Classification of normal toric varieties over a valuation ring of rank one
aa r X i v : . [ m a t h . AG ] J a n Classification of normal toric varietiesover a valuation ring of rank one
Walter Gubler and Alejandro SotoMarch 20, 2018
Abstract
Normal toric varieties over a field or a discrete valuation ring are classified by rationalpolyhedral fans. We generalize this classification to normal toric varieties over an arbitraryvaluation ring of rank one. The proof is based on a generalization of Sumihiro’s theorem tothis non-noetherian setting. These toric varieties play an important role for tropicalizations.
MSC2010: 14M25 , 14L30, 13F30
Contents
Toric varieties over a field have been studied since the 70’s. Their geometry is completelydetermined by the convex geometry of rational polyhedral cones. This gives toric geometry animportant role in algebraic geometry for testing conjectures. There are many good referencesfor them, for instance Cox–Little–Schenk [6], Ewald [7], Fulton [8], Kempf–Knudsen–Mumford–Saint-Donat [15] and Oda [21]. Although in these books, toric varieties are defined over analgebraically closed field, the main results hold over any field.Motivated by compactification and degeneration problems, Mumford considered in [15, Chap-ter IV] normal toric varieties over discrete valuation rings. A similar motivation was behindSmirnov’s paper [24] on projective toric varieties over discrete valuation rings. In the paper ofBurgos–Philippon–Sombra [5], toric varieties over discrete valuation rings were considered forapplications to an arithmetic version of the famous Bernstein–Kusnirenko–Khovanskii theorem1n toric geometry. The restriction to discrete valuation rings is mainly caused by the use ofstandard methods from algebraic geometry requiring noetherian schemes.At the beginning of the new century, tropical geometry emerged as a new branch of math-ematics (see [18] or [19]). We fix now a valued field (
K, v ) with value group Γ := v ( K × ) ⊂ R .The Bieri–Groves theorem shows that the tropicalization of a closed d -dimensional subvarietyof a split torus T := ( G nm ) K over K is a finite union of Γ-rational d -dimensional polyhedra in R n . Moreover, this tropical variety is the support of a weighted polyhedral complex of puredimension d such that the canonical tropical weights satisfy a balancing condition in every faceof codimension 1. The study of the tropical weights leads naturally to toric schemes over thevaluation ring K ◦ of K (see [14] for details).A T -toric scheme Y over K ◦ is an integral separated flat scheme over K ◦ containing T as adense open subset such that the translation action of T on T extends to an algebraic action ofthe split torus T := ( G nm ) K ◦ on Y . If a T -toric scheme is of finite type over K ◦ , then we call ita T -toric variety. There is an affine T -toric scheme V σ associated to any Γ-admissible cone σ in R n × R + . This construction is similar to the classical theory of toric varieties over a field, whereevery rational polyhedral cone in R n containing no lines gives rise to an affine T -toric variety.The additional factor R + takes the valuation v into account and Γ-admissible cones are conescontaining no lines satisfying a certain rationality condition closely related to the Γ-rationalityin the Bieri–Groves theorem. If Σ is a fan in R n × R + of Γ-admissible cones, then we call Σ aΓ-admissible fan. By using a gluing process along common subfaces, we get an associated T -toricscheme Y Σ over K ◦ with the open affine covering ( V σ ) σ ∈ Σ . We refer to § T -toric schemes Y Σ are studied in [14]. Many of the properties of toric varietiesover fields hold also for Y Σ .Rohrer considered in [22] toric schemes X Π over an arbitrary base S associated to a rationalfan Π in R n containing no lines. If we restrict to the case S = Spec( K ◦ ), then Rohrer’s toricschemes are a special case of the above T -toric schemes as we have X Π = Y Π × R + . Note thatthe cones of Π × R + are preimages of cones in R n with respect to the canonical projection R n × R + → R n and hence the fans Π × R + form a very special subset of the set of Γ-admissiblefans in R n × R + . As a consequence, Rohrer’s toric scheme X Π is always obtained by base changefrom a corresponding toric scheme over Spec( Z ) while this is in general not true for Y Σ . Thegeneric fibre of Y Σ is the toric variety over K associated to the fan formed by the recessioncones of all σ ∈ Σ, but the special fibre of Y Σ need not be a toric variety. In fact, the specialfibre of Y Σ is a union of toric varieties corresponding to the vertices of the polyhedral complexΣ ∩ ( R n ×{ } ). On the other hand, every fibre of the toric scheme X Π is a toric variety associatedto the same fan Π.This leads to the natural question if every normal T -toric variety Y over the valuation ring K ◦ is isomorphic to Y Σ for a suitable Γ-admissible fan Σ in R n × R + . This classification ispossible in the classical theory of normal toric varieties over a field and also in the case of normaltoric varieties over a discrete valuation ring. First, one shows that every affine normal T -toricvariety over a field or a discrete valuation ring is of the form V σ for a rational cone σ in R n × R + containing no lines and then one uses Sumihiro’s theorem which shows that every point in Y hasa T -invariant affine open neighbourhood (see [15]). Sumihiro proved his theorem over a field in[25]. In [15, Chapter IV], the arguments were extended to the case of a discrete valuation ring.The proof of Sumihiro’s theorem relies on noetherian techniques from algebraic geometry.Now we describe the structure and the results of the present paper. For the generalizationof the above classification to normal T -toric varieties Y over an arbitrary valuation ring K ◦ ofrank 1, one needs a theory of divisors on varieties over K ◦ . This will be done in §
2. First, werecall some basic facts about normal varieties over K ◦ due to Knaf [17]. Then we define theWeil divisor associated to a Cartier divisor D and more generally a proper intersection product2f D with cycles. This is based on the corresponding intersection theory of Cartier divisors onadmissible formal schemes over K ◦ given in [12]. In §
3, we recall the necessary facts for toricvarieties over K ◦ which were proved in [14]. In §
4, we show the following classification for affinenormal toric varieties:
Theorem 1. If v is not a discrete valuation, then the map σ V σ defines a bijection betweenthe set of those Γ -admissible cones σ in R n × R + for which the vertices of σ ∩ ( R n × { } ) arecontained in Γ n × { } and the set of isomorphism classes of normal affine T -toric varieties overthe valuation ring K ◦ . Similarly to the classical case, the proof uses finitely generated semigroups in the characterlattice of T and duality of convex polyhedral cones. The new ingredient here is an approximationargument concluding the proof of Theorem 1. The additional condition for the vertices of theΓ-admissible cone σ is equivalent to the property that the affine T -toric scheme V σ is of finitetype over K ◦ meaning that V σ is a T -toric variety over K ◦ (see Proposition 3.3). If v is a discretevaluation, then V σ is always a T -toric variety over K ◦ and hence the condition on the verticeshas to be omitted to get the bijective correspondence in Theorem 1.For the globalization of the classification, the main difficulty is the generalization of Sumihiro’stheorem. The proof follows the same steps working in the case of fields or discrete valuation rings(see [15], proof of Theorem 5 in Chapter I and § §
5, we show that for every non-emptyaffine open subset U of a normal T -toric variety Y over the valuation ring K ◦ of rank one,the smallest open T -invariant subset U containing U has an effective Cartier divisor D withsupport equal to U \ U . This is rather tricky in the non-noetherian situation and it is preciselyhere, where we use the results on divisors from § §
6, we use the Cartier divisor D constructed in the previous section to show that O ( D ) is anample invertible sheaf with a T -linearization. This leads to a T -equivariant immersion of U intoa projective space over K ◦ on which T -acts linearly. It remains to prove Sumihiro’s theorem forprojective T -toric subvarieties of a projective space over K ◦ on which T -acts linearly. This variantof Sumihiro’s theorem will be proved in § T -toric varieties given in [14, § Sumihiro’stheorem : Theorem 2.
Let v be a real valued valuation with valuation ring K ◦ and let Y be a normal T -toric variety over K ◦ . Then every point of Y has an affine open T -invariant neighborhood. As an immediate consequence, we will obtain our main classification result:
Theorem 3. If v is not a discrete valuation, then the map Σ Y Σ defines a bijection betweenthe set of fans in R n × R + , whose cones are as in Theorem 1, and the set of isomorphism classesof normal T -toric varieties over K ◦ . If v is a discrete valuation, then we have to omit the additional condition on the vertices ofthe cones again to get a bijective correspondence in Theorem 3.The authors thank the referee for his comments and suggestions. Notation
For sets, in A ⊂ B equality is not excluded and A \ B denotes the complement of B in A .The set of non-negative numbers in Z , Q or R is denoted by Z + , Q + or R + , respectively. Allthe rings and algebras are commutative with unity. For a ring A , the group of units is denotedby A × . A variety over a field k is an irreducible and reduced scheme which is separated and offinite type over k . See § valued field ( K, v ) which means here that v is a valuation onthe field K with value group Γ := v ( K × ) ⊂ R . Note that K is not required to be algebraicallyclosed or complete and that its valuation can be trivial. We have a valuation ring K ◦ := { x ∈ K | v ( x ) ≥ } with maximal ideal K ◦◦ := { x ∈ K | v ( x ) > } and residue field e K := K ◦ /K ◦◦ .We denote by K an algebraic closure of K .Let M be a free abelian group of rank n with dual N := Hom( M, Z ). For u ∈ M and ω ∈ N ,the natural pairing is denoted by h u, ω i := ω ( u ) ∈ Z . For an abelian group G , the base changeis denoted by M G := M ⊗ Z G ; for instance M R = M ⊗ Z R . The split torus over K ◦ of rank n with generic fiber T = Spec( K [ M ]) is given by T = Spec( K ◦ [ M ]), therefore M can be seenas the character lattice of T and N as the group of one parameter subgroups. For u ∈ M , thecorresponding character is denoted by χ u . The goal of this section is to recall some facts about divisors on varieties over the valuation ring K ◦ of the valued field ( K, v ) with value group Γ ⊂ R . The problem here is that K ◦ need not benoetherian and so we cannot use the usual constructions from algebraic geometry. Instead wewill adapt the intersection theory with Cartier divisors on admissible formal schemes from [12]to our algebraic framework.We will start with some topological considerations of varieties over K ◦ . As we mainly focuson normal varieties in this paper, we will gather some results of Knaf relating the local rings ofsuch varieties to valuation rings. This will be useful to define the Weil divisor associated to aCartier divisor. If the variety is not normal then we will pass to the formal completion alongthe special fibre to define the associated Weil divisor according to [12]. This construction willbe explained in 2.7–2.9 and is rather technical. In Proposition 2.11, we will show that bothconstructions agree for normal varieties over K ◦ . As a consequence, we will obtain a properintersection theory of Cartier divisors with cycles on a (normal) variety over K ◦ . The relevantproperties are listed in 2.10–2.16.This intersection theory will be used in the proof of the generalization of Sumihiro’s theoremgiven in § K ◦ which will be helpful to reduce Sumihiro’s theorem to an easier projective variant. A variety over K ◦ is an integral scheme which is of finite type and separated over K ◦ . By[14, Lemma 4.2] such a variety Y is flat over K ◦ . We have Spec( K ◦ ) = { η, s } , where the genericpoint η (resp. the special point s ) is the zero-ideal (resp. the maximal ideal) in K ◦ . We get thegeneric fibre Y η as a variety over K and the special fibre Y s as a separated scheme of finite typeover e K . The variety Y is called normal if all the local rings O Y ,y are integrally closed. Proposition 2.2.
A variety Y over K ◦ is a noetherian topological space. If d := dim( Y η ) , thenevery irreducible component of the special fibre has also dimension d . If Y s is non-empty and if v is non-trivial, then the topological dimension of Y is d + 1 . If Y s is empty or if v is trivial,then Y = Y η .Proof. The set Y over K ◦ is the union of Y η and Y s . This proves the first claim. The secondclaim follows from flatness of Y over K ◦ . The other claims are now obvious.The following facts about normal varieties over a valuation ring follow from a paper by Knaf[17]. Proposition 2.3.
Let Y be a normal variety over K ◦ . Then the following properties hold: a) For y ∈ Y , the local ring O Y ,y is a valuation ring if and only if y is either a dense pointof a divisor of the generic fibre Y η or y is a generic point of Y s or Y η .(b) If y is a dense point of a divisor of the generic fibre, then O Y ,y is a discrete valuation ring.(c) If y is a generic point of the special fibre Y s , then O Y ,y is the valuation ring of a real-valuedvaluation v y extending v such that Γ is of finite index in the value group of v y .(d) If Y = Spec( A ) , then A = T y O Y ,y , where y ranges over all points from (b) and (c).Proof. Since Y η is a normal variety over the field K , it is regular in codimension 1 and hence (b)follows. The claims (a) and (c) follow from [17, Theorem 2.6], where the set of valued points ofa normal variety over an arbitrary valuation ring are classified. It remains to prove (d). By [17,Theorem 2.4], the integral domain A is integrally closed and coherent. It follows from [17, 1.3]that A is a Pr¨ufer v -multiplication ring and hence (d) is a consequence of [17, 1.5]. Let Y be a variety over K ◦ with generic fibre Y . A horizontal cycle Z on Y is a cycle on Y ,i.e. Z is a Z -linear combination of closed subvarieties W of Y . The support supp( Z ) is the unionof all closures W in Y , where W ranges over all closed subvarieties with non-zero coefficients.Such W ’s are called prime components of the horizontal cycle Z . If the closure of every primecomponent of Z in Y has dimension k (resp. codimension p ), then we say that the horizontalcycle Z of Y has dimension k (resp. codimension p ).A vertical cycle V on Y is a cycle on Y s with real coefficients, i.e. V is an R -linear combinationof closed subvarieties W of Y s . The support and prime components are defined as usual. We saythat the vertical cycle V of Y has dimension k (resp. codimension p ) if every prime componentof V is a closed subvariety of Y s of dimension k (resp. of codimension p in Y ).A cycle Z on Y is a formal sum of a horizontal cycle Z and a vertical cycle V . The support of Z is supp( Z ) := supp( Z ) ∪ supp( V ). If the horizontal part Z and the vertical part V of Z both have dimension k (resp. codimension p ), then we say that Z has dimension k (resp. codimension p ). We say that a cycle is effective if the multiplicities in all its prime componentsare positive. If ϕ : Y ′ → Y is a flat morphism of varieties over K ◦ , then we define the pull-back ϕ ∗ ( Z )of a cycle Z on Y by using flat pull-back of the horizontal and vertical parts. The resultingcycle ϕ ∗ ( Z ) of Y ′ keeps the same codimension as Z . Similarly, we define the push-forward ofa cycle with respect to a proper morphism of varieties over K ◦ . This preserves the dimension ofthe cycles. We recall that the support supp( D ) of a Cartier divisor D on Y is the complement of theunion of all open subsets U for which D is given by an invertible element in O ( U ). Clearly,supp( D ) is a closed subset of Y . We say that the Cartier divisor D intersects a cycle Z of Y properly , if for every prime component W of Z , we havecodim(supp( D ) ∩ W , Y ) ≥ codim( W , Y ) + 1 . We are going to define the associated Weil divisor cyc( D ) of a Cartier divisor D on thevariety Y over K ◦ . The horizontal part of cyc( D ) is the Weil divisor corresponding to the Cartierdivisor D | Y on the generic fibre Y of Y . Thus we just need to construct the vertical part ofcyc( D ). This will be done by using the corresponding construction for admissible formal schemesover the completion of K ◦ given in [12]. This is technically rather demanding and we will freelyuse the terminology and the results from [12]. The reader might skip the details below in a firstread trusting that the algebraic intersection theory with Cartier divisors works in the usual way.In fact, we are dealing mostly with normal varieties over K ◦ in this paper and then one can use5roposition 2.11 to define the multiplicities ord( D, V ) of cyc( D ) in an irreducible component V of Y s without bothering about admissible formal schemes.To define the vertical part of cyc( D ), we may assume that v is non-trivial. Since the specialfibre remains the same by base change to the completion of K ◦ , we may also assume that v iscomplete. Let ˆ Y be the formal completion of Y along the special fibre (see [26, § K ◦ with special fibre equal to Y s . We will denote its genericfibre by Y ◦ which is an analytic subdomain of the analytification Y an of Y . Note that Y ◦ maybe seen as the set of potentially integral points (see [14, 4.9–4.13] for more details). We have amorphism ˆ Y → Y of locally ringed spaces and using pull-back, we see that the Cartier divisor D induces a Cartier divisor ˆ D on ˆ Y . By [12, Definition 3.10], we have a Weil divisor cyc( ˆ D )on ˆ Y . It follows from [12, Proposition 6.2] that the analytification of the Weil divisor cyc( D | Y )restricts to the horizontal part of cyc( ˆ D ). We define the vertical part of cyc( D ) as the verticalpart of cyc( ˆ D ).The multiplicity of cyc( D ) in an irreducible component V of Y s is denoted by ord( D, V ).Since ord(
D, V ) is linear in D , the map D cyc( D ) is a homomorphism from the group ofCartier divisors to the group of cycles of codimension 1. It follows from the definitions thatcyc( D ) is an effective cycle if D is an effective Cartier divisor. For convenience of the reader, werecall the definition of ord( D, V ) in more details to make these statements obvious.
First, we assume that K is algebraically closed and that v is complete. We repeat thatwe (may) assume v non-trivial. To define ord( D, V ), we may restrict our attention to an affineneighbourhood of the generic point ζ V of V where D is given by a single rational function f .Hence we may assume Y = Spec( A ) and D = div( f ). Then ˆ Y is the formal affine spectrum ofthe ν -adic completion ˆ A of A for any non-zero element ν in the maximal ideal of K ◦ . We notethat A := ˆ A ⊗ K ◦ K is a K -affinoid algebra and we have that Y ◦ is the Berkovich spectrum M ( A ) of A . Since Y ◦ is an analytic subdomain of Y an , we conclude that Y ◦ is reduced (see[1, Proposition 3.4.3] for a proof). Let A ◦ be the K ◦ -subalgebra of power bounded elements in A . Then Y ′′ := Spf( A ◦ ) is an admissible formal affine scheme over K ◦ with reduced specialfibre and we have a canonical morphism Y ′′ → ˆ Y . The restriction of this morphism to thespecial fibres is finite and surjective (see [14, 4.13] for the argument). In particular, there is ageneric point y ′′ of Y ′′ s over ζ V . It follows from [1, Proposition 2.4.4] that there is a unique ξ ′′ in the generic fibre Y ◦ of Y ′′ with reduction y ′′ . Recall that the elements of Y ◦ are boundedmultiplicative seminorms on A . Since y ′′ is a generic point of the special fibre of Y ′′ = Spf( A ◦ ),the seminorm corresponding to ξ ′′ is in fact an absolute value with valuation ring equal to O Y ′′ ,y ′′ .This follows from [3, Proposition 6.2.3/5], where necessary and sufficient conditions are given forthe supremum seminorm to be a valuation. We use it to defineord( D, V ) := − X y ′′ [ e K ( y ′′ ) : e K ( V )] log | f ( ξ ′′ ) | , (1)where y ′′ is ranging over all generic points of Y ′′ s mapping to the generic point ζ V of V . If K is not algebraically closed, then we perform base change to C K . The latter is thecompletion of the algebraic closure of the completion of K . This is the smallest algebraicallyclosed complete field extending the valued field ( K, v ), and the residue field e C K is the algebraicclosure of e K [3, § Y = Spec( A ) and D = div( f ) for a rationalfunction f on Y . Let Y ′ be the base change of Y to the valuation ring C ◦ K of C K . Let ( Y ′ j ) j =1 ,...,r be the irreducible components of Y ′ . Our goal is to define ord( D, V ) in the irreducible component V of Y s . The definition will be determined by the two guidelines that cyc( D ) should be invariantunder base change to C ◦ K and that this base change should be linear in the irreducible components6 ′ j . Since we do not assume that a variety is geometrically reduced, the multiplicity m ( Y ′ j , Y ′ ) ofthe generic fibre Y ′ j of Y ′ j in the generic fibre Y ′ of Y ′ has to be considered. Note also that theabsolute Galois group Gal( e C K / e K ) acts transitively on the irreducible components of the basechange V e C K and hence the multiplicity m ( V ′ , V e C K ) is independent of the choice of an irreduciblecomponent V ′ of V e C K .We choose an irreducible component V ′ of V e C K . It is also an irreducible component of Y ′ s and hence there is an irreducible component Y ′ j containing the generic point ζ V ′ of V ′ . For Y ′ j = Spec( A ′ j ), we proceed as in 2.8. We get an admissible formal scheme Y ′′ j = Spf(( A ′′ j ) ◦ )over C ◦ K with reduced generic fibre ( Y ′ j ) ◦ = M ( A ′′ j ) and reduced special fibre ( Y ′′ j ) s with asurjective finite map onto ( Y ′ j ) s . Hence there is at least one generic point y ′′ j of ( Y ′′ j ) s mappingto ζ V ′ . Again, there is a unique point ξ ′′ j ∈ ( Y ′ j ) ◦ with reduction y ′′ j . Then ξ ′′ j extends to anabsolute value with valuation ring O Y ′′ j ,y ′′ j and (1) leads to the definitionord( D, V ) := − m ( V ′ , V e C K ) X j m ( Y ′ j , Y ′ ) X y ′′ j [ e C K ( y ′′ j ) : e C K ( V ′ )] log | f ( ξ ′′ j ) | , (2)where Y ′ j ranges over the irreducible components of Y ′ and y ′′ j ranges over the generic points of( Y ′′ j ) s lying over the generic point ζ V ′ of V ′ . Using the action of Gal( e C K / e K ), we see that thedefinition is independent of the choice of the irreducible component V ′ of V e C K . It follows fromcompatibility of base change and passing to the formal completion along the special fibre that P V ord( D, V ) V is indeed the vertical part of cyc( D ) as defined in 2.7.The Weil divisor associated to a Cartier divisor has all the expected properties. The proofsfollow from the corresponding properties in [12] or [13]. This is illustrated in the proof of thefollowing projection formula : Proposition 2.10.
Let ϕ : Y ′ → Y be a proper morphism of varieties over K ◦ and let D be aCartier divisor on Y such that supp( D ) does not contain ϕ ( Y ′ ) . As usual, we define [ Y ′ : Y ] to be the degree of the extension of the fields of rational functions if this degree is finite and otherwise. Then we have ϕ ∗ (cyc( ϕ ∗ ( D ))) = [ Y ′ : Y ]cyc( D ) . Proof.
The projection formula holds in the generic fibre [9, Proposition 2.3]. We have an inducedproper morphism ˆ ϕ : ˆ Y ′ → ˆ Y of admissible formal schemes over the completion of K ◦ andhence the projection formula follows for vertical parts from the projection formula for propermorphisms of admissible formal schemes given in [12, Proposition 4.5] and from the compatibilityof push-forward and passing to the analytification given in [12, Proposition 6.3]. Proposition 2.11.
Assume that v is non-trivial, let Y be a normal variety over K ◦ and let V be an irreducible component of Y s . If the Cartier divisor D on Y is given by the rationalfunction f in a neighbourhood of the generic point y = ζ V of V , then O Y ,y is a valuation ringfor a unique real-valued valuation v y extending v and we have ord( D, V ) = v y ( f ) .Proof. We may assume that Y = Spec( A ) and D = div( f ) for a rational function f on Y .The first claim follows from Proposition 2.3. In the following, we use the notation and theresults from 2.9. We have a generic point y ′′ j of the special fibre of the admissible formal scheme Y ′′ j := Spf(( A ′′ j ) ◦ ) lying over y . We have seen in 2.9 that the unique point ξ ′′ of the generic fibreof Y ′′ j mapping to y ′′ j extends to an absolute value with valuation ring O Y ′′ j ,y ′′ j . We conclude thatthe valuation ring O Y ′′ j ,y ′′ j dominates the valuation ring O Y ,y . Since valuation rings are maximal7ith respect to dominance of local rings in a given field, we conclude that − log | f ( ξ ′′ ) | = v y ( f )and hence (2) simplifies toord( D, V ) := v y ( f ) m ( V ′ , V e C K ) X j m ( Y ′ j , Y ′ ) X y ′′ j [ e C K ( y ′′ j ) : e C K ( V ′ )] , (3)where Y ′ j ranges over the irreducible components of Y ′ and y ′′ j ranges over the generic points of( Y ′′ j ) s lying over the generic point ζ V ′ of V ′ . By [13, Lemma 4.5], we have m ( V ′ , V e C K ) = X j m ( Y ′ j , Y ′ ) X y ′′ j [ e C K ( y ′′ j ) : e C K ( V ′ )]proving the claim. Corollary 2.12.
The following properties hold for a Cartier divisor D on a normal variety Y over K ◦ .(a) supp( D ) = supp(cyc( D )) .(b) The Cartier divisor D is effective if and only if cyc( D ) is an effective cycle.(c) The map D cyc( D ) is an injective homomorphism from the group of Cartier divisors on Y to the group of cycles of codimension on Y .Proof. It follows easily from the definitions that supp(cyc( D )) ⊂ supp( D ) and that the Weildivisor associated to an effective Cartier divisor is an effective cycle without assuming normality.If v is trivial, the claims are classical results for divisors on normal varieties over K and so wemay assume that v is non-trivial. Then (b) follows from Propositions 2.11 and 2.3. To prove(a), the above shows that by passing to the open subset Y \ supp(cyc( D )), we may assume thatcyc( D ) = 0 and hence (a) follows from (b). Similarly, (c) is a consequence of (b). The construction of the Weil divisor associated to a Cartier divisor allows us to define aproper intersection product of a Cartier divisor with a cycle. Indeed, let D be a Cartier divisorintersecting the cycle Z on Y properly. Then we define the proper intersection product D. Z asa cycle on Y in the following way: By linearity, we may assume that Z is a prime cycle W . If W is vertical, then D restricts to a Cartier divisor on W and we define D.W := cyc( D | W ) usingalgebraic intersection theory on the variety W . If W is horizontal, then D restricts to a Cartierdivisor on the closure of W in Y and we define D.W as the associated Weil divisor. Obviously,this proper intersection product is bilinear.
Proposition 2.14.
Let D and E be properly intersecting Cartier divisors on Y which means codim(supp( D ) ∩ supp( E ) , Y ) ≥ . Then we have D. cyc( E ) = E. cyc( D ) .Proof. For the horizontal parts, this follows from algebraic intersection theory. For the verticalparts, this follows from the corresponding statement for Cartier divisors on admissible formalschemes given in [12, Theorem 5.9]. Recall that the special fiber doesn’t change after passing tothe formal completion of Y along Y s . Proposition 2.15.
Let ϕ : Y ′ → Y be a flat morphism of varieties over K ◦ and let D be aCartier divisor on Y . Then we have ϕ ∗ (cyc( D )) = cyc( ϕ ∗ ( D )) .Proof. Since ϕ is flat, the pull-back of D is well-defined as a Cartier divisor and the claimfollows from [13, Proposition 4.4(d)], where it is proved for Cartier divisors on admissible formalschemes. 8 .16. We say that two cycles D and D of codimension 1 on the variety Y over K ◦ are rationallyequivalent if there is a non-zero rational function f on Y such that D − D = cyc(div( f )). The first Chow group CH ( Y ) of Y is defined as the group of cycles of codimension 1 modulo rationalequivalence. It follows from Proposition 2.15 that rational equivalence is compatible with flatpull-back.Two Cartier divisors D and D on Y are said to be linearly equivalent if there is a non-zerorational function f on Y such that D − D = div( f ). The group of Cartier divisors modulolinear equivalence is isomorphic to Pic( Y ) using the map D O ( D ).We may use rational equivalence to define a refined intersection theory with pseudo divisorson a variety Y over K ◦ with the same properties as in [9, Chapter 2]. The proofs follow directlyfrom [12] and [13, § In this section, (
K, v ) is a valued field with valuation ring K ◦ , residue field e K and value groupΓ := v ( K × ) ⊂ R . As usual, T = Spec( K ◦ [ M ]) is the split torus of rank n with generic fibre T and N is the dual of the free abelian group M .We will start with the definition of a T -toric scheme over K ◦ . Then we will recall from [14]the basic construction of a normal T -toric scheme associated to a Γ-admissible fan. The overallgoal of this paper is to show that every normal toric variety over K ◦ arises in this way. At theend of this section, we will study projective toric varieties over K ◦ with a linear action of thetorus. They can also be understood purely in combinatorial terms, but they are not necessarilynormal. We will use them in § Definition 3.1. A T -toric scheme over the valuation ring K ◦ is an integral separated flatscheme Y over K ◦ such that the generic fiber Y η contains T as an open subset and such thatthe translation action T × K T → T extends to an algebraic action T × K ◦ Y → Y over K ◦ .A homomorphism (resp. isomorphism ) of T -toric schemes is an equivariant morphism (resp.isomorphism) which restricts to the identity on T . A T -toric scheme of finite type over K ◦ iscalled a T -toric variety .Note that if Y is a T -toric variety over K ◦ , then Y η is a T -toric variety over K . In order toconstruct examples of T -toric schemes and to see how they can be described by the combinatoricsof some objects in convex geometry, we need to introduce and to study the following algebrasassociated to Γ-admissible cones. A cone σ ⊂ N R × R + is called Γ -admissible if it can be written as σ = k \ i =1 { ( ω, s ) ∈ N R × R + | h u i , ω i + sc i ≥ } , u , . . . , u k ∈ M, c , . . . , c k ∈ Γ , and does not contain a line. For such a cone σ , we define K [ M ] σ := { X u ∈ M α u χ u ∈ K [ M ] | cv ( α u ) + h u, ω i ≥ ∀ ( ω, c ) ∈ σ } and V σ := Spec( K [ M ] σ ). It is easy to see that K [ M ] σ is an M -graded K ◦ -algebra and hence wehave a canonical T -action on V σ . 9 roposition 3.3. Let σ be a Γ -admissible cone in N R × R + . Then V σ is a normal T -toric schemeover K ◦ . If v is a discrete valuation, then V σ is always a T -toric variety. If v is not a discretevaluation, then V σ is a T -toric variety over K ◦ if and only if the vertices of σ ∩ ( N R × { } ) arecontained in N Γ × { } .Proof. Normality is proven in [14, Proposition 6.10]. If the valuation is discrete, then [14,Proposition 6.7] shows that V σ is a T -toric variety. If v is not discrete, then the equivalencein the last claim is proved in [14, Propositions 6.9]. A Γ -admissible fan
Σ in N R × R + is a fan consisting of Γ-admissible cones. Given a Γ-admissible fan Σ, we glue the normal affine T -toric schemes V σ , σ ∈ Σ, along the open subschemescoming from their common faces. The result is a normal T -toric scheme Y Σ . Similarly to theclassical case of toric varieties over a field, the properties of the T -toric schemes Y Σ may bedescribed by the combinatorics of the cones Σ. For details, we refer to [14].Now we review the construction of projective T -toric schemes which are not necessarily normal(see [14, § K ◦ butjust those which have a linear action of the torus, see [14, Proposition 9.8]. For the correspondingprojective toric varieties over a field, we refer to Cox–Little–Schenk [6, § § Given R ∈ Z + , we choose projective coordinates on the projective space P RK ◦ . Let A =( u , . . . , u R ) ∈ M R +1 and y = ( y : · · · : y R ) ∈ P R ( K ). The height function of y is defined as a : { , ..., R } → Γ ∪ {∞} , j a ( j ) := v ( y j ) . The action of T on P RK ◦ is given by( t, x ) ( χ u ( t ) x : · · · : χ u R ( t ) x R ) . We define the projective toric variety Y A,a to be the closure of the orbit T y . The generic fiber Y A,a is a toric variety respect to the torus
T /
Stab( y ). It follows from [14, 9.2] that Y A,a is a T -toric variety over K ◦ with respect to the split torus over K ◦ with generic fiber T /
Stab( y ).The weight polytope Wt( y ) is defined as the convex hull of A ( y ) := { u j | a ( j ) < ∞} . The weight subdivision Wt( y , a ) is the polytopal complex with support Wt( y ) obtained by projectingthe faces of the convex hull of { ( u j , λ j ) ∈ M R × R + | j = 0 , . . . , R ; λ j ≥ a ( j ) } . We will see in thenext result that the orbits of Y A,a can be read off from the weight subdivision.
Proposition 3.6.
There is a bijective order preserving correspondence between faces Q of theweight subdivision Wt ( y , a ) and T -orbits Z of the special fiber of Y A,a given by Z = { x ∈ ( Y A,a ) s | x j = 0 ⇔ u j ∈ A ( y ) ∩ Q } . Proof.
This is the content of [14, Proposition 9.12].
We recall that (
K, v ) is a valued field with valuation ring K ◦ , residue field e K and value groupΓ ⊂ R . Let T = Spec( K ◦ [ M ]) be the split torus over K ◦ with generic fiber T . The free abeliangroup M of rank n is isomorphic to the character group of T . For an element u ∈ M , thecorresponding character is denoted by χ u . Let N = Hom( M, Z ) be the dual abelian group of M .10s we have seen in the previous section, a Γ-admissible cone σ in N R × R + induces a normalaffine T -toric scheme V σ = Spec( K [ M ] σ ). This is a T -toric variety if and only if the vertices of σ ∩ ( N R × { } ) are contained in N Γ × { } or if v is discrete.In this section, we will show that every normal affine T -toric variety Y = Spec( A ) has thisform proving Theorem 1. We may assume that the valuation is non-trivial as in the classical caseof normal toric varieties over a field, the statement is well known (see [15, ch. I, Theorem 1]).The T -action induces an M -grading A = L m ∈ M A m on the K ◦ -algebra A . Since T is an opendense orbit of Y , we may and will assume that A is a subalgebra of the quotient field K ( M ) of K [ M ].The proof of Theorem 1 will be done in three steps along the classical proof dealing with theadditional complications of a non-trivial value group and of a non-noetherian ring A . In Lemma4.1, we will associate to the M -graded subalgebra A a semigroup S in M × Γ. Then in a secondstep, we will show in Lemmata 4.2 and 4.3 that S gives rise to a Γ-admissible cone σ in N × R + .Finally, we will prove in Proposition 4.4 that Y = V σ by using an approximation argument. Lemma 4.1.
The set S := { ( m, v ( a )) ∈ M × Γ | aχ m ∈ A \{ }} is a saturated semigroup in M × Γ .Proof. Obviously, the set S is a semigroup. Let k ( m, v ( a )) ∈ S for m ∈ M , a ∈ K \ { } and k ∈ Z + \ { } , i.e. ( aχ m ) k ∈ A . By normality of A , we get aχ m ∈ A and hence ( m, v ( a )) ∈ S . Lemma 4.2.
There are M -homogeneous generators a χ m , . . . , a k χ m k of A . Moreover, thesemigroup S from Lemma 4.1 and the set { (0 , , ( m i , v ( a i )) | i = 1 , . . . , k } generate the samecone in M R × R .Proof. Since Y is a variety over K ◦ , it is clear that A is a finitely generated K ◦ -algebra. Usingthat A is an M -graded algebra, we find generators a χ m , . . . , a k χ m k of A . Obviously, every( m i , v ( a i )) is contained in the cone generated by S which we denote by cone( S ). Since thevaluation v is non-trivial, it is clear that (0 , ∈ cone( S ).It remains to show that the cone generated by S is contained in the cone generated by { (0 , , ( m i , v ( a i )) | i = 1 , . . . , k } . An element of cone( S ) is a finite sum P j α j ( u j , v ( b j )) with α j ∈ R + and ( u j , v ( b j )) ∈ S with b j χ u j ∈ A . Using the above generators, we get b j χ u j = λ ( j ) ( a χ m ) l ( j )1 · · · ( a k χ m k ) l ( j ) k for λ ( j ) ∈ K ◦ \ { } , l ( j )1 , . . . , l ( j ) k ∈ Z + . This implies v ( b j ) = v ( λ ( j ) ) + k X i =1 l ( j ) i v ( a i ) u j = k X i =1 l ( j ) i m i . We conclude that the element P j α j ( u j , v ( b j )) of cone( S ) is equal to X j α j k X i =1 l ( j ) i m i , v ( λ ( j ) ) + k X i =1 l ( j ) i v ( a i ) ! = X j α j (0 , v ( λ ( j ) )) + X j X i α j l ( j ) i ( m i , v ( a i ))= (0 , λ ) + X i λ i ( m i , v ( a i ))with λ := P j α j v ( λ ( j ) ) ∈ R + and λ i := P j α j l ( j ) i ∈ R + . This proves the lemma.11 emma 4.3. The set σ := { ( ω, s ) ∈ N R × R | h u, ω i + ts ≥ ∀ ( u, t ) ∈ S } is a Γ -admissible conein N R × R + .Proof. By definition, σ is the dual cone of the cone generated by S . From Lemma 4.2, we have σ = k \ i =1 { ( ω, s ) ∈ N R × R + | h m i , ω i + sv ( a i ) ≥ } . It remains to show that σ doesn’t contain a line. Suppose σ contains a line. Then we have R · ( ω, t ) ⊂ σ for some ( ω, t ) ∈ N R × R + . Since σ ⊂ N R × R + , we must have t = 0. Therefore theline is of the form R · ( ω, ⊂ N R × { } . For any aχ u ∈ A \ { } , we have ( u, v ( a )) ∈ S and hence0 ≤ h ( u, v ( a )) , ( λω, i = λ h u, ω i ∀ λ ∈ R . This proves u ∈ ω ⊥ . Choosing a basis { u , . . . , u n } for M such that u , . . . , u n − ∈ ω ⊥ , we get A ⊂ K [ χ ± u , . . . , χ ± u n − ]. On the other hand, Y is a T -toric variety and hence the quotientfield of A is K ( χ ± u , . . . , χ ± u n ). This is a contradiction and hence σ doesn’t contain any line.We conclude that σ is Γ-admissible. Proposition 4.4.
Let Y = Spec( A ) be an affine normal T -toric variety over K ◦ . Then Y = V σ for the Γ -admissible cone σ defined in Lemma 4.3.Proof. We have to show K [ M ] σ = A . Take any aχ m ∈ A \ { } . Since ( m, v ( a )) ∈ S , we get h m, ω i + t · v ( a ) ≥ ω, t ) ∈ σ and hence aχ m ∈ K [ M ] σ . This proves A ⊂ K [ M ] σ .To prove the reverse inclusion, we take aχ m ∈ K [ M ] σ \ { } . By definition, ( m, v ( a )) is in thedual cone ˇ σ of σ . Using biduality of convex polyhedral cones (see [8, § m, v ( a )) is contained in the cone in M R × R generated by S . By Lemma 4.2, we get( m, v ( a )) = κ (0 ,
1) + k X i =1 λ i ( m i , v ( a i )) , κ, λ i ∈ R + . From this, we deduce the following equivalent system of equations m = X i λ i m i (4) v ( a ) = κ + X i λ i v ( a i ) . (5)Now we show that it is always possible to choose all λ i ∈ Q + . We may assume that λ i > i , otherwise we omit these coefficients. Let b , . . . , b s be a basis in Q k for the solutions of thehomogeneous equation associated to (4) and let µ ∈ Q k be a particular solution for (4). We willuse the coordinates ( b (1) j , . . . , b ( k ) j ) for the vector b j of the basis. The space of solutions L is givenby L = { µ + s X j =1 ρ j b j | ρ j ∈ R , j = 1 , . . . , s } . Since λ := ( λ i ) ∈ R k + is a solution of (4), there exist ρ j ∈ R ( j = 1 , . . . , s ) such that λ = µ + X j ρ j b j . ρ j ∈ Q close to ρ j , i.e. ρ j = ˆ ρ j + ǫ j with | ǫ j | small. Then ˆ λ = µ + X j ˆ ρ j b j is also a solution of (4) in Q k which is close to λ . In particular, we may choose | ε j | so small thatall ˆ λ i >
0. Explicitly we have λ ... λ k = ˆ λ ...ˆ λ k + s X j =1 ǫ j b j . Inserting this in (5), we get v ( a ) = κ + X i ˆ λ i + X j ǫ j b ( i ) j v ( a i )= κ + X i ˆ λ i v ( a i ) + X i X j ǫ j b ( i ) j v ( a i ) . With α := P i (cid:16)P j ǫ j b ( i ) j (cid:17) v ( a i ) = P j ǫ j P i b ( i ) j v ( a i ), we get v ( a ) = κ + α + X i ˆ λ i v ( a i ) . It is easy to see that we may choose ǫ , . . . , ǫ s in a small neighbourhood of 0 such that κ + α ≥ m, v ( a )) = ( κ + α )(0 ,
1) + k X i =1 ˆ λ i ( m i , v ( a i )) , κ + α ∈ R + , ˆ λ i ∈ Q + . (6)The above shows that ( m, v ( a )) = (0 , κ ) + P i λ i ( m i , v ( a i )) with λ i ∈ Q + and κ ∈ R + . Let R be a positive integer such that Rλ i ∈ Z + for i = 1 , . . . , k . Then we get R ( m, v ( a )) = R (0 , κ ) + X i Rλ i ( m i , v ( a i )) . This proves in particular that Rκ ∈ Γ. Since (0 , Rκ ) , ( m i , v ( a i )) ∈ S ( i = 1 , . . . , k ) and Rλ i ∈ Z + ,we conclude that ( Rm, Rv ( a )) is also in the semigroup S . This means that there is a b ∈ K with v ( b ) = v ( a R ) such that bχ Rm ∈ A . Since a R = ub for a unit in K ◦ , we get ( aχ m ) R ∈ A . Bynormality of A , this implies that aχ m ∈ A . We conclude that K [ M ] σ = A and hence Y = V σ . Proof of Theorem 1.
We assume that v is not a discrete valuation. We have seen in Propositions4.4 and 3.3 that the map σ V σ from the set of those Γ-admissible cones in N R × R + for whichthe vertices of σ ∩ ( N R × { } ) are contained in N Γ × { } to the set of isomorphism classes of affinenormal T -toric varieties over K ◦ is surjective. By [14, Proposition 6.24], we can reconstruct thecone σ from the T -toric scheme V σ by applying the tropicalization map to the set of integralpoints of T ∩ V σ and hence the correspondence is indeed bijective. If v is a discrete valuation, thenthe same argument works if we omit the additional condition on the vertices of the cones.13 Construction of the Cartier divisor
Let (
K, v ) be a valued field with valuation ring K ◦ , value group Γ = v ( K × ) ⊂ R and residuefield e K . Let T be the split torus of rank n over K ◦ . In this section, we consider a non-emptyaffine open subset U in a normal T -toric variety Y over K ◦ .The main goal is to construct an effective Cartier divisor D on the smallest T -invariant opensubset U of Y containing U with support equal to U \ U . This will be achieved in Proposition5.12. We will start by noting in Proposition 5.1 that normality of Y yields that the complementof U is a Weil divisor. Using the divisorial intersection theory from §
2, we will deduce inProposition 5.9 that all translates of this Weil divisor are rationally equivalent on U . This willbe enough to deduce that the Weil divisor is actually a Cartier divisor on U proving Proposition5.12. In fact, it will also follow in Corollary 5.14 that O ( D ) is generated by global sections. Thiswill be important in the next section, where we will show that D is ample and gives rise to anequivariant immersion of U to a projective space with a linear T -action. This will allow us in § Proposition 5.1.
Let U be a non-empty affine open subset of a normal variety Y over K ◦ .Then every irreducible component of Y \ U has codimension in Y .Proof. By removing the irreducible components of Y \ U of codimension 1, we may assume that Y \ U has no irreducible components of codimension 1. Then we have to prove U = Y . We mayassume that Y is an affine variety Spec( A ). Using Proposition 2.3(d), we get O ( U ) = O ( Y )and hence the affine varieties U and Y are equal.In the following result, we will use the notions introduced in Section 2. Proposition 5.2.
Let p be the canonical projection of T × K ◦ Y onto the variety Y over K ◦ and let D be a cycle of codimension in T × K ◦ Y . Then there is a cycle D ′ on Y of codimension such that p ∗ ( D ′ ) is rationally equivalent to D .Proof. We note that irreducible components of ( T × K ◦ Y ) s are given by T s × e K V with V rangingover the irreducible components of Y s . We conclude that every vertical cycle of codimension 1in T × K ◦ Y is the pull-back of a vertical cycle of codimension 1 in Y . This reduces the claimto the horizontal parts where it is a standard fact from algebraic intersection theory on varietiesover a field [9, Proposition 1.9]. The generic fibre of T is denoted by T . Let T ◦ be the affinoid torus in T an given by { x ∈ T an | | x ( x ) | = · · · = | x n ( x ) | = 1 } in terms of torus coordinates x , . . . , x n and let Y be avariety over K ◦ with generic fibre Y . For t ∈ T ◦ ( K ), the reduction ˜ t ∈ T s ( e K ) is well-defined. Let i t : Y → T × K ◦ Y be the embedding over Y induced by the integral point of T corresponding to t . We are going to define the pull-back i ∗ t ( Z ) for every cycle Z on T × K ◦ Y which satisfies thefollowing flatness condition : We assume that every component of the horizontal (resp. vertical)part of Z is flat over T (resp. T s ).Since i t induces a regular embedding of Y into T × K Y (resp. of Y s into T s × e K Y s ), thepull-back of the horizontal part (resp. vertical part) of Z is a well-defined cycle on Y (resp. Y s )(see [9, Chapter 6]). We define i ∗ t ( Z ) as the sum of these two pull-backs. Clearly, this pull-backkeeps the codimension and is linear in Z . For t ∈ T ◦ ( K ) with coordinates t := x ( t ) , . . . , t n := x n ( t ), let D t j be the Cartier divisoron T × K ◦ Y given by pull-back of div( x j − t j ) with respect to the canonical projection onto14 = ( G nm ) K ◦ . Let Z be a cycle on T × K ◦ Y satisfying the flatness condition from 5.3. Then wemay use the proper intersection product with Cartier divisors from 2.13 to get( i t ) ∗ ( i ∗ t ( Z )) = D t . . . D t n . Z . Indeed, the flatness condition ensures that the right hand side is a proper intersection productand hence the claim follows from [9, Example 6.5.1]. By Proposition 2.14, the proper intersectionproduct on the right is symmetric with respect to the Cartier divisors.Let Y , Y ′ be varieties over K ◦ and let ϕ : T × K ◦ Y → T × K ◦ Y ′ be a flat morphismover T . The point t ∈ T ◦ ( K ) corresponds to an integral point of T inducing a flat morphism ϕ t : Y → Y ′ by base change from ϕ . We recall from 2.5 that we have also introduced thepull-back with respect to flat morphisms. The following result shows some functoriality with theabove pull-backs. We will use the canonical projection p : T × K ◦ Y ′ → Y ′ . Proposition 5.5.
Under the hypothesis above, let Z ′ be a cycle of Y ′ . Then the cycle ϕ ∗ ( p ∗ ( Z ′ )) satisfies the flatness condition from 5.3 and we have i ∗ t ( ϕ ∗ ( p ∗ ( Z ′ ))) = ϕ ∗ t ( Z ′ ) .Proof. The cycle p ∗ ( Z ) satisfies the flatness condition. Using that ϕ is a flat morphism over T ,we deduce that ϕ ∗ ( p ∗ ( Z ′ )) also fulfills the flatness condition . Since the pull-backs are definedfor horizontal and vertical parts in terms of the corresponding operations for varieties over fields,the claim follows from [9, Proposition 6.5], where the functoriality for pull-backs of cycles ofvarieties has been proved. Lemma 5.6.
Let Y be a variety over K ◦ and let t ∈ T ◦ ( K ) . Suppose that g is a rationalfunction on T × K ◦ Y such that every irreducible component of the support of the restriction of div( g ) to the generic fibre is flat over T . Then g ( t, · ) is a rational function on Y and we have i ∗ t (cyc(div( g ))) = cyc(div( g ( t, · ))) .Proof. The flatness assumption yields the first claim immediately. Note that the vertical com-ponents of cyc(div( g )) are automatically flat over T s and hence we get a well-defined cycle i ∗ t (cyc(div( g ))) on Y . The second claim follows easily from the fact that we may write i ∗ t as an n -fold proper intersection product with Cartier divisors (see 5.4) and from Proposition 2.14. Proposition 5.7.
Let Y be a variety over K ◦ . Then pull-back with respect to the canonicalprojection p : T × K ◦ Y → Y induces an isomorphism p ∗ : CH ( Y ) → CH ( T × K ◦ Y ) .Proof. By 2.16, p ∗ is compatible with rational equivalence and hence it is well-defined on theChow groups. Surjectivity follows from Proposition 5.2. Suppose that D is a cycle of codimension1 on Y such that p ∗ ( D ) is rationally equivalent to 0 on T × K ◦ Y . Using Lemma 5.6 for the unitelement e in T ◦ ( K ), we deduce that D is rationally equivalent to 0. This proves injectivity.We have a similar statement for Picard group as pointed out by Qing Liu and C. P´epin. Proposition 5.8.
Let Y be a normal variety over K ◦ . Then pull-back with respect to p inducesan isomorphism Pic( Y ) → Pic( T × K ◦ Y ) .Proof. This statement was proved in [14, Remark 9.6] based on an argument of Qing Liu and C.P´epin.Now let Y be a normal T -toric variety over K ◦ and let D be a cycle of codimension 1 in Y .Note that t ∈ T ◦ ( K ) acts on Y and we denote by D t the pull-back of D with respect to this flatmorphism. Proposition 5.9.
Under the hypothesis above, D t is rationally equivalent to D . roof. Let σ : T × K ◦ Y → Y be the torus action on Y . It follows from Propositions 5.2 that σ ∗ ( D ) is rationally equivalent to p ∗ ( D ′ ) for a cycle D ′ of codimension 1 in Y . Then Lemma 5.6and Proposition 5.5, applied for the unit element e , show that D is rationally equivalent to D ′ .By 2.16, we conclude that σ ∗ ( D ) is rationally equivalent to p ∗ ( D ). If we apply Proposition 5.5again, but now in t instead of e , we get the claim. Lemma 5.10.
Let U be a non-empty open subset of the T -toric variety Y over K ◦ and let U := S t ∈ T ◦ ( K ) t U . Then U is the smallest T -invariant (open) subset containing U .Proof. Consider the subset S of T such that translation with its elements leaves U invariant.The subset S ∩ T s is equal to the stabilizer of Y s \ U s and hence it is an algebraic subgroup of T s . By construction, it contains T s ( e K ) and hence it is equal to T s . We use the same argumentfor the points of S contained in the generic fibre T = T η . Again, S ∩ T is an algebraic subgroupcontaining T ◦ ( K ). Since T ◦ is an n -dimensional affinoid torus, we conclude that T ◦ ( K ) is Zariskidense in T and hence the algebraic subgroup is the torus T over K . We conclude that U is T -invariant. This proves the claim immediately. Since the torus T s acts continuously on the discrete set of the generic points of Y s , everysuch generic point is fixed under the action. We conclude that every irreducible component of Y s is invariant under the T -action. This means that the special fibres of U and U have thesame generic points. We have seen in Proposition 5.1 that U \ U is a union of irreduciblecomponents of codimension 1 and hence every such irreducible component is horizontal. Let D be the horizontal cycle on U given by the formal sum of these irreducible components. Proposition 5.12.
Under the hypothesis above, there is a unique Cartier divisor D on U suchthat D = cyc( D ) . Moreover, this Cartier divisor is effective.Proof. For t ∈ T ◦ ( K ), Proposition 5.9 yields a non-zero rational function f t on U such that D − D t = cyc(div( f t )). Since U \ t − U is equal to the support of D t , we deduce that therestriction of D to t − U is the Weil divisor given by the rational function f t on t − U . ByCorollary 2.12, the Cartier divisor on a normal variety is uniquely determined by its associatedWeil divisor. This yields immediately that { ( t − U , f t ) | t ∈ T ◦ ( K ) } is a Cartier divisor on U with associated Weil divisor D and uniqueness follows as well. By Corollary 2.12, the Cartierdivisor D is effective. Proposition 5.13.
Let σ : T × K ◦ Y → Y be the torus action of the normal T -toric variety Y over K ◦ and let D be the Cartier divisor from Proposition 5.12. Then σ ∗ ( D ) is linearlyequivalent to p ∗ ( D ) .Proof. The unit element e in T ◦ ( K ) induces the section i e of σ and p . Then the claim followsfrom Proposition 5.8. Another way to deduce the claim is to use the corresponding statementfor cycles of codimension 1 (see Proposition 5.7) together with Corollary 2.12. Corollary 5.14.
Let D be the Cartier divisor from Proposition 5.12 and let D t be its pull-backwith respect to translation by t ∈ T ◦ ( K ) . Then the invertible sheaves O ( D t ) and O ( D ) on U are isomorphic. Moreover, O ( D ) is generated by global sections.Proof. The first claim follows from Proposition 5.13 by applying i ∗ t . By Proposition 5.12, s D t isa global section with support U \ t − U and hence the second claim follows from the first.16 Linearization and immersion into projective space
Let T be the split torus of rank n over K ◦ and let Y be a normal T -toric variety over K ◦ . Wedenote by µ : T × K ◦ T → T the multiplication map, by σ : T × K ◦ Y → Y the group actionand by p : T × K ◦ Y → Y the second projection. As in the previous section, we considera non-empty affine open subset U of Y and the smallest T -invariant open subset U of Y containing U . In Proposition 5.12, we have constructed an effective Cartier divisor D on U with supp( D ) = U \ U such that cyc( D ) is a horizontal cycle with all multiplicities equal to1. In this section, we will see that O ( D ) has a T -linearization (Proposition 6.3) and is ample(Proposition 6.7) leading in 6.8 to a T -equivariant immersion into a projective space with linear T -action. This will be summarized in Proposition 6.9 which reduces the proof of Sumihiro’stheorem to an easier projective variant shown in the next section. Definition 6.1.
First, we recall the definition of a T -linearization of a line bundle L on a toricvariety (see [20] for details). Geometrically, a T -linearization is a lift of the torus action on Y to an action on L such that the zero section is T -invariant. In terms of the underlying invertiblesheaf L , a T -linearization is an isomorphism φ : σ ∗ L → p ∗ L , of sheaves on T × K ◦ Y satisfying the cocycle condition p ∗ φ ◦ (id T × σ ) ∗ φ = ( µ × id Y ) ∗ φ, (7)where p : T × K ◦ T × K ◦ Y → T × K ◦ Y is the projection to the last two factors.We need the following application of a result of Rosenlicht. Lemma 6.2.
For every f ∈ O ( T × K ◦ U ) × , there is a character χ on T and a g ∈ O ( U ) × suchthat f = χ · g .Proof. Note that O ( T ) × is the set of characters on T multiplied by units in K ◦ . Then the claimfollows from [23, Theorem 2], where it is proved in the case of fields. Proposition 6.3.
The invertible sheaf O ( D ) has a T -linearization.Proof. By Proposition 5.13, we have an isomorphism φ : σ ∗ L → p ∗ L for the invertible sheaf L := O ( D ). Both sides of (7) are isomorphisms between the sameinvertible sheaves on T × K ◦ T × K ◦ U and hence there is a unique f ∈ O ( T × K ◦ T × K ◦ U ) × suchthat that the left hand side is obtained by multiplying the right hand side with f . We may choose φ such that we have the canonical isomorphism over { e } × U and hence we get f ( e, · , · ) = 1and f ( · , e, · ) = 1. By Lemma 6.2, there are characters χ , χ on T and g ∈ O ( U ) × such that f ( t , t , u ) = χ ( t ) χ ( t ) g ( u ) for all t , t ∈ T ( K ) and u ∈ U ( K ). Since f ( e, e, u ) = 1, we get g = 1. Therefore f ( t , t , u ) = χ ( t ) χ ( t ) = f ( t , e, u ) f ( e, t , u ) = 1 . By density of the K -rational points, we get f = 1 and (7) holds.17 .4. The T -linearization on L = O ( D ) induces a dual action of T on the space H ( U , L ) ofglobal sections, given by the composition ˆ σ of the canonical K ◦ -linear maps H ( U , L ) → H ( T × K ◦ U , σ ∗ L ) → H ( T × K ◦ U , p ∗ L ) → H ( T , O T ) ⊗ K ◦ H ( U , L ) , where the last isomorphism comes from the K¨unneth formula (see [16]). We refer to [20, Chapter1, Definition 1.2] for the definition of a dual action. This was written for vector spaces over abase field, but the same definition applies in case of a free K ◦ -module. Since V := H ( U , L )is a torsion free K ◦ -module, V is indeed free (see [14, Lemma 4.2] for a proof). A dual actionmeans that the torus T acts linearly on the possibly infinite dimensional projective space P ( V ) =Proj( K ◦ [ V ]).The dual action ˆ σ induces an action of t ∈ T ◦ ( K ) on V which we denote by s t · s . For s ∈ V = H ( U , L ), the action is geometrically given by ( t · s )( u ) = t − ( s ( tu )), u ∈ U , where t − operates on the underlying line bundle using the linearization. Lemma 6.5.
Let x , . . . , x k be affine coordinates of U considered as rational functions on U .Then there exists ℓ ∈ Z + such that for every i ∈ { , . . . , k } , the meromorphic section s i := x i s ℓD of O ( ℓD ) is in fact a global section. Here, s ℓD denotes the canonical global section of O ( ℓD ) .Proof. Using the theory of divisors from §
2, we get the identitycyc(div( x i )) = X j m ij Z j + V of cycles on U , where Z j are the irreducible components of U \ U and where V is an effectivecycle of codimension 1 in U which meets U . By construction (see Proposition 5.12), we havecyc( D ) = D = P j Z j . For ℓ := − min ij { m ij , } , we getcyc(div( x i )) + ℓ D = X j m ij Z j + ℓ D + V = X j ( m ij + ℓ ) Z j + V ≥ . Therefore the Weil divisor div( x i ) + ℓD is effective. By Corollary 2.12, we conclude that x i s ℓD is a global section of O ( ℓD ). By Proposition 2.2, Y is a noetherian topological space and therefore U is quasicompact.Using Lemma 5.10, there is a finite subset S of T ◦ ( K ) such that U = S t ∈ S t − U . We haveseen in Lemma 6.5 that the affine coordinates x , . . . , x k of U induce global sections s , . . . , s k of O ( ℓD ). Then the dual action from 6.4 gives global sections t · s , . . . , t · s k of O ( ℓD ) induced byaffine coordinates of t − U . We conclude that ( t · s j ) t ∈ S,j =1 ,...,k generate O ( ℓD ). By construction,the global section t · s D has support U \ t − U . We get a morphism ψ : U → P R ′ K ◦ , u ( · · · : t · s ( u ) : · · · : t · s k ( u ) : t · s ℓD ( u ) : · · · ) t ∈ S with R ′ := | S | ( k + 1) −
1. Note that this map is well defined because O ( ℓD ) is generated bythese global sections, and we have ψ ∗ ( O P R ′ K ◦ (1)) ≃ O ( ℓD ). Proposition 6.7.
The morphism ψ is an immersion and hence L is ample.Proof. For t ∈ S , the support of the Cartier divisor div( t · s D ) = D t is equal to U \ t − U . Let y j be the coordinate of P R ′ K ◦ corresponding to t · s ℓD with respect to the morphism ψ . Then weget ψ − { y j = 0 } = t − U . Since t · x , . . . , t · x k are affine coordinates on t − U , we concludeeasily that ψ restricts to a closed immersion of t − U into the open subvariety { y j = 0 } of P R ′ K ◦ .Since these open subvarieties form an open covering of P R ′ K ◦ , we may use [11, Corollaire 4.2.4] toconclude that the morphism ψ is an immersion and hence O ( D ) is ample.18 .8. Let V ℓ be the submodule of V ℓ := H ( U , L ℓ ) which is generated by the global sections( t · s j ) t ∈ S,j =1 ,...,k and ( t · s ℓD ) t ∈ S used in the definition of ψ in 6.6. Since O ( ℓD ) has a T -linearization, we get a dual action ˆ σ of T on V ℓ similarly to 6.4.A K ◦ -submodule W of V ℓ is called invariant under the dual action of T if ˆ σ ( W ) ⊂ A ⊗ K ◦ W for A := H ( T , O T ) = K ◦ [ M ]. The lemma on p. 25 of [20] generalizes in a straightforward mannerto our setting and hence there is a finitely generated submodule W of V ℓ which is invariant underthe dual action of T and contains V ℓ . Since W is K ◦ -torsion free, we conclude that W is a free K ◦ -module of finite rank R + 1.We get a morphism i : U → P ( W ) with i ∗ ( O P ( W ) (1)) ∼ = O ( ℓD ). The dual action of T on W induces a linear action of T on the projective space P ( W ). By construction, i is T -equivariant.Since i factorizes through ψ , we deduce from Proposition 6.7 that i is an immersion.Recall that T = Spec( K ◦ [ M ]) is the split torus of rank n . We summarize our findings: Proposition 6.9.
Let U be a non-empty affine open subset of the normal T -toric variety Y over K ◦ and let U be the smallest T -invariant open subset of Y containing U . Then there isa T -equivariant open immersion of U into a projective T -toric variety Y A,a given by A ∈ M R +1 and height function a as in 3.5.Proof. Let i : U → P ( W ) be the T -equivariant immersion from 6.8. Then the closure Y of i ( U )in P ( W ) is a projective T -toric variety over K ◦ on which T -acts linearly. We choose a K -rationalpoint y in the open dense orbit of i ( U ). By [14, Proposition 9.8], there are suitable coordinateson P ( W ) and A ∈ M R +1 such that Y = Y A,a for the height function a of y defined in 3.5. In this section, we will finally prove Sumihiro’s theorem for normal toric varieties over K ◦ asannounced in Theorem 2. Sumihiro’s theorem is wrong for arbitrary non-normal toric varietieseven over a field (see [6, Example 3.A.1] for a projective counterexample). However, we willshow first in this section that Sumihiro’s theorem holds for open invariant subsets of projectivetoric varieties over K ◦ with a linear torus action. Note that such projective toric varieties arenot necessarily normal. As a consequence of Proposition 6.9, we will obtain Sumihiro’s theoremfor normal toric varieties over K ◦ . At the end, Theorem 3 will easily follow from Sumihiro’stheorem similarly as in the classical case of normal toric varieties over a field.In this section, ( K, v ) will be a valued field with value group Γ = v ( K × ) ⊂ R . Moreover, T = Spec( K ◦ [ M ]) is a split torus over the valuation ring K ◦ of rank n and we consider aprojective T -toric variety over K ◦ with a linear T -action. By [14, Proposition 9.8] the latter is atoric subvariety Y A,a of P RK ◦ as in 3.5 for A ∈ M R +1 , height function a and suitable projectivecoordinates x , . . . , x R . We fix a point z ∈ Y A,a and a closed T -invariant subset Y of Y A,a with z Y . Since Y isa closed subset of the ambient projective space P RK ◦ , there is a k ∈ Z + and s ∈ H ( P RK ◦ , O ( k ))such that s | Y = 0 and s ( z ) = 0.Obviously, V := H ( P RK ◦ , O ( k )) is a free K ◦ -module of finite rank. The linear T -action on P RK ◦ induces a linear representation of T on V , i.e. a homomorphism S : T → GL ( V ) of groupschemes over K ◦ . We say that s ∈ V is semi-invariant if there is u ∈ M such that S t ( s ) = χ u ( t ) s for every t ∈ T and for the character χ u of T associated to u . In the following, the K ◦ -submodule W := { s ∈ H ( P RK ◦ , O ( k )) | ∃ λ ∈ K ◦ \{ } s. t. λs | Y = 0 } V will be of interest. Note that W is equal to the set of global sections s of O ( k ) whichvanish on the generic fiber Y η . Since Y is T -invariant, it is clear that W is invariant under the T -action. The multiplicative torus T = T K is split over K and hence the vector space W K has asimultaneous eigenbasis for the T -action. This is well-known in the theory of toric varieties overa field and follows from [2, Proposition III.8.2]. Note that this K -basis is semi-invariant. We willshow in the next lemma that such a basis exists as a basis of W over K ◦ . Lemma 7.2. W is a free K ◦ -module of finite rank which has a semi-invariant basis.Proof. A valuation ring is a Pr¨ufer domain. Since W is a saturated K ◦ -submodule of the freemodule V of finite rank, we conclude that W is free of finite rank r (see [4, ch. VI, §
4, Exercise16]). We have seen above that W K has a simultaneous eigenbasis w , . . . , w r for the T -action.For j = 1 , . . . , r , we have S t ( w j ) = χ u j ( t ) · w j for all t ∈ T ( K ) and some u j ∈ M . Let E u j bethe corresponding eigenspace. Then W u j := E u j ∩ V is a saturated K ◦ -submodule of W . Thesame argument as above shows that W u j is a free K ◦ -module of finite rank. We may choose thesimultaneous eigenbasis w , . . . , w r above in such a way that a suitable subset is a K ◦ -basis of W u j for every j = 1 , . . . , r . Note that every w j is semi-invariant.For t in the subgroup U := T ( K ◦ ) = T ◦ ( K ) of T ( K ), we have S t ∈ GL ( V, K ◦ ) and hence theeigenvalues χ u j ( t ) have valuation 0. Using reduction modulo the maximal ideal K ◦◦ of K ◦ , the U -action becomes a T e K -operation on f W := W ⊗ K ◦ e K . We note that the reduction of a K ◦ -basisin W u j is linearly independent in f W . Using that eigenvectors for distinguished eigenvalues arelinearly independent, we conclude that the reduction of w , . . . , w r is a a simultaneous eigenbasisfor the T e K -action on f W . By Nakayama’s Lemma, it follows that w , . . . , w r is a K ◦ -basis for W . We can now prove the following quasi-projective version of Sumihiro’s theorem. Proposition 7.3.
Let U be a T -invariant open subset of Y A,a . Then every point of U has a T -invariant open affine neighbourhood in U .Proof. Let z ∈ U and let Y := Y \ U . Since Y is T -invariant, we are in the setting of 7.1 and wewill use the notation from there. In particular, we have s ∈ H ( P RK ◦ , O ( k )) such that s | Y = 0and s ( z ) = 0. Using Lemma 7.2, we conclude that there is a semi-invariant s ∈ H ( P RK ◦ , O ( k ))with s ( z ) = 0 and λs | Y = 0 for some λ ∈ K ◦ \ { } .To construct the affine invariant neighborhood of z , we assume first that z is contained in thegeneric fibre of U over K ◦ . Then U := { x ∈ Y A,a | λs ( x ) = 0 } is an affine open subset of U that contains z . Since s is semi-invariant, it follows that U is T -invariant proving the claim.Now we suppose that z is contained in the special fibre U s . Let ζ be a generic point of anirreducible component Z of Y s . Since Y is T -invariant, ζ is the generic point of an orbit whoseclosure Z does not contain z . Using the orbit–face correspondence from Proposition 3.6, thereis a projective coordinate x i ( ζ ) such that x i ( ζ ) ( Z ) = 0 but x i ( ζ ) ( z ) = 0. By definition of Y A,a ,we may view x i ( ζ ) as a semi-invariant global section of O (1) on P RK ◦ . Letting ζ varying overthe generic points of the irreducible components of Y s , we get a semi-invariant global section s := s · Q ζ s i ( ζ ) of a suitable tensor power of O (1) on P RK ◦ with s ( z ) = 0 and s | Y = 0. Then U := { x ∈ Y A,a | s ( x ) = 0 } is a T -invariant affine open neighbourhood of z in U . Proof of Theorem 2.
We are now ready to prove Sumihiro’s theorem for a normal T -toric variety Y over K ◦ . Every point z ∈ Y has an affine open neighbourhood U . Let U be the smallest T -invariant open subset of Y containing U . By Proposition 6.9, there is an equivariant openimmersion i : U → Y A,a for suitable A ∈ M R +1 and height function a . By Proposition 7.3,20here is a T -invariant open neighbourhood U of z in i ( U ). We conclude that i − ( U ) is anaffine T -invariant open neighbourhood of z in U proving Sumihiro’s theorem.Finally in order to complete the picture which gives rise to the interplay between toric geom-etry and convex geometry, we prove Theorem 3 which give us a bijective correspondence betweennormal T -toric varieties and Γ-admissible fans. Proof of Theorem 3.
We assume first that v is not a discrete valuation. For simplicity, we fixtorus coordinates on the split torus T of rank n . Let Y be a normal T -toric variety. By Theorem2, Y has an open covering { V i } i ∈ I by affine T -varieties V i . By Theorem 1, we have V i ∼ = V σ i for a Γ-admissible cone σ i in R n × R + for which the vertices of σ i ∩ ( R n × { } ) are containedin Γ n × { } . Since Y is separated, V ij := V i ∩ V j is affine for every i, j ∈ I . We conclude that V i ∩ V j is an affine normal T -toric variety and hence Theorem 1 again shows V ij ∼ = V σ ij for aΓ-admissible cone σ ij in R n × R + . Applying the orbit–face correspondence from [14, Proposition8.8] to the open immersions V ij → V i and V ij → V j , it follows that σ ij is a closed face of σ i and σ j . Moreover, the same argument shows that σ ij = σ i ∩ σ j and hence the closed faces of all σ i form a Γ-admissible fan Σ in R n × R + with Y Σ ∼ = Y . From Theorem 1, we get now easilythe desired bijection. If v is a discrete valuation, then the same argument works if we omit theadditional condition on the vertices of the cones. References [1] Vladimir G. Berkovich.
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