A prin is the universal torsor over X ′ . The above cited resultsshow the Cox ring of X ′ is a UFD, so the upper cluster algebra with principal coefficientsis also a UFD. (cid:3) The X variety in the rank ǫ = 2 case In this section we will fix seed data as usual, with the same assumptions as in theprevious section, namely that there are no frozen variables and that no row (or column)of ǫ is zero. We will assume furthermore that rank ǫ = 2, i.e., rank K = rank N −
2. Inthis case, the morphism
X → T K ∗ is a flat family of two-dimensional schemes (flatnessfollowing from the fact that the maps X s → T K ∗ are flat for each seed). We can usethe description of the X variety given in § K ⊥ ⊆ M is a saturated rank two sublattice by the assumption on the rank of ǫ . Furthermore, d i v i ∈ K ⊥ for each i and these vectors are non-zero by the assumptionon ǫ . Choose a complete non-singular fan ¯Σ in K ⊥ such that each − d i v i generates aray of ¯Σ. Via the inclusion K ⊥ ⊆ M , ¯Σ also determines a fan in M , which we denoteby Σ. Note Σ contains the fan of one-dimensional cones Σ s , X . Then the projection M → K ∗ ∼ = M/K ⊥ induces a map¯ λ : TV(Σ) → T K ∗ , each of whose fibres is a complete toric surface TV( ¯Σ); we in fact have non-canonicallyTV(Σ) ∼ = TV( ¯Σ) × T K ∗ , arising from a choice of splitting M = K ⊥ ⊕ K ∗ .Let D i denote the divisor of TV(Σ) corresponding to the ray generated by − d i v i .For each i we obtain a (possibly non-reduced) hypersurface Z i ⊆ D i given by Z i := D i ∩ V (cid:0) (1 + z e i ) ind d i v i (cid:1) as in (3.2). Lemma 5.1.
The underlying closed subset of Z i is the image of a section q i : T K ∗ → TV(Σ) of ¯ λ if and only if the image of e i in N/K is primitive.Proof.
A choice of splitting M = K ⊥ ⊕ K ∗ gives a dual splitting N ∼ = N/K ⊕ K .Write e i = ( e ′ i , e ′′ i ) under this splitting. The monomial z e i is non-vanishing on D i as IRATIONAL GEOMETRY OF CLUSTER ALGEBRAS 39 h e i , − d i v i i = 0. Then restricting z e i to ¯ λ − ( φ ) ∩ D i for some φ ∈ T K ∗ , we obtaina monomial ( z e ′′ i ( φ )) · z e ′ i ∈ k [( d i v i ) ⊥ ], where ( d i v i ) ⊥ is a sublattice of N/K . Thus Z i ∩ ¯ λ − ( φ ) ∩ D i consists of a single point if and only if e ′ i is primitive, i.e., the imageof e i is primitive in N/K . (cid:3) The following is an enhanced restatement of Theorem 3.9.
Lemma 5.2.
Let
Y → T K ∗ be the flat family of surfaces obtained by blowing up thesubschemes Z i ⊆ TV(Σ) in some order. Let
D ⊆ Y be the proper transform of thetoric boundary of
TV(Σ) , λ : Y \ D → T K ∗ the induced map. Then (1) X ft isomorphic to Y \ D off of a codimension ≥ set. (2) If φ ∈ T K ∗ is very general (i.e., in the complement of a countable union ofproper closed subsets), then λ − ( φ ) is isomorphic to the fibre X ft φ of X ft → T K ∗ away from codimension two.Proof. (1) is immediate from Theorem 3.9, observing that blowing up the Z i in someorder differs only in codimension two with the blow-up of the subscheme S i Z i . For(2), we first use the explicit description of X as described at the beginning of § X is obtained by gluing together tori U i , 0 ≤ i ≤ n as described explicitly inthe proof of Theorem 4.1. Denote by U i,φ , U ij,φ the fibres of U i , U ij → T K ∗ over φ . If Z i ∩ Z j ∩ ¯ λ − ( φ ) = ∅ , it is then easy to see that the maximal open set of the domainfor which the map ϕ ij | U i,φ : U i,φ U j,φ is an isomorphism is precisely U ij,φ . Thus X φ is constructed as the space X is in Lemma 3.5. The schemes Z i in that constructioncoincide with the schemes ¯ λ − ( φ ) ∩ Z i . Thus, provided φ does not lie in ¯ λ ( Z i ∩ Z j )for any i, j , Lemma 3.5 applies to show that there is an open immersion X φ → λ − ( φ )which is an isomorphism off of a codimension two subset of λ − ( φ ).To complete the argument, we follow the proofs of Lemma 3.8 and Theorem 3.9. If s ′ = µ k ( s ) and X ′ , Y ′ , Z ′ i etc. are constructed using the seed s ′ , then the argument ofLemma 3.8 shows that provided φ λ ( Z i ∩ Z j ) , λ ′ ( Z ′ i ∩ Z ′ j ) for any pair i = j , λ − ( φ )is isomorphic to ( λ ′ ) − ( φ ) off codimension two. Thus X φ and X ′ φ are isomorphic off aset of codimension two, and the argument is finished as in Theorem 3.9. (cid:3) Thus the family X ft → T K ∗ can be thought of, away from codimension two, as afamily of surfaces obtained by blowing up a collection of points on the boundary ofa toric variety, and then deleting the proper transform of the boundary. In general,since these points are being blown up with multiplicity, Y \ D can be singular. We willfirst see that any surface obtained via blowups on the boundary of a toric surface isdeformation equivalent to any surface in the family
Y → T K ∗ constructed using someseed. Construction 5.3.
Let ¯ Y be a complete non-singular toric surface, with toric bound-ary ¯ D , given by a fan ¯Σ in a lattice N ∼ = Z . Choose a collection of irreducible boundarydivisors ¯ D , . . . , ¯ D n (possibly with repetitions) and let w i ∈ N be the primitive gener-ator of the ray corresponding to ¯ D i . Fix positive integers ν i , 1 ≤ i ≤ n . Suppose that w , . . . , w n generate N .We will use this data to construct seed data, as follows. Set N = Z n with basis { e i } , M the dual lattice as usual. Define a map ψ : N → N by e i w i . By assumption, ψ is surjective. Choose an isomorphism V N ∼ = Z . The map ϕ : N → M given by¯ n ( n ψ ( n ) ∧ ¯ n ) gives a primitive embedding of the lattice N into M by surjectivityof ψ . Let ν = gcd( ν , . . . , ν n ). We then obtain an integer-valued skew-symmetric form {· , ·} on N by { n , n } = νψ ( n ) ∧ ψ ( n ) ∈ Z . Note that ker ψ coincides with K = { n ∈ N | { n, ·} = 0 } . Set d i = ν i /ν . This gives usseed data { e i } for the fixed data N = N uf , {· , ·} and { d i } .We now analyze the family Y → T K ∗ arising from this seed data. Using the inclusion ϕ , we write Σ for the fan ¯Σ as a fan in M . We write D i for the toric divisor of TV(Σ)corresponding to the ray generated by w i . We note that with v i = p ∗ ( e i ) = { e i , ·} = − νϕ ( ψ ( e i )) , we have − d i v i = ν i ϕ ( w i ). As ψ is surjective, N/K ∼ = N , and the image of each e i in N/K is primitive, being w i ∈ N . Thus by Lemma 5.1, the closed sets Z i are imagesof sections of D i → T K ∗ . It then follows by Lemma 5.2 that the general fibre of λ : Y → T K ∗ is obtained by blowing up ¯ Y at a collection of points p , . . . , p n , with p i ∈ ¯ D i taken with multiplicity ν i .We now consider the special case that all ν i = 1. First we note: Proposition 5.4.
Giving • fixed data with rank N = n , no frozen variables, d i = 1 for all i and such that {· , ·} has rank two and the induced non-degenerate skew-symmetric pairing on N/K is unimodular, and • a seed s for this fixed data such that the image of each e i in N/K is primitive;is equivalent to giving primitive vectors w , . . . , w n ∈ N where N is a rank two latticeand for which w , . . . , w n generate N .Proof. Construction 5.3 explains how to pass from the data of the w i ’s to the fixedand seed data. Here we take ν i = 1 for all i in that construction. Conversely, givenfixed and seed data as in the proposition, we take N = N/K , w i the image of each e i . The only unimodular integral skew-symmetric pairing on N , up to sign, is given IRATIONAL GEOMETRY OF CLUSTER ALGEBRAS 41 by { n , n } = n ∧ n , after a choice of identification V N ∼ = Z . Thus after makinga suitable choice of identification, the given pairing {· , ·} on N agrees with the onedescribed in Construction 5.3. (cid:3) Continuing with the above notation, with ν = · · · = ν n = 1, consider the family λ : Y → T K ∗ of blown up toric surfaces. In this case, a general fibre λ − ( φ ) for φ ∈ T K ∗ is obtained byblowing up reduced points on the non-singular part of the toric boundary of ¯ Y . A fibre( Y, D ) = ( Y φ , D φ ) is what we call a Looijenga pair in [GHK12]. Given a cyclic orderingof the irreducible components of D = D ′ + · · · + D ′ r one gets a canonical identificationof the identity component Pic ( D ) of Pic D with G m , see [GHK12], Lemma 2.1. (Wenote the divisors D , . . . , D n are a possibly proper subset of D ′ , . . . , D ′ r , and the formeroccur with repetitions and need not be cyclically ordered). Furthermore, we define D ⊥ ⊆ Pic( Y ) by D ⊥ := { H ∈ Pic( Y ) | H · D ′ i = 0 ∀ i } . Then the period point of (
Y, D ) is the element of Hom( D ⊥ , Pic ( D )) given by restriction. Theorem 5.5.
Let π : Y → ¯ Y be the blow-up describing Y , with exceptional divisors E , . . . , E n over Z ∩ ¯ λ − ( φ ) , . . . , Z n ∩ ¯ λ − ( φ ) . Then there is a natural isomorphism K → D ⊥ X a i e i π ∗ C − X a i E i where C is the unique divisor class such that (5.1) C · D ′ j = X i : D i = D ′ j a i . Under this identification, and the canonical identification
Pic ( D ) ∼ = G m , the periodpoint of ( Y, D ) in Hom( D ⊥ , Pic ( D )) coincides with φ ∈ T K ∗ = Hom( K, G m ) .Proof. Recall the standard description of the second homology group of the toric variety¯ Y : 0 → H ( ¯ Y , Z ) → Z r → N → , where the map Z r → N takes the i th generator of Z r to the primitive generator ofthe ray of ¯Σ corresponding to the divisor ¯ D ′ i . The inclusion H ( ¯ Y , Z ) ֒ → Z r is givenby α ( α · D ′ i ) ≤ i ≤ r . Since ¯ Y is a non-singular proper rational surface, we have H ( ¯ Y , Z ) ∼ = Pic( ¯ Y ). In particular, if P a i e i ∈ K , then ( P j : D j = D ′ i a j ) ≤ i ≤ r ∈ H ( ¯ Y , Z ).Thus there is a unique element C ∈ H ( ¯ Y , Z ) ∼ = Pic( ¯ Y ) satisfying (5.1). It is then clearthat π ∗ C − P a i E i ∈ D ⊥ . That this is an isomorphism is easily checked. We now need to calculate O Y ( π ∗ C − P a i E i ) | D . As the identification Pic ( D ) with G m requires a choice of cyclic ordering of the D ′ i , or equivalent of the ¯ D ′ i , we order w ′ , . . . , w ′ r clockwise as defined using the choice of isomorphism V N → Z . In particu-lar, this choice of isomorphism also allows an identification of N with M = Hom( N , Z ),via n ∈ ¯ N acts by n ′ ( n ′ ∧ n ). Thus, in particular, z w ′ i can be viewed as a coordinateon D ′ i which is zero on p i,i +1 , the intersection point of D ′ i and D ′ i +1 , and infinite at p i − ,i (all indices taken modulo r ).We next note that O ¯ Y ( C ) | ¯ D was calculated in the proof of [GHK12], Lemma 2.6,(1). Let m i ∈ D ′ i be the point where z w ′ i takes the value −
1. Then O ¯ Y ( C ) | ¯ D = O ¯ D ( r X j =1 ( C · ¯ D ′ j ) m j ) . Thus we have the same identity for the restriction of O Y ( π ∗ C ) to D . So if E i ∩ D = p i ,we then have L := O Y ( π ∗ C − X a i E i ) | D ∼ = O D (cid:0) − n X i =1 a i p i + r X j =1 ( C · ¯ D ′ j ) m j (cid:1) . This line bundle is described under the isomorphism Pic ( D ) ∼ = G m of [GHK12],Lemma 2.6, as follows. We have L| D ′ j = O D ′ j (( C · ¯ D ′ j ) m j − P i : D i = D ′ j a i p i ). View-ing this trivial line bundle as a subsheaf of the sheaf of rational functions, and using asplitting M = K ⊥ ⊕ K ∗ , N = N/K ⊕ K as in the proof of Lemma 5.1, a trivializingsection is given by the rational function σ j := Q i : D i = D ′ j ( z w ′ j · z e ′′ i ( φ ) + 1) a i ( z w ′ j + 1) C · ¯ D ′ j since Z i is given by the equation z e i + 1 = 0 and under the choice of splitting e i =( w i , e ′′ i ), with w i = w ′ j if D i = D ′ j . The image of the line bundle in G m is r Y j =1 σ j +1 ( p j,j +1 ) /σ j ( p j,j +1 ) = n Y i =1 ( z e ′′ i ( φ )) a i . Remembering that we are viewing φ ∈ Hom( K, G m ), we see that z e ′′ i ( φ ) = φ ( e ′′ i ). Notethat if P a i e i ∈ K , we have P a i e i = P a i e ′′ i . Thus we see that the element of G m corresponding to our line bundle is precisely φ ( P a i e i ). Thus φ is the period point of( Y, D ). (cid:3) This shows that the families
Y → T K ∗ agree with the universal families of Looijengapairs constructed in [GHK12].We can also use the above observations to define an unexpected mutation invariantin the situation of Proposition 5.4. IRATIONAL GEOMETRY OF CLUSTER ALGEBRAS 43
Theorem 5.6.
Given fixed data and seed data satisfying the conditions of Proposition5.4, the isomorphism K ∼ = D ⊥ of Theorem 5.5 induces a symmetric integral pairingon K via the intersection pairing on D ⊥ ⊆ Pic( Y ) . This symmetric pairing on K isindependent of mutation.Proof. It is enough to check the independence under a single mutation µ k . So supposegiven seeds s , s ′ = µ k ( s ). These two seeds give rise to families Y , Y ′ → T K ∗ . Fix φ ∈ T K ∗ sufficiently general so that the fibres Y φ and Y ′ φ are both blowups of toricvarieties ¯ Y , ¯ Y ′ at distinct points. These toric varieties are related by an elementarytransformation as follows. Let ψ : N → N = N/K be the quotient map. If ¯Σ , ¯Σ ′ arethe fans in N defining ¯ Y and ¯ Y ′ respectively, then ¯Σ ′ is obtained from ¯Σ as follows.First, we can assume that both ¯Σ and ¯Σ ′ contain rays generated by w k and − w k . Thenwe can assume the remaining rays of ¯Σ ′ are obtained by applying the piecewise lineartransformation T k : ¯ n ¯ n + [¯ n ∧ w k ] + w k . Note in particular that this is compatible with (2.3).The map N → N / R w k defines P -fibrations g : ¯ Y → P , g ′ : ¯ Y ′ → P .By Lemma 3.6, the seed mutation µ k : T N T N extends to a birational map µ k : Y → Y ′ which is an isomorphism off of sets of codimension ≥
2. In fact, one checkseasily from the details of the proof that this birational map restricts to a biregularisomorphism between Y and Y ′ . Specifically, this isomorphism is described as follows.Let p , . . . , p n be the points of ¯ Y blown up to obtain Y , and p ′ , . . . , p ′ n the points of ¯ Y ′ blown up to obtain Y ′ . Then if Y k , Y ′ k denote the blowup of ¯ Y , ¯ Y ′ at p k , p ′ k respectively,we already have an isomorphism µ k : Y k → Y ′ k , and p ′ i = µ k ( p i ) for i = k . Theisomorphism µ k : Y → Y ′ is then obtained by further blowing up the p i , p ′ i for i = k .Furthermore, the composition Y k µ k −→ Y ′ k → ¯ Y ′ contracts the proper transform of thecurve F k = g − ( g ( p k )) to p ′ k . In particular, the curve class F k − E k ∈ Pic( Y ) is mappedto E ′ k .We now need to check that the composition of isomorphisms ( D ′ ) ⊥ ∼ = K ∼ = D ⊥ given in Theorem 5.5 coincides with µ ∗ k : ( D ′ ) ⊥ → D ⊥ . To do so, suppose given P a i e i = P a ′ i e ′ i ∈ K . Then a i = a ′ i for i = k and a ′ k = − a k + P i [ ǫ ik ] + a i by (2.3).These determine classes C ∈ Pic( ¯ Y ), C ′ ∈ Pic( ¯ Y ′ ) as in Theorem 5.5. It is enough tocheck that if π : Y k → ¯ Y , π ′ : Y ′ k → ¯ Y ′ are the blowups at p k , p ′ k respectively, then µ ∗ k (( π ′ ) ∗ ( C ′ ) − a ′ k E ′ k ) = π ∗ ( C ) − a k E k . Call these two divisors C ′ and C respectively.Since the Picard group of Y k is easily seen to be generated by the proper transforms ofthe toric divisors of Y k , and µ k takes the boundary divisor of Y k corresponding to ρ ∈ ¯Σto the boundary divisor of Y ′ k corresponding to T k ( ρ ) ∈ ¯Σ ′ , it is enough to check that C ′ and C have the same intersection numbers with the boundary divisors of Y k . It is clear that C ′ and C have the same intersection numbers with all boundary divisors exceptpossibly those corresponding to the rays ± R ≥ w k . Call these two divisors D k, ± ⊆ Y k .Then C · D k, + = X i : w i = w k ,i = k a i = C ′ · D k, + , while C · D k, − = X i : w i = − w k ,i = k a i = C ′ · D k, − . This proves the result. (cid:3) Examples of non-finitely generated upper cluster algebras
We will now use the material of the previous two sections to construct examplesof non-finitely generated upper cluster algebras with principal coefficients and withgeneral coefficients. These examples are a generalization of the example of Speyer[Sp13]. They fail to be finitely generated as a consequence of the following:
Lemma 6.1.
Let A be a ring, R an M = Z n -graded A -algebra, R = L m ∈ M R m . If R = A and R m is not a finitely generated A -module for some m ∈ M , then R is notNoetherian.Proof. Let I be the homogeneous ideal of R generated by R m . We show I is not finitelygenerated as an ideal. Suppose to the contrary that homogeneous f , . . . , f p ∈ R generate I . Necessarily f i = P r ij f ij for some r ij ∈ R and f ij ∈ R m , so we can assume I is generated by a finite number of f ij ∈ R m . But R m = I ∩ R m is the A = R -modulegenerated by the f ij , contradicting the assumption that R m is not finitely generated asan A -module. (cid:3) In what follows, suppose given fixed data and seed data satisfying the hypothesesof Proposition 5.4. This gives rise to the family λ : ( Y , D ) → T K ∗ of Loojienga pairs,obtained by blowing up a sequence of centers Z , . . . , Z n ⊆ ¯ Y × T K ∗ in some order, for¯ Y a toric surface. Theorem 6.2.
Let ( Y, D ) be the general fibre of ( Y , D ) → T K ∗ . Suppose that everyirreducible component of D has self-intersection − . Then (1) Γ( A prin , O A prin ) is non-Noetherian. (2) For t ∈ T M very general, A t the corresponding cluster variety with generalcoefficients, then Γ( A t , O A t ) is non-Noetherian.Proof. Let X be as usual the subset of X obtained by gluing together the initialseed torus X s and adjacent seed tori X µ k ( s ) . By Corollary 4.6, Γ( A prin , O A prin ) = IRATIONAL GEOMETRY OF CLUSTER ALGEBRAS 45 L m ∈ M ◦ Γ( X, L m ), and this gives a M ◦ -grading on this algebra. In addition, by Lemma5.2, X and Y \ D agree off of a codimension ≥ X and Y \ D are non-singular. So Pic X ∼ = Pic( Y \ D ). Thus for each m ∈ M ◦ , L m can be viewed as a linebundle on Y \ D , and L m has the same space of sections regardless of whether L m isviewed as a bundle on X or on Y \ D . So, by the lemma, it will suffice to show thatΓ(
Y \ D , O Y ) = A := k [ K ] and find some line bundle L on Y \ D such that Γ(
Y \ D , L )is not a finitely generated A -module.To show Γ( Y \ D , O Y ) = A , it is sufficient to show that a regular function on Y \ D must be constant on the very general fibre of λ : Y \ D → T K ∗ = Spec k [ K ].So consider the fibre ( Y, D ) of ( Y , D ) → T K ∗ over φ ∈ T K ∗ . The space of regularfunctions on Y \ D can be identified with lim → H ( Y, O Y ( nD )). Consider the longexact cohomology sequence associated to0 → O Y ( nD ) → O Y (( n + 1) D ) → O Y (( n + 1) D ) | D → . Note that D ∈ D ⊥ since all components of D have square − φ ∈ T K ∗ =Hom( D ⊥ , Pic ( D )), φ ( D ) is the normal bundle of D in Y by Theorem 5.5. Thus as φ is very general, φ ( D ) is not torsion. So H ( D, O D ( nD )) = 0 for all n >
0, and we seethat H ( Y, O Y ( nD )) = k for all n ≥
0. Thus the only regular functions on Y \ D areconstant.Now let E be the exceptional divisor over the last center Z i blown up in constructing Y , so that E is a P -bundle over T K ∗ . Then we claim Γ( Y \ D , O Y ( E )) is not a finitelygenerated A -module. Note thatΓ( Y \ D , O Y ( E )) = lim −→ n ≥ Γ( Y , O Y ( E + n D )) . Since each of these groups is an A -module, it is sufficient to show that the increasingchain of A -modules(6.1) Γ( Y , O Y ( E )) ⊆ Γ( Y , O Y ( E + D )) ⊆ Γ( Y , O Y ( E + 2 D )) ⊆ · · · does not stabilize. We have a long exact sequence0 −→ H ( Y , O Y ( E + ( n − D )) i −→ H ( Y , O Y ( E + n D )) −→ H ( D , O D ( E + n D )) −→ H ( Y , O Y ( E + ( n − D )) −→ H ( Y , O Y ( E + n D )) −→ H ( D , O D ( E + n D )) . If (
Y, D ) is any fibre of ( Y , D ) → T K ∗ , one checks easily that H ( Y, O Y ( E ∩ Y )) = 0(as E ∩ Y is an irreducible − H ( D, O D (( E ∩ D ) + nD )) = 0 (usingthat E ∩ D consists of one point). It then follows from cohomology and base changealong with the fact that T K ∗ is affine that H ( D , O D ( E + n D )) = 0 for all n ≥ H ( Y , O Y ( E )) = 0. Inductively from the above long exact sequence one sees H ( Y , O Y ( E + n D )) = 0 for all n ≥
0. Thus the cokernel of the inclusion i in the above long exact sequence is H ( D , O D ( E + n D )). Now λ ∗ O D ( E + n D ) is a line bundle on T K ∗ , again by cohomology and base change, and since T K ∗ is an algebraic torus, thisline bundle must be trivial. Thus H ( D , O D ( E + n D )) = A , and we see the chain (6.1)never stabilizes.The argument for A t for t very general is identical but easier, as we have already donethe relevant cohomology calculations on ( Y, D ) a very general fibre of ( Y , D ) → T K ∗ .Then one makes use of Corollary 4.6, (3). (cid:3) Example 6.3.
Using Construction 5.3 it is easy to produce many examples satisfyingthe hypotheses of the above theorem. For example, let ¯Σ be the fan for P , with raysgenerated by w = w = w = (1 , w = w = w = (0 ,
1) and w = w = w =( − , − ν i = 1. Thus a general ( Y, D ) involves blowing three points on eachof the coordinate lines of P , so D is a cycle of three − P , w = (1 , w = (0 ,
1) and w = ( − , − ν i = 3. The surface ( Y, D ) will be constructed byperforming a weighted blowup of one point on each of three coordinate lines on P .Then D is still a cycle of three − Y is in fact singular (having three A singularities). Remark . In fact there is a much broader range of counterexamples: suppose that theblowup ( Y , D ) → ¯ Y × T K ∗ factors through ( Y , D ) → ( Y ′ , D ′ ), such that a very generalfibre ( Y ′ , D ′ ) of ( Y ′ , D ′ ) → T K ∗ has the property that every irreducible componentof D ′ has self-intersection −
2. Then the argument above shows that the Cox ring of Y ′ \ D ′ is non-Noetherian, and Y ′ \ D ′ is an open subset of Y \ D . If U ⊆ V , thenthe Cox ring of V surjects onto the Cox ring of U , so the fact that the Cox ring of Y ′ \ D ′ is non-Noetherian implies the Cox ring of Y \ D is non-Noetherian. A similarbut slightly more delicate argument also applies to the principal coefficient case.In fact, we expect that whenever the intersection matrix of D is negative definite,the Noetherian condition fails.7. Counterexamples to the Fock-Goncharov dual bases conjecture [FG09] gave an explicit conjecture about the existence of k -bases for the X and A cluster algebras. We will state it loosely here, under the assumption that all d i = 1,so that M = M ◦ , N = N ◦ . This merely allows us to avoid discussing Langlands dualseeds.Fock and Goncharov conjecture IRATIONAL GEOMETRY OF CLUSTER ALGEBRAS 47
Conjecture 7.1 ([FG09], 4.1) . N parameterizes a canonical basis of H ( X , O X ) , and M parameterizes a canonical basis of sections of H ( A , O A ) . In fact, the conjecture as stated in [FG09] is much stronger, giving an explicit con-jectural description of the bases as the set of positive universal Laurent polynomialswhich are extremal, i.e., not a non-trivial sum of two other positive universal Lau-rent polynomials. This strongest form of the conjecture has now been disproven in[LLZ13], in which examples are given where the set of all extremal positive universalLaurent polynomials are not linearly independent. Here we give a much more basiccounterexample to a much weaker form of the conjecture.We shall again restrict to the case that that there are no frozen variables. We willmerely assume the conjectured basis is compatible with the T K action on A given byRemark 2.2, (3), and the map λ : X → T K ∗ in the natural way. We assume that thecanonical basis element of Γ( X , O X ) corresponding to n ∈ K is λ ∗ ( z n ). Furthermore,for π : M → K ∗ the natural projection dual to the inclusion K → N , we assumethat the set π − ( m ) parameterizes a basis of the subspace of H ( A , O A ) of weight m eigenvectors for the T K action.We indicate now why a basis with these properties cannot exist in general. We con-sider the rank 2 cluster algebras produced by Construction 5.3, following the notationof the construction, taking all ν i = 1. The general fibre of λ : X → T K ∗ is isomorphicup to codimension two subsets to the general fibre of λ : Y \ D → T K ∗ . A fibre ofthe latter map is of the form U := Y \ D , where ( Y, D ) is a Looijenga pair with amap Y → ¯ Y obtained by blowing up points on the toric boundary. Since the initialdata of the w i in Construction 5.3 can be chosen arbitrarily, and in particular the w i ’s may be repeated as many times as we like, we can easily find examples for which D ⊆ Y is analytically contractible, i.e., there is an analytic map ( Y, D ) → ( Y ′ , p ) withexceptional locus D and p ∈ Y ′ a single point. Further, U = Y \ D = Y ′ \ { p } , andso H ( U, O U ) = H ( Y ′ , O Y ′ ) = k . Even if D is not contractible but rather a cycleof − Y \ D will only have constantfunctions. It follows that H ( X , O X ) = H ( T K ∗ , O T K ∗ ) = k [ K ] . Thus there are no functions for points of N \ K to parameterize.Consider the conjecture in the opposite direction. Assume for simplicity that, as inCorollary 4.7, Pic( X ) = Pic( X t ) = M/p ∗ ( N ) is torsion free. Then ( M/p ∗ ( N )) ∗ = K =Ker( p ∗ : N → M ). The Fock-Goncharov conjecture for A prin implies the analogousresult for very general A t , i.e. the existence of a canonical basis of the upper cluster algebra with very general coefficients H ( A t , O ), parameterized by X trop ( Z ). We have H ( A t , O A t ) = Cox( X i ( t ) ) = M m ∈ K ∗ =Pic( X i ( t ) ) H ( X i ( t ) , L m )by Corollary 4.6. Here L m is a line bundle representing the isomorphism class givenby m . Assuming the canonical bases are compatible with the natural torus actions,then for m ∈ K ∗ , π − ( m ) ⊂ X trop ( Z ) restricts to a basis for the weight m -eigenspace H ( X i ( t ) , L m ) of H ( A t , O ) under the T K action, for π : X trop ( Z ) → K ∗ = M/p ∗ ( N ),the natural map induced by the fibration X → T K ∗ . But any choice of seed identifies X trop ( Z ) with M and each fibre of π with a p ∗ ( N ) = N/K torsor. Thus the conjectureimplies all line bundles on X t have isomorphic spaces of sections, with basis parame-terized (after choice of seed) by an N/K affine space. This is a very strong condition— most varieties have line bundles with no non-trivial sections, and rather than anaffine space one would expect (for example by comparison with the toric case) sectionsparameterized by integer points of a polytope. Explicitly, the example of Theorem 6.2clearly has line bundles with non-isomorphic spaces of sections.This reasoning suggests to us the conjecture can only hold if X is affine up to flops : Conjecture 7.2.
If the Fock-Goncharov conjecture holds then H ( X , O X ) is finitelygenerated, and the canonical map X →
Spec( H ( X , O X )) is an isomorphism, outsideof codimension two. The results of § X → T K ∗ is affine, which istrue iff the canonical symmetric form on K given by Theorem 5.6 is negative definite.Indeed, the generic fibre of X → T K ∗ is isomorphic, up to codimension two, to a surface Y \ D as in Theorem 5.5. But if Y \ D is affine, then D supports an ample divisor, andby the Hodge index theorem, D ⊥ is negative definite. Conversely, if D ⊥ is negativedefinite, there must be some integers a i such that ( P a i D i ) >
0. The result thenfollows from [GHK11], Lemma 6.8 and the fact that (
Y, D ) is chosen generally in thefamily.
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