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LIFTING TROPICAL INTERSECTIONS

BRIAN OSSERMAN AND SAM PAYNE

Abstract. We show that points in the intersection of the tropicalizations ofsubvarieties of a torus lift to algebraic intersection points with expected multi-plicities, provided that the tropicalizations intersect in the expected dimension.We also prove a similar result for intersections inside an ambient subvariety ofthe torus, when the tropicalizations meet inside a facet of multiplicity 1. Theproofs require not only the geometry of compactified tropicalizations of subva-rieties of toric varieties, but also new results about the geometry of finite typeschemes over non-noetherian valuation rings of rank 1. In particular, we provesubadditivity of codimension and a principle of continuity for intersections insmooth schemes over such rings, generalizing well-known theorems over reg-ular local rings. An appendix on the topology of finite type morphisms mayalso be of independent interest.

Contents

1. Introduction 22. Preliminaries 42.1. Initial degenerations 42.2. Tropicalization 52.3. Tropical multiplicities 62.4. Minkowski weights and the fan displacement rule 72.5. Toric intersection theory 82.6. Stable tropical intersections 92.7. Compatibility of toric and stable tropical intersections 103. The trivial valuation case 133.1. Topology of extended tropicalizations 133.2. Tropical subadditivity 143.3. Lower bounds on multiplicities 154. Geometry over valuation rings of rank 1 174.1. Subadditivity and lifting over valuation rings of rank 1 184.2. A principal ideal theorem 194.3. Proofs of subadditivity and lifting 214.4. Intersection multiplicities over valuation rings of rank 1 235. Lifting tropical intersections with multiplicities 275.1. Intersections of two subvarieties 275.2. Intersections of three or more subvarieties 295.3. Non-proper intersections 306. Examples 31Appendix A. Initial degenerations, value groups, and base change 36Appendix B. Topology of finite type morphisms 39References 40

1

2 BRIAN OSSERMAN AND SAM PAYNE

1. Introduction

The tropicalization of the intersection of two subvarieties of a torus T over anonarchimedean field is contained in the intersection of their tropicalizations, butthis containment is sometimes strict. For instance, if X is smooth in characteristiczero, and X ′ is the translate of X by a general torsion point t in T , then X and X ′

are either disjoint or meet transversely, but their tropicalizations are equal. Thedimension of X∩X ′ is then strictly less than the dimension of Trop(X)∩Trop(X ′),so there are tropical intersection points that do not lift to algebraic intersectionpoints. In other cases, such as Example 6.2 below, the tropicalizations meet in apositive dimensional set that does not contain the tropicalization of any positivedimensional variety, so at most finitely many of the tropical intersection points canlift to algebraic intersection points. Our main result, in its most basic form, saysthat the dimension of the tropical intersection is the only obstruction to such lifting.

Following standard terminology from intersection theory in algebraic geometry,say that Trop(X) and Trop(X ′) meet properly at a point w if Trop(X)∩Trop(X ′)has pure codimension codim X + codim X ′ in a neighborhood of w.1

Theorem 1.1. Suppose Trop(X) meets Trop(X ′) properly at w. Then w is con-tained in the tropicalization of X ∩X ′.

In other words, tropicalization commutes with intersections when the intersectionshave the expected dimension. This generalizes a well-known result of Bogart,Jensen, Speyer, Sturmfels and Thomas, who showed that tropicalization commuteswith intersections when the tropicalizations meet transversely [BJS+07, Lemma 3.2].Our proof, and the proofs of all of the other main results below, involves a reduc-tion to the case where the base field is complete with respect to its nonarchimedeannorm and the point w is rational over the value group. The reduction is not diffi-cult, and should be a standard step in the rigorous application of tropical methodsto algebraic geometry. We emphasize that the theorems hold in full generality, overan arbitrary algebraically closed nonarchimedean field, and at any real point in theintersection of the tropicalizations.

Although Theorem 1.1 improves significantly on previously known results, it isstill too restrictive for the most interesting potential applications. Frequently, Xand X ′ are subvarieties of an ambient variety Y inside the torus. In this case, onecannot hope that Trop(X) and Trop(X ′) will meet properly in the above sense.Instead, we say that the tropicalizations Trop(X) and Trop(X ′) meet properly inTrop(Y ) at a point w if the intersection

Trop(X) ∩ Trop(X ′) ⊂ Trop(Y )

has pure codimension codimY X + codimY X ′ in a neighborhood of w. We extendTheorem 1.1 to proper intersections at suitable points of Trop(Y ), as follows.

Recall that Trop(Y ) is the underlying set of a polyhedral complex of pure di-mension dim Y , with a positive integer multiplicity assigned to each facet. We saythat a point in Trop(Y ) is simple if it lies in the interior of a facet of multiplicity1.

1Although this is not necessary for the proofs of tropical lifting theorems, we do show that codi-mension is subadditive for intersections of tropicalizations, which means that Trop(X)∩Trop(X′)has codimension at most j + j′, if it is nonempty. See Proposition 3.2.1.

LIFTING TROPICAL INTERSECTIONS 3

Theorem 1.2. Suppose Trop(X) meets Trop(X ′) properly in Trop(Y ) at a simplepoint w. Then w is contained in the tropicalization of X ∩X ′.

Every point in the tropicalization of the torus is simple, so Theorem 1.1 is the specialcase of Theorem 1.2 where Y is the full torus T . The hypothesis that w be simpleis necessary; see Section 6. We strengthen Theorem 1.2 further by showing thatwhere Trop(X) and Trop(X ′) meet properly, the facets of the tropical intersectionappear with the expected multiplicities, suitably interpreted. See Theorem 5.1.1for a precise statement.

The proof of Theorem 1.2 is in two steps; roughly speaking, we lift first fromtropical points to points in the special fiber of a suitable model over the valuationring, and then from the special fiber to the generic fiber. Recall that the tropical-ization of a subvariety X of T is the set of weight vectors w such that the initialdegeneration Xw is nonempty in the torus torsor Tw over the residue field.

Theorem 1.3. If Trop(X) meets Trop(X ′) properly in Trop(Y ) at a simple pointw, then Xw meets X ′

w properly at a smooth point of Yw.

If w is a simple point of Trop(Y ) then Yw is smooth, and standard arguments showthat Xw meets X ′

w properly at every point in the intersection; the main content ofthe theorem is that (X ∩ X ′)w is nonempty. The proof, given in Section 3, usesextended tropicalizations and the intersection theory of toric varieties. In Section 4,we develop geometric techniques over valuation rings of rank 1 that permit liftingof such proper intersections at smooth points from the special fiber to the genericfiber.

Theorem 1.4. Suppose that Xw meets X ′w properly at a smooth point x of Yw.

Then x is contained in (X ∩X ′)w.

In particular, if Xw meets X ′w properly at a smooth point of Yw then w is contained

in Trop(X ∩X ′). If w is defined over the value group of the nonarchimedean fieldthen surjectivity of tropicalization says that x can be lifted to a point of X ∩X ′.The dimension theory over valuation rings of rank 1 developed in Section 4 alsogives a new proof of surjectivity of tropicalization, as well as density of tropicalfibers.

Theorem 1.4 is considerably stronger than Theorems 1.1 and 1.2. It often hap-pens that initial degenerations meet properly even when tropicalizations do not.The proof of this theorem follows standard arguments from dimension theory, butthe dimension theory that is needed is not standard because the valuation rings weare working with are not noetherian. We develop the necessary dimension theorysystematically in Section 4, using noetherian approximation to prove a version ofthe Krull principal ideal theorem and then deducing subadditivity of codimensionfor intersections in smooth schemes of finite type over valuation rings of rank 1.Furthermore, we prove a principle of continuity, showing that intersection numbersare well-behaved in families over valuation rings of rank 1. In Section 5, we applythis principle of continuity to prove a stronger version of Theorem 1.2 with multi-plicities and then extend all of these results to proper intersections of three or moresubvarieties of T .

Remark 1.5. The main results of this paper can be interpreted as correspondencetheorems for points, in the spirit of Mikhalkin’s correspondence theorem for planecurves [Mik05]. Mikhalkin considers curves of fixed degree and genus subject


to constraints of passing through specified points. He shows, roughly speaking,that a tropical plane curve that moves in a family of the expected dimensionpassing through specified points lifts to a predictable number of algebraic curvespassing through prescribed algebraic points. Here, we consider points subject tothe constraints of lying inside subvarieties X and X ′ and prove the analogouscorrespondence—if a tropical point moves in a family of the expected dimensioninside Trop(X)∩Trop(X ′) then it lifts to a predictable number of points in X∩X ′.One hopes that Mikhalkin’s correspondence theorem will eventually be reprovedand generalized using Theorem 1.2 on suitable tropicalizations of moduli spaces ofcurves. Similarly, one hopes that Schubert problems can be answered tropicallyusing Theorem 1.2 on suitable tropicalizations of Grassmannians and flag varieties.The potential for such applications underlines the importance of having the flexi-bility to work with intersections inside ambient subvarieties of the torus.

Acknowledgments. We thank B. Conrad, D. Eisenbud, W. Heinzer, D. Rydh,and B. Ulrich for helpful conversations.

The first author was partially supported by a fellowship from the National ScienceFoundation. The second author was supported by the Clay Mathematics Institute.Part of this research was carried out during the special semesters on AlgebraicGeometry and Tropical Geometry at MSRI, in Spring and Fall 2009.

2. Preliminaries

Let K be an algebraically closed field, and let ν : K∗ → R be a valuation withvalue group G. Let R be the valuation ring, with maximal ideal m, and residuefield k = R/m. Since K is algebraically closed, the residue field k is algebraicallyclosed and the valuation group G is divisible. In particular, G is dense in R unlessthe valuation ν is trivial, in which case G is zero. Typical examples of such nonar-chimedean fields in equal characteristic are given by the generalized power seriesfields K = k((tG)), whose elements are formal power series with coefficients in thealgebraically closed field k and exponents in G, where the exponents occurring inany given series are required to be well-ordered [Poo93].

Let T be an algebraic torus of dimension n over R, with character lattice M ∼=Zn, and let N = Hom(M, Z) be the dual lattice. We write NG for N ⊗ G, andNR for N ⊗ R, the real vector space of linear functions on the character lattice.We treat M and N additively and write xu for the character associated to a latticepoint u in M , which is a monomial in the Laurent polynomial ring K[M ]. We writeT for the associated torus over K.

In this section, we briefly review the basic properties of initial degenerationsand tropicalizations, as well as the relationship between tropical intersections andintersections in suitable toric compactifications of T .

2.1. Initial degenerations. Each vector w ∈ NR determines a weight functionon monomials, where the w-weight of axu is ν(a) + 〈u, w〉. The tilted group ringR[M ]w ⊂ K[M ] consists of the Laurent polynomials

∑auxu in which every mono-

mial has nonnegative w-weight. We then define T w = Spec R[M ]w. If ν is non-trivial, then T w is an integral model of T , which means that it is a scheme over Rwhose generic fiber T w ×Spec R Spec K is naturally identified with T . In general,


the scheme T w carries a natural T -action, and is a T -torsor if w is in NG.2 It T w

is reduced and irreducible, and contains T as a dense open subscheme. We defineTw to be the closed subscheme cut out by monomials of strictly positive w-weight.If the valuation is nontrivial, then Tw is the special fiber of T w.

Let X be a closed subscheme of T , of pure dimension d. Let Xw denote theclosure of X in T w.

Definition 2.1.1. The initial degeneration Xw is the closed subscheme of Tw

obtained by intersecting with Xw.

The terminology reflects the fact that Xw is cut out by residues of initial terms(lowest-weight monomials) of Laurent polynomials in the ideal of X. If the valuationis nontrivial, then Xw is an integral model of X, and Xw is the special fiber of thismodel.

One consequence of Grobner theory is that the space of weight vectors NR canbe decomposed into finitely many polyhedral cells so that the initial degenerationsare essentially invariant on the relative interior of each cell [Spe05, Theorem 2.2.1].Roughly speaking, these cells are cut out by inequalities whose linear terms haveinteger coefficients, and whose constant terms are in the value group G. Moreprecisely, the cells are integral G-affine polyhedra, defined as follows.

Definition 2.1.2. An integral G-affine polyhedron in NR is the solution set ofa finite number of inequalities

〈u, v〉 ≤ b,

with u in the lattice M and c in the value group G.

An integral G-affine polyehdral complex Σ is a polyhedral complex consisting en-tirely of integral G-affine polyhedra. In other words, it is a finite collection ofintegral G-affine polyhedra such that every face of a polyhedron in Σ is itself in Σ,and the intersection of any two polyhedra in Σ is a face of each. Note that if Gis zero, then an integral G-affine polyhedron is a rational polyhedral cone, and anintegral G-affine polyhedral complex is a fan.

2.2. Tropicalization. Following Sturmfels, we define the tropicalization of X tobe

Trop(X) = {w ∈ NR | Xw is nonempty}.The foundational theorems of tropical geometry, due to the work of many authors,are the following.3

(1) The tropicalization Trop(X) is the underlying set of an integral G-affinepolyhedral complex of pure dimension d.

(2) The integral G-affine polyhedral structure on Trop(X) can be chosen sothat the initial degenerations Xw and Xw′ are Tk-affinely equivalent forany w and w′ in NG in the relative interior of the same face.

2If ν is nontrivial and w is not in NG, or if ν is trivial and w does not span a rational ray, thenT w is not of finite type over Spec R. Some additional care is required in handling these schemes,but no major difficulties arise for the purposes of this paper. The special fiber T w

k is still finite

type over k, and is a torsor over a quotient torus of Tk. See Appendix A.3See [BG84], [SS04], and Propositions 2.1.4, Theorem 2.2.1, and Proposition 2.4.5 of [Spe05]

for proofs of (1) and (2). The first proposed proof of (3), due to Speyer and Sturmfels, containedan essential gap that is filled by Theorem 4.2.5, below. Other proofs of (3) have appeared in[Dra08, Theorem 4.2] and [Pay09b]. The latter includes a discussion of the gap in the first proof.


(3) The image of X(K) under the natural tropicalization map trop : T (K) →NR is exactly Trop(X) ∩NG.

Here, two subschemes of the Tk-torsors Tw and Tw′ are said to be Tk-affinelyequivalent if they are identified under some Tk-equivariant choice of isomorphismTw

∼= Tw′ . In fact, a stronger statement holds: for any w,w′ ∈ NG, there existsa Tk-equivariant isomorphism Tw

∼= Tw′ which sends Xw to Xw′ for all X suchthat w,w′ are in the relative interior of the same face of Trop(X). If the valuationis nontrivial, then it follows from (3) that Trop(X) is the closure of the image ofX(K) under the tropicalization map.

We also note that, for any extension of valued fields L|K, the tropicalization ofthe base change Trop(XL) is exactly equal to Trop(X) [Pay09a, Proposition 6.1]. Inparticular, for any extension L|K with nontrivial valuation, Trop(X) is the closureof the image of X(L). Furthermore, if we extend to some L that is complete withrespect to its valuation then the tropicalization map on X(L) extends naturallyto a continuous map on the nonarchimedean analytification of XL, in the sense ofBerkovich [Ber90], whose image is exactly Trop(X) [Pay09a]. It follows, by reducingto the case of a complete field, that Trop(X) is connected if X is connected, sincethe analytification of a connected scheme over a complete nonarchimedean field isconnected.

Remark 2.2.1. The natural tropicalization map from T (K) to NR takes a point tto the linear function u 7→ ν ◦ evt xu, and can be understood as a coordinatewisevaluation map, as follows. The choice of a basis for M induces isomorphismsT ∼= (K∗)n and NR ∼= Rn. In such coordinates, the tropicalization map sends(t1, . . . , tn) to (ν(t1), . . . , ν(tn)).

There is no canonical choice of polyhedral structure on Trop(X) satisfying (2)in general, but any refinement of such a complex again satisfies (2). Throughoutthe paper, we assume that such a polyhedral complex with underlying set Trop(X)has been chosen. We refer to its faces and facets as faces and facets of Trop(X),and refine the complex as necessary.

The tropicalization of a subscheme of a torus torsor over K is well-defined up totranslation by NG. In particular, if the value group is zero, then the tropicalizationis well-defined, and is the underlying set of a rational polyhedral fan. An importantspecial case is the tropicalization of an initial degeneration. The valuation ν inducesthe trivial valuation on the residue field k, and by Proposition 2.2.3 of [Spe05] thereis a natural identification of Trop(Xw) with the star of w in Trop(X). This star is,roughly speaking, the fan one sees looking out from w in Trop(X); it is constructedby translating Trop(X) so that w is at the origin and taking the cones spanned byfaces of Trop(X) that contain w.

2.3. Tropical multiplicities. If w ∈ NG is in the relative interior of a facet σof Trop(X) then Xw is invariant under translation by the d-dimensional subtorusTσ ⊂ Tk whose lattice of one parameter subgroups is parallel to the affine spanof σ [Spe05, Proposition 2.2.4]. This subtorus acts freely, so a geometric quotientexists and Xw is a Tσ-torsor over the zero-dimensional k-scheme Xw/Tσ [MFK94,Sections 0.4 and 1.3]. The initial degeneration Xw′ at any other point w′ ∈ NG inthe relative interior of σ is Tk-affinely equivalent to Xw, so Xw/Tσ and Xw′/Tσ areisomorphic and hence have the same length.


Definition 2.3.1. In the case of nonitrivial valuation, the tropical multiplicitym(σ) is the length of Xw/Tσ for w ∈ NG in the relative interior of σ. If the valua-tion is trivial, we define the multiplicity by extending K to a field with nontrivialvaluation.

Remark 2.3.2. For the trivial valuation case, the multiplicity defined above is in-dependent of choice of field extension by the results of Appendix A. In fact, it ispossible to define multiplicity directly at all points in the relative interior of a facetof Trop(X) by giving the suitable generalization of the above statements in the casethat w 6∈ NG, and the result will agree with our definition, again using AppendixA.

For w ∈ NG, the multiplicities on the facets of Trop(Xw) agree with those onthe facets of Trop(X) that contain w. This is because the initial degeneration ofXw at a point in the relative interior of a given facet is Tk-affinely equivalent tothe initial degeneration of X at a point in the relative interior of the correspondingfacet of Trop(X).

Points in the relative interiors of facets of multiplicity 1 will be particularlyimportant for our purposes.

Definition 2.3.3. A simple point in Trop(X) is a point in the relative interiorof a facet of multiplicity 1.

If w ∈ NG is a simple point, then the initial degeneration Xw is isomorphic to a d-dimensional torus, and initial degenerations at arbitrary simple points are smooth.The importance of these simple points, roughly speaking, is that intersections oftropicalizations of subvarieties of X at simple points behave just like intersectionsin NR of tropicalizations of subvarieties of T . See, for instance, the proof of Theo-rem 1.2 in Section 5.

2.4. Minkowski weights and the fan displacement rule. We briefly reviewthe language of Minkowski weights and their ring structure, in which the product isgiven by the fan displacement rule. These tools are the basis of both toric intersec-tion theory, as developed in [FS97], and the theory of stable tropical intersections[RGST05, Mik06], discussed in the following sections. We refer the reader to theseoriginal papers for further details. Although the theory of Minkowski weights ex-tends to arbitrary complete fans, we restrict to unimodular fans, corresponding tosmooth toric varieties, which suffice for our purposes. Fix a complete unimodularfan Σ in NR.

Definition 2.4.1. A Minkowski weight of codimension j on Σ is a function c thatassigns an integer c(σ) to each codimension j cone σ in Σ, and satisfies the followingbalancing condition at every codimension j + 1 cone.

Balancing Condition. Let τ be a cone of codimension j +1 in Σ. Say σ1, . . . , σr

are the codimension j cones of Σ that contain τ , and let vi be the primitive generatorof the image of σi in NR/ span(τ). Then c is balanced at τ if

c(σ1)v1 + · · ·+ c(σr)vr = 0

in NR/ span(τ).

Tropicalizations are one interesting source of Minkowski weights. If X is a subvari-ety of codimension j in T and Trop(X) is a union of cones in Σ, then the function


c given by

c(σ) ={

m(σ) if σ ⊂ Trop(X).0 otherwise.

saitsfies the balancing condition, and hence is a Minkowski weight of codimension j.This correspondence between tropicalizations and Minkowski weights is compatiblewith intersection theory in the toric variety Y (Σ). See Section 2.7, below.

We write Minkj(Σ) for the group of Minkowski weights of codimension j on Σ.The direct sum

Mink∗(Σ) =⊕

j

Minkj(Σ)

naturally forms a graded ring, whose product is given by the fan displacement rule,as follows. Let v be a vector in NR that is sufficiently general so that for any twocones σ and σ′ in Σ, the displaced cone σ′ + v meets σ properly. The set of suchvectors is open and dense in NR; it contains the complement of a finite union oflinear spaces. For each cone σ in Σ, let Nσ denote the sublattice of N generated byσ ∩N . Note that if σ intersects the displaced cone σ′ + v then σ and σ′ togetherspan NR, so the index [N : Nσ + Nσ′ ] is finite.

Fan Displacement Rule. The product c·c′ is the Minkowski weight of codimensionj + j′ given by

(c · c′)(τ) =∑σ,σ′

[N : Nσ + Nσ′ ] · c(σ) · c′(σ′),

where the sum is over cones σ and σ′ containing τ , of codimensions j and j′ re-spectively, such that the intersection σ ∩ (σ′ + v) is nonempty.

The balancing conditions on c and c′ ensure that the product is independent of thechoice of displacement vector v, and also satisfies the balancing condition.

2.5. Toric intersection theory. We briefly recall the basics of intersection theoryin smooth complete toric varieties. As in the previous section, we fix a completeunimodular fan Σ in NR. Let Y (Σ) be the associated smooth complete toric variety.A codimension j cycle in Y (Σ) is a formal sum of codimension j subvarieties, withinteger coefficients, and we write Aj(Y (Σ)) for the Chow group of codimension jcycles modulo rational equivalence. Let X be a subvariety of codimension j inY (Σ). We write [X] for the class it represents in Aj(Y (Σ)).

Intersection theory gives a natural ring structure on the direct sum

A∗(Y (Σ)) =⊕

j

Aj(Y (Σ)).

If X and X ′ have complementary codimension, which means that codim X +codim X ′ = dim Y (Σ), then the product [X] · [X ′] is a zero-dimensional cycle class.The degree of this class, which is the sum of the coefficients of any representativecycle, is denoted deg([X] · [X ′]). In particular, if σ is a cone of codimension j inΣ, with V (σ) the associated closed T -invariant subvariety, then X and V (σ) havecomplementary codimension. We write cX for the induced function on codimensionj cones, given by

cX(σ) = deg([X] · [V (σ)]).The fact that the intersection product respects rational equivalence ensures that cX

satisfies the balancing condition and is therefore a Minkowski weight of codimension


j on Σ. The main result of [FS97] then says that there is a natural isomorphism ofrings

A∗(Y (Σ)) ∼−→ Mink∗(Σ)taking the Chow class [X] to the Minkowski weight cX .

The theory of refined intersections says that the product [X]·[X ′] of the classes oftwo subvarieties of a smooth variety Y is not only a well-defined cycle class in thatsmooth variety, but also the Gysin push-forward of a well-defined cycle class in theintersection X∩X ′. If X and X ′ meet properly, then this class has codimension zero,and hence can be written uniquely as a formal sum of the components of X ∩X ′.The coefficient of a component Z of the intersection is called the intersectionmultiplicity of X and X ′ along Z, and is denoted i(Z,X ·X ′;Y ). It is a positiveinteger, less than or equal to the length of the scheme theoretic intersection of Xand X ′ along the generic point of Z, and can be computed by Serre’s alternatingsum formula

i(Z,X ·X ′;Y ) =∑

j

(−1)j lengthOY,ZTorj

OY,Z(OX ,OX′).

See [Ful98, Chapter 8] for further details on the theory of refined intersections insmooth varieties.

Definition 2.5.1. If X and X ′ meet properly in a smooth variety Y , then therefined intersection cycle is

X ·X ′ =∑Z

i(Z,X ·X ′;Y ) Z,

where the sum is over all irreducible components Z of X ∩X ′.

We will consider such refined intersections inside the torus T , as well as in its smoothtoric compactifications. Finally, it is useful to work not only with tropicalizationsof subschemes of T , but also with tropicalizations of refined intersection cycles, sowe define:

Definition 2.5.2. Let∑

Z aZZ be a pure-dimensional cycle in T . Then its trop-icalization Trop(

∑Z aZZ) is by definition the union of the tropicalizations of the

finitely many varieties Z for which aZ 6= 0, with the multiplicity on each facet τdefined by

m(τ) =∑

aZmZ(τ).

2.6. Stable tropical intersections. Many papers have introduced and studiedanalogues of intersection theory in tropical geometry, including [AR09, Kat09a,Mik06, RGST05, Tab08]. Here we are interested in only one basic common fea-ture of these theories, the stable tropical intersection, and its compatibility withintersection theory in toric varieties. The key point is that stable tropical inter-sections are defined combinatorially, depending only on the polyhedral geometry ofthe tropicalizations, by a local displacement rule. Compatibility with the multipli-cation rule for Minkowski weights and algebraic intersections of generic translatesis discussed in the following section.

Given subvarieties X and X ′ of T , of codimensions j and j′, respectively, thestable intersection Trop(X) ·Trop(X ′) is a polyhedral complex of pure codimensionj + j′, with support contained in the set-theoretic intersection Trop(X)∩Trop(X ′),and with appropriate multiplicities on its facets, satisfying the balancing condition.


Many different subvarieties of T may have the same tropicalization, but the stableintersection depends only on the tropicalizations.

Suppose the valuation is trivial, so Trop(X) and Trop(X ′) are fans, and let τ be aface of Trop(X)∩Trop(X ′) of codimension j+j′ in NR. The tropical intersectionmultiplicity of Trop(X) and Trop(X ′) along τ , is given by the fan displacementrule

i(τ,Trop(X) · Trop(X ′)) =∑σ,σ′

[N : Nσ + Nσ′ ] ·m(σ) ·m′(σ′),

where the sum is over facets σ and σ′ of Trop(X) and Trop(X ′), respectively,such that the intersection σ ∩ (σ′ + v) is nonempty, and v ∈ NR is a fixed genericdisplacement vector.

In the general case, where the valuation may be nontrivial, the tropical inter-section multiplicities are defined similarly, by a local displacement rule, in whichthe generic displacement vector is required to be very small. Let τ be a face ofcodimension j + j′ in Trop(X)∩Trop(X ′), and for each face σ containing τ , let Nσ

be the sublattice of N parallel to the affine span of σ.

Definition 2.6.1. The tropical intersection multiplicity i(τ,Trop(X) · Trop(X ′))is given by the local displacement rule

i(τ,Trop(X) · Trop(X ′)) =∑σ,σ′

[N : Nσ + Nσ′ ] ·m(σ) ·m′(σ′),

where the sum is over facets σ and σ′ of Trop(X) and Trop(X ′), respectively, suchthat the intersection σ ∩ (σ′ + εv) is nonempty for ε sufficiently small and positive.

Definition 2.6.2. The stable tropical intersection, denoted Trop(X)·Trop(X ′),is the union of those faces τ such that i(τ,Trop(X) ·Trop(X ′)) is positive, weightedby their tropical intersection multiplicities.

2.7. Compatibility of toric and stable tropical intersections. As discussedin Section 2.4, tropical multiplicities satisfy the balancing condition, and hence giveMinkowski weights. The stable tropical intersections are compatible with multipli-cation of Minkowski weights, and algebraic intersections of generic translates in thetorus, as follows.

Let X be a subvariety of codimension j in T . Suppose the valuation is trivial,so Trop(X) is a fan. After subdividing, we may assume this fan is unimodular, andextends to a complete unimodular fan Σ in NR. Let Y (Σ) be the associated smoothcomplete toric variety, and let X be the closure of X in Y (Σ). Then X meets atorus orbit Oτ if and only if τ is contained in Trop(X), and in this case X meets Oτ

properly. Furthermore, for any facet σ of Trop(X), the tropical multiplicity m(σ)is equal to the intersection number deg([X] · V (σ)) [ST08, Lemma 3.2]. In otherwords,

cX(σ) ={

m(σ) if σ ⊂ Trop(X).0 otherwise.

Now consider the general case, where the valuation may be nontrivial, and letw ∈ NG be a point of Trop(X). After subdividing, we may assume that the starof w in Trop(X) is unimodular, and extends to a complete unimodular fan Σ. Wewrite Xw for the closure of Xw in Y (Σ). For a facet σ of Trop(X) that contains w,let σw be the corresponding cone in Σ, which is exactly R≥0(σ − w).


Lemma 2.7.1. Let σ be a face of codimension j in Σ. Then

cXw(σ) =

{m(σ′) if σ = σ′w, for some σ′ ⊂ Trop(X).0 otherwise.

Proof. This follows from the case of the trivial valuation, since m(σ) is equal to thetropical multiplicity m(σw) in Trop(Xw). �

This correspondence between tropical multiplicities and Minkowski weights iscompatible with intersections, roughly speaking, with stable tropical intersectionscorresponding to products of Minkowski weights. Suppose X ′ is a subvariety ofcodimension j′ in T , respectively, and τ is a face of codimension j +j′ in Trop(X)∩Trop(X ′) that contains w. We may assume that the star of w in Trop(X ′) is asubfan of the unimodular fan Σ, and we write X

′w for the closure of X ′

w in Y (Σ).

Proposition 2.7.2. Let τ be a cone of codimension j + j′ in Σ. Then

(cXw· cX

′w)(τ) =

{i(τ ′,Trop(X) · Trop(X ′)) if τ = τ ′w,0 otherwise,

Proof. Suppose τ = τ ′w. We claim that the displacement rules giving (cXw·cX

′w)(τ)

and i(τ ′,Trop(X)·Trop(X ′)) agree, term by term. If σ and σ′ are facets of Trop(X)and Trop(X ′), respectively, that contain τ ′, then σ meets σ′ + εv for small positiveε if and only if σw meets σ′w + v. Then we have an equality of summands in thedisplacement rules,

[N : Nσ + Nσ′ ] ·m(σ) ·m′(σ′) = [N : Nσw + Nσ′w ] · cXw(σw) · cX

′w(σ′w),

because Nσ+N ′σ, m(σ), and m′(σ′) are equal to Nσw +Nσ′w , cXw

(σw), and cX′w(σ′w),

respectively. The Minkowski weights cXwand cX

′w

vanish on cones that do notcome from facets of Trop(X) and Trop(X ′), respectively. Therefore, all nonzerosummands in the displacement rules are accounted for in the equality above, andthe proposition follows. �

It follows that the star of w in the stable tropical intersection Trop(X) ·Trop(X ′) isexactly the union of the faces of Σ on which (cXw

·cX′w) is positive, with multiplicities

given by this product of Minkowski weights.

Proposition 2.7.3. Suppose the valuation is trivial, and assume X meets X ′ prop-erly in T , and every component of X ∩X

′is the closure of a component of X ∩X ′.

Then, for any face τ of Trop(X) ∩ Trop(X ′) of codimension j + j′ in NR, thetropical intersection multiplicity along τ is equal to the weighted sum of multiplicitiesof tropicalizations

i(τ,Trop(X) · Trop(X ′)) =∑Z

i(Z,X ·X ′;T )mZ(τ),

where the sum is over components Z of X ∩X ′ such that Trop(Z) contains τ , withtropical multiplicity mZ(τ).

Proof. First of all, by the preceding proposition,

i(τ,Trop(X) · Trop(X ′)) = (cX · cX′)(τ).

Next, the isomorphism from Mink∗(Σ) to A∗(Y (Σ)) identifies (cX ·cX′)(τ) with the

intersection number deg([X] · [X ′] · V (τ)). By hypothesis, X meets X

′properly in


Y (Σ), and the generic point of each component of the intersection lies in T , so wehave the refined intersection

X ·X ′=

∑Z

i(Z,X ·X ′;T )[Z],

where the sum is taken over components Z of X ∩X ′. Therefore,

deg([X] · [X ′] · [V (τ)]) =

∑Z

i(Z,X ·X ′;T ) deg([Z] · [V (τ)]).

The proposition follows, since the intersection number deg([Z] · [V (τ)]) is equal tothe tropical multiplicity mZ(τ). �

The following proposition may be seen as a geometric explanation for the exis-tence of well-defined stable tropical intersections, in the case of the trivial valua-tion; they are exactly the tropicalizations of intersections of generic translates inthe torus.

Proposition 2.7.4. Suppose the valuation is trivial, and let t be a general point inT . Then X meets tX ′ properly and the tropicalization of the intersection schemeX ∩ tX ′ is exactly the stable tropical intersection Trop(X) · Trop(X ′).

Proof. First, an easy Bertini argument shows that X meets tX ′ properly when t isgeneral. Furthermore, since Y (Σ) is smooth, subadditivity of codimension says thatevery component of X ∩ tX

′has codimension at most j + j′. Say Z is a component

of this intersection, and let Oσ be the torus orbit in Y (Σ) that contains the genericpoint of Z. Now, X and tX

′meet Oσ properly, since their tropicalizations are

subfans of Σ [Kat09b, Section 8], and X ∩ Oσ meets tX′ ∩ Oσ properly, since t is

general. Therefore Z has codimension j + j′ + codim Oσ, and hence Oσ must bethe dense torus T .

Next, since T acts transitively on itself, a transversality theorem of Miller andSpeyer [MS08] says that the Tor groups

TorjOY (Σ),Z

(OX ,OtX′)

vanish for j > 0, so the intersection multiplicity i(Z,X ·tX ′;T ) is equal to the lengthof the intersection of X and tX ′ along the generic point of Z. Proposition 2.7.3then says that


length(OZ,X∩tX′) mZ(τ),

and the right hand side is equal to mX∩tX′(τ), as required. �

Remark 2.7.5. We apply this proposition in the proof of Lemma 3.3.3. All that isneeded for that application is the standard fact that the intersection multiplicityat a point of proper intersection is strictly positive. In particular, the vanishing ofhigher Tor groups is not needed for the main lifting theorems.

In Section 5, we extend Proposition 2.7.4 to the general case, where the valuationmay be nontrivial, as an application of tropical lifting theorems. See Theorem 5.3.2.


3. The trivial valuation case

The first step in our approach to tropical lifting theorems is to understand howthe intersection of Trop(X) and Trop(X ′) near a point w relates to the intersectionof the initial degenerations Xw and X ′

w. If w ∈ NG then the stars of w in Trop(X)and Trop(X ′) are the tropicalizations of Xw and X ′

w, respectively, with respect tothe trivial valuation, so this amounts to understanding how tropicalization relatesto intersections when the tropicalizations meet properly and the valuation is trivial.Our proof that tropicalization commutes with intersection in this case uses boththe theory of extended tropicalizations as projections of nonarchimedean analytifi-cations and a new result about the support of stable intersections of tropicalizations(Lemma 3.3.3). We begin by proving two necessary lemmas about the topology ofextended tropicalizations. We state and prove these lemmas in the general case,where the valuation is not necessarily trivial, since the arguments are identical tothose in the case where the valuation is trivial. Similarly, for the sake of con-text we show that tropical intersections satisfy the same codimension subadditivityproperty as algebraic intersections, again for an arbitrary valuation.

3.1. Topology of extended tropicalizations. The extended tropicalizations ofY (Σ) and its subvarieties were introduced by Kajiwara [Kaj08], and their basicproperties were studied further in Section 3 of [Pay09a], to which we refer thereader for further details. We recall the definition and prove that the extendedtropicalization of an irreducible subscheme X that meets the dense torus T is theclosure of the ordinary tropicalization of X ∩ T . We also characterize the closurein the extended tropicalization Trop(Y (Σ)) of a cone σ in Σ.

Let Σ be a fan in NR, and let Y (Σ) be the associated toric variety. Recallthat each cone σ in Σ corresponds to an affine open subvariety Uσ of Y (Σ) whosecoordinate ring is the semigroup ring K[σ∨ ∩ M ] associated to the semigroup oflattice points in the dual cone σ∨ in MR. Let R be the real line extended in thepositive direction

R = R ∪ {+∞},which is a semigroup under addition, with identity zero. Then

Trop(Uσ) = Hom(σ∨ ∩M,R)

is the space of all semigroup homomorphisms taking zero to zero, with its naturaltopology as a subspace of Rσ∨∩M . For each face τ of Σ, let N(τ) be the real vectorspace

N(τ) = NR/ span(τ).

Then Trop(Uσ) is naturally a disjoint union of the real vector space N(τ), forτ � σ, where N(τ) is identified with the subset of semigroup homomorphisms thatare finite exactly on the intersection of τ⊥ with σ∨ ∩M .

The tropicalization Trop(Y (Σ)) is a space obtained by gluing the spaces Trop(Uσ)for σ ∈ Σ along the natural open inclusions Trop(Uτ ) ⊂ Trop(Uσ) for τ � σ. It isa disjoint union

Trop(Y (Σ)) =⊔

σ∈Σ

N(σ),

just as Y (Σ) is the disjoint union of the torus orbits Oσ. Points in N(σ) may be seenas weight vectors on monomials in the coordinate ring of Oσ, and the tropicalization


of a closed subscheme X in Y (Σ) is defined to be

Trop(X) =⊔

σ∈Σ

{w ∈ N(σ) | (X ∩Oσ)w is not empty}.

In other words, Trop(X) is the disjoint union of the tropicalizations of its inter-sections with the T -orbits in Y (Σ). This space is compact when Σ is complete.

Lemma 3.1.1. If X meets the dense torus T , then Trop(X) is the closure inTrop(Y (Σ)) of the ordinary tropicalization Trop(X ∩ T ).

Proof. Since tropicalizations are invariant under extensions of valued fields [Pay09a,Proposition 6.1], and the completion of the algebraically closed field K is alge-braically closed [BGR84, Proposition 3.4.1.3], we may assume the field K is com-plete with respect to its valuation.

The tropicalization map from X(K) to Trop(Y (Σ)) extends to a proper con-tinuous map on the nonarchimedean analytification Xan whose image is Trop(X)[Pay09a, Section 2]. Since the open subset (X ∩T )an is dense in Xan [Ber90, Corol-lary 3.4.5] and maps onto Trop(X ∩ T ), the extended tropicalization Trop(X) iscontained in the closure of Trop(X ∩ T ). The lemma follows, since Trop(X) isclosed. �

If σ is a face of Σ that contains another face τ , then we write στ for the imageof σ in N(τ). It is a cone of dimension dim(σ)− dim(τ). We will use the followinglemma in the proof of Proposition 3.3.2.

Lemma 3.1.2. Let σ and τ be faces of Σ, and let σ be the closure of σ inTrop(Y (Σ)). Then

σ ∩N(τ) = στ

if σ contains τ , and σ is disjoint from N(τ) otherwise.

Proof. Let v be a point in N(τ), and let π : Trop(Uτ ) → N(τ) be the continuousmap that restricts to the canonical linear projections from N(γ) onto N(τ) forγ � τ . If v is not in στ , then the preimage under π of N(τ) r στ is an openneighborhood of v that is disjoint from σ, so v is not in σ.

For the converse, suppose v is in στ and let w be a point in σ that projects tov. Let w′ be a point in the relative interior of τ . Then w + R≥0w

′ is a path in σwhose limit is v, so v is in σ. �

3.2. Tropical subadditivity. Although the following proposition is not used any-where in the paper, it is perhaps helpful to know that subadditivity holds forintersections of tropicalizations, which means that Trop(X) and Trop(X ′) neverintersect in less than the expected dimension. The proof is by a diagonal projec-tion argument, similar to the proof of Lemma 3.3.3. The proof generalizes directlyto multiple intersections, without the need for an induction.

Proposition 3.2.1. Let X and X ′ be subvarieties of codimension j and j′, respec-tively, in T . If Trop(X) ∩ Trop(X ′) is nonempty then it has codimension at mostj + j′.

Proof. Let T ′ be the quotient of T ×T by the diagonal subtorus, with π : X×X ′ →T ′ the induced projection, and p : Trop(X) × Trop(X ′) → N ′

R the tropicalizationof π. Since Trop(X) meets Trop(X ′), the point zero is in the image of p, whichis exactly the tropicalization of the closure of the image of π. Since the fibers of


tropicalization are Zariski dense [Pay09b], there must be some point y in π(X×X ′)whose tropicalization is zero in N ′. Now, the fiber π−1(y) has dimension at leastdim X + dim X ′ − dim T ′, and its tropicalization is contained in p−1(0). Sincep−1(0) is naturally identified with Trop(X)∩Trop(X ′), it follows that the tropicalintersection has dimension at least dim X ′ + dim X − dim T , as required. �

3.3. Lower bounds on multiplicities. For the remainder of this section, weassume the valuation ν is trivial. Let X and X ′ be subvarieties of codimension jand j′ in T , respectively. After subdividing, we choose a complete unimodular fanΣ in NR such that each face of Trop(X) and each face of Trop(X ′) is a face of Σ.We write X and X

′for the closures of X and X ′ in the smooth complete toric

variety Y (Σ).Our goal is to prove the following version of Theorem 1.1 in the special case

of the trivial valuation, with lower bounds on the multiplicities of the facets inTrop(X ∩X ′). These lower bounds are extended to the general case in Section 5.

Theorem 3.3.1. Suppose ν is trivial and Trop(X) meets Trop(X ′) properly. Then

Trop(X ∩X ′) = Trop(X) ∩ Trop(X ′).

Furthermore, the multiplicity of any facet τ is bounded below by the tropical inter-section multiplicity

mX∩X′(τ) ≥ i(τ,Trop(X) · Trop(X ′)),

and both are strictly positive.

Given the compatibility of toric and stable tropical intersections in Section 2.7, themain step in the proof of Theorem 3.3.1 is showing that every component of X∩X

′

is the closure of a component of X ∩ X ′. This step can fail spectacularly whenTrop(X) and Trop(X ′) do not meet properly. In such cases, X ∩X

′may contain

components of larger than expected dimension, even if X and X ′ meet properly inT, and it is difficult to predict what Trop(X ∩X ′) will look like.

Proposition 3.3.2. Suppose Trop(X) meets Trop(X ′) properly, and let τ be a faceof Σ. Then

(1) The tropicalizations of X ∩Oτ and X′ ∩Oτ meet properly in N(τ).

(2) The subvarieties X ∩ V (τ) and X′ ∩ V (τ) meet properly in V (τ).

(3) The closures X and X′

meet properly in Y (Σ), and every component ofX ∩X

′is the closure of a component of X ∩X ′.

Proof. First, if the tropicalizations of X ∩Oτ and X′ ∩Oτ meet properly in N(τ),

then the intersections themselves must meet properly in Oτ . If this holds for allcones σ containing τ , then X ∩ V (τ) meet X

′ ∩ V (τ) properly in V (τ). Therefore(1) implies (2). Furthermore, if (2) holds then every component of X ∩X

′ ∩ V (τ)has codimension j + j′ in V (τ). In particular, if Oτ is not the dense torus T , thenV (τ) does not contain a component of X ∩X

′. If this holds for all τ , then every

component of X ∩ X′

meets the dense torus T . This shows that (2) implies (3).We now prove (1).

The tropicalization of X ∩Oτ is the intersection of the extended tropicalizationTrop(X) with N(τ) and, by Lemma 3.1.1, this extended tropicalization is theclosure of Trop(X). Therefore, by Lemma 3.1.2, Trop(X ∩Oτ ) is the union of the


projected faces στ such that σ is in Trop(X) and contains τ . Similarly, Trop(X′ ∩

Oτ ) is the union of the projected faces σ′τ such that σ′ is in Trop(X ′) and containsτ . Since the intersection of στ and σ′τ is exactly (σ∩σ′)τ , the codimension of στ∩σ′τin N(τ) is equal to the codimension of σ ∩ σ′ in NR. In particular, if Trop(X) andTrop(X ′) meet properly at τ , then Trop(X ∩Oτ ) and Trop(X

′∩Oτ ) meet properlyin N(τ), as required. �

The following lemma says that the support of the stable tropical intersectionTrop(X) · Trop(X ′) is equal to Trop(X) ∩ Trop(X ′) when the two meet properly.

Lemma 3.3.3. Suppose Trop(X) and Trop(X ′) meet properly and let τ be a facet oftheir intersection. Then the tropical multiplicity i(τ,Trop(X) ·Trop(X ′)) is strictlypositive.

Proof. First, we reduce to the case where X and X ′ have complementary dimen-sion. Recall that tropical intersection multiplicities agree with algebraic intersectionnumbers, as discussed in Section 2.7, and the intersection number deg(X ·X ′ ·V (τ))in Y (Σ) is equal to the intersection number

deg((X · V (τ)) · (X ′ · V (τ))

)in V (τ) [Ful98, Example 8.1.10], because V (τ) is smooth. Because X and X

′

meet all torus orbits properly, it follows that the tropical intersection multiplicityi(τ,Trop(X) ·Trop(X ′)) is equal to the tropical intersection multiplicity at zero ofTrop(X ·Oτ ) and Trop(X

′ ·Oτ ) inside N(τ). By Proposition 3.3.2, the intersectioncycles X · Oτ and X

′ · Oτ have complementary dimension in Oτ and their tropi-calizations meet properly in N(τ). The same is then true for each of the reducedcomponents of X · Oτ and X

′ · Oτ , and it suffices to prove the proposition in thiscase.

Assume X and X ′ have complementary dimension. Since their tropicalizationsmeet properly, by hypothesis, their intersection must be the single point zero. Con-sider the quotient T ′ of T ×T by its diagonal subtorus, and the induced projection

π : X ×X ′ → T ′.

By Proposition 2.7.4, the tropical intersection number Trop(X) ·Trop(X ′) is equalto the length of X ∩ tX ′ for a general point t ∈ T , which is equal to the length ofthe general fiber of π. In particular, to prove the proposition, it is enough to showthat π is dominant. Now, since the valuation is trivial, the tropicalization of anyfiber of π is contained in the preimage of zero under the induced projection

Trop(X)× Trop(X ′) → N ′R,

which is the single point zero, by hypothesis. In particular, every nonempty fiberof π is zero-dimensional, and hence π is dominant, as required. �

We can now easily prove the desired result.

Proof of Theorem 3.3.1. Let τ be a facet of Trop(X) ∩ Trop(X ′). By Proposi-tion 2.7.3, the tropical intersection multiplicity i(τ,Trop(X)·Trop(X ′)) is a weightedsum over components of X ∩X ′


i(Z,X ·X ′;T )mZ(τ).


By Lemma 3.3.3, this multiplicity is strictly positive, so there is at least one com-ponent Z of X ∩X ′ such that Trop(X) contains τ . This proves that τ is containedin Trop(X ∩ X ′), and hence Trop(X ∩ X ′) is equal to Trop(X) ∩ Trop(X ′), set-theoretically.

To see the inequality on multiplicities, note that the multiplicity of τ in Trop(X∩X ′) is given by

mX∩X′(τ) =∑Z

lengthOT,Z(OX∩X′,Z)mZ(τ).

The inequality mX∩X′(τ) ≥ i(τ,Trop(X) ·Trop(X ′)) follows by comparing the twosummation formulas term by term. The length lengthOT,Z

(OX∩X′,Z) is at least aslarge as the intersection multiplicity i(Z,X · X ′;T ) [Ful98, Proposition 8.2], andthe theorem follows. �

4. Geometry over valuation rings of rank 1

In [Nag66], Nagata began extending certain results from dimension theory tonon-noetherian rings. Here we continue in Nagata’s spirit with a view towardtropical geometry, focusing on valuation rings of rank 1 and showing that theserings have many of the pleasant properties of regular local rings.

Recall that a valuation ring of rank 1 is an integral domain R whose field offractions K admits a nontrivial valuation ν : K∗ → R such that the nonzeroelements of R are exactly the elements of K∗ of nonnegative valuation. It is a localring with exactly two prime ideals, the zero ideal and the maximal ideal. If K isalgebraically closed or, more generally, if the valuation is not discrete, then R isnot noetherian, because the maximal ideal is not finitely generated.

Throughout this section, R is a valuation ring of rank 1, with fraction field Kand residue field k. In this section only, we allow the possibility that K is notalgebraically closed, because the results of this section hold equally for nonclosedfields and the greater generality creates no additional difficulties. We fix

S = Spec R,

and use calligraphic notation for schemes over S. If X is a scheme over S, we writeXK = X ×S Spec K for the generic fiber and Xk = X ×S Spec k for the special fiber.

A module is flat over R if and only if it is torsion free [Mat89, Exercise 10.2],so an integral scheme X is flat over S if and only if it is dominant. In particular,if X is an integral scheme that is surjective and of finite type over S then thespecial fiber Xk is of pure dimension equal to the dimension of the generic fiberXK ; this follows from [Gro66, Theorem 12.1.1(i)], applied to the generic point ofeach component of the special fiber. We will use this equality of fiber dimensionsrepeatedly throughout the section, without further mention.

Dimension theory over valuation rings of rank 1 involves some subtleties. Forinstance, if R is not noetherian then the formal power series ring R[[x]] has infiniteKrull dimension [Arn73]. Because dimensions are well-behaved under specializationover R, such pathologies are always visible in at least one of the fibers. For X =Spec R[[x]], the special fiber is Spec k[[x]], which is one dimensional, but R[[x]]⊗Kis the subring of K[[x]] consisting of formal power series whose coefficients havevaluations bounded below. This ring, and hence the generic fiber of Spec R[[x]], isinfinite dimensional. Nevertheless, for a scheme of finite type over S, the generic


fiber and special fiber are schemes of finite type over K and k, respectively, andsuch pathologies do not occur.

Many of the results of this section have natural generalizations to more generalbase schemes. See Appendix B for details. We repeatedly use, without furthercomment, the result of Raynaud-Gruson, which states that a scheme which is flatand locally of finite type over an integral scheme is locally of finite presentation.See [RG71, Theorem 3.4.6 and Corollary 3.4.7].

4.1. Subadditivity and lifting over valuation rings of rank 1. Serre famouslyproved that codimension is subadditive under intersections in regular schemes[Ser65, Theorem V.3]. In other words, for any irreducible closed subschemes Xand X ′ of a regular scheme Y , and any irreducible component Z of X ∩ X ′, wehave

codimY (X ∩X ′) ≤ codimY X + codimY X ′.

The same is then necessarily true for schemes smooth over Y . We extend Serre’stheorem to smooth schemes over rank 1 valuation rings, as follows. Recall that wehave fixed S = Spec R, where R is a valuation ring of rank 1.

Theorem 4.1.1. Let Y be smooth over S. Then codimension is subadditive underintersection in Y.

In this respect, valuation rings of rank 1 behave like regular local rings.

Remark 4.1.2. Subadditivity of codimension is often used to deform points in fam-ilies using dimension counting arguments. Such techniques are essential, for in-stance, in the theory of limit linear series developed by Eisenbud and Harris [EH86].Here, we use subadditivity of codimension to lift intersection points from the specialfiber to the generic fiber in schemes of finite type over valuation rings of rank 1.4

As an application of Theorem 4.1.1 we prove the following.

Theorem 4.1.3. Let Y be smooth over S with subschemes X and X ′ of purecodimension j and j′, respectively. Suppose the special fibers Xk and X ′

k intersectin codimension j + j′ at a point x in Yk. Then

(1) There is a point in XK ∩ X ′K specializing to x.

(2) If x is closed then there is a closed point in XK ∩ X ′K specializing to x.

In Section 4.4 we also prove a principle of continuity for intersection multiplicitiesin families over S.

Remark 4.1.4. The theorem reduces easily to the case where X and X ′ are reducedand irreducible and then, using subadditivity, it is not difficult to show that bothmust meet the generic fiber YK and are therefore flat over S. Our methods alsogive a more general lifting theorem for intersections of flat subschemes over an arbi-trary base scheme (Theorem B.4), even when subadditivity fails. Here we proceedthrough subadditivity because it eliminates the need for a flatness hypothesis andmakes the lifting arguments particularly transparent, especially for those familiarwith similar arguments over regular local rings.

4The assumption that schemes are of finite type over S is crucial for the main results of thissection. In the tropical setting, if ν is nontrivial and w is not in NG, then closed points of Xw

never lift to closed points of X. However, this does not contradict Theorem 4.2.5 because in thiscase Xw is not of finite type. See Appendix A.


4.2. A principal ideal theorem. Recall that Krull’s principal ideal theorem,translated into geometric terms, says that every component of a Cartier divisor ona noetherian scheme has codimension 1. Here we prove the following generalizationto valuation rings of rank 1.

Theorem 4.2.1. Let X be of finite type over S, and let Z be a locally principalsubscheme of X . Then every irreducible component of Z has codimension at most1 in each component of X that contains it.

The same is not true for valuation rings that are not of rank 1. If A is such a ringthen Spec A has principal subschemes of codimension greater than 1. A key stepin the proof of the theorem is the following technical proposition, which we proveby noetherian approximation.

Proposition 4.2.2. Let X be irreducible, finite type, and flat over S. Suppose thatD is a locally principal closed subscheme in X that does not meet the generic fiberXK . Then every irreducible component of Dk is an irreducible component of Xk.

Proof. Since X is flat and of finite type over S, it is of finite presentation over S.Therefore, there exists a finitely generated subring R′ ⊂ R, with models X ′ and D′

over S′ = Spec R′ of X and D, respectively. Then S′ is irreducible by construction,and we may assume X ′ is irreducible as well.

Let s′ be the image in S′ of the closed point in S. Since XK and Xk are bothpure of the same dimension d, the fibers of X ′ over the generic point and s′ arealso pure of dimension d. Similarly, Dk is pure of dimension d if and only if D′

s′ is,so to prove the proposition it is enough to show that D′

s′ is pure of dimension d.Let Z be a component of D′ that meets the fiber over s′. We will show that

Zs′ has pure dimension d. Let s′′ be the image in S′ of the generic point of Z.Then s′′ is a specialization of the generic point of S′ and specializes to s′, so uppersemicontinuity of fiber dimension [Gro66, Theorem 13.1.3] implies that X ′

s′′ haspure dimension d. We claim that Zs′′ is a component of X ′

s′′ . To see this, notethat s′′ is not the generic point of S′, since D does not meet the generic fiber ofX , and hence every component of X ′

s′′ has codimension at least 1 in X ′. Since X ′

is noetherian, Krull’s principal ideal theorem says that every component of D′ hascodimension at most 1 in X ′, and hence Zs′′ is a component of X ′

s′′ , as claimed. Inparticular, Zs′′ has pure dimension d. Since s′′ specializes to s′ and X ′

s′ has puredimension d, it follows by upper semicontinuity that Zs′ has pure dimension d, asrequired. �

The following two lemmas are special cases of a more general altitude formulaover valuation rings of finite rank [Nag66, Theorem 2]. These codimension formulasare used in the proof of the principal ideal theorem, and again in the proof ofsubadditivity of codimension over valuation rings of rank 1. We include proofs forthe reader’s convenience.

Lemma 4.2.3. Suppose X is irreducible, finite type, and flat over S with Z ⊂ Xk

an irreducible closed subset of the special fiber. Then

codimX Z = codimXkZ + 1.

Proof. A maximal chain of irreducible closed subsets between Z and Xk can beextended by adding X at the end, so codimX Z is at least codimXk

Z + 1. We nowshow that codimX Z is at most codimXk

Z + 1.


If W ⊂ W ′ is a strict inclusion of irreducible closed subsets of X that meetthe special fiber then the dimension of Wk is less than or equal to the dimensionof W ′

k. This inequality is strict if W and W ′ are both contained in the specialfiber. Similarly, if W and W ′ both meet the generic fiber then dim WK is less thandim W ′

K , and the dimensions of the special fibers are equal to the dimensions ofthe respective generic fibers, by flatness. Therefore, if dim Wk is equal to dim W ′

k

then W is contained in the special fiber and W ′ is not. This can happen at mostonce in any chain of inclusions, so any chain between Z and X has length at mostdimXk − dim Z + 1, as required. �

Lemma 4.2.4. Suppose X is irreducible, finite type, and flat over S, and let Z ⊂ Z ′be irreducible closed subsets. Then

codimX Z = codimX Z ′ + codimZ′ Z.

Proof. If Z meets the generic fiber of X then this is a classical formula for codi-mension of varieties over K. Suppose Z is contained in the special fiber. If Z ′meets the generic fiber then codimX Z ′ = dimXK − dimZ ′K , and codimZ′ Z =dimZ ′k − dimZ + 1, by Lemma 4.2.3. Now dimXK and dimZ ′K are equal todimXk and dimZ ′k, respectively, so adding these two equations gives

codimX Z ′ + codimZ′ Z = dimXk − dimZ + 1,

and Lemma 4.2.3 says that the right hand side is equal to codimX Z. The proof inthe case where Z ′ is contained in the special fiber is similar. �

We now prove the principal ideal theorem over valuation rings of rank 1.

Proof of Theorem 4.2.1. We may assume X is reduced and irreducible. If X issupported in the special fiber, the theorem reduces to the classical principal idealtheorem over k. Otherwise X is dominant, and hence flat, over S. If Z meetsthe generic fiber XK , then the theorem follows from the classical principal idealtheorem for XK . On the other hand, if Z is contained in the special fiber, then itmust be a union of components of the special fiber, by Proposition 4.2.2, and hencehas codimension 1 in X, by Lemma 4.2.3. �

We apply the principal ideal theorem to prove the following result on lifting closedpoints in the special fiber to closed points in the generic fiber. Our argument is inthe spirit of Katz’s proof of Lemma 4.15 in [Kat09b].

Theorem 4.2.5. Let X be a scheme locally of finite type over S, and let x be aclosed point of the special fiber Xk. If x is in the closure of XK then there is aclosed point x′ in XK specializing to x. Moreover, the set of all such x′ is Zariskidense in every component of XK with closure containing x.

Remark 4.2.6. For the initial degeneration of a subscheme of a torus T over Kassociated to a weight vector w ∈ NG, the theorem says that every point in Xw(k)lifts to a point in X(K), and the set of such lifts is Zariski dense. In particular, thisgives an algebraic proof of surjectivity of tropicalization, along the lines suggestedby Speyer and Sturmfels in [SS04], and also yields a new proof of the density oftropical fibers [Pay09b].

Proof. We may assume X is integral and affine. Then x is a specialization of thegeneric point of X . If XK is zero-dimensional, then the generic point is closed, sowe can take that to be x′. We proceed by induction on dimXK .


Suppose XK has positive dimension d. Let f be a regular function on X thatvanishes at x but not on any component of Xk, and let D ⊂ X be the principalsubscheme cut out by f . Passing to a smaller neighborhood of x, if necessary,we may assume every component of D contains x. By Proposition 4.2.2, somecomponent Z of D meets the generic fiber, and hence ZK has dimension d− 1. Byinduction, ZK contains a closed point x′ specializing to x, and the first assertionof the theorem follows.

In order to prove the Zariski density assertion, it suffices to prove that given achoice of the restriction f |Xk

, and given any proper closed subset W ⊂ XK , wecan choose f having the given restriction to Xk and such that f does not vanishon any component of W . Using a closed immersion X ⊂ An

S for some n, we maylift f |Xk

to a polynomial over k of some degree d, and since the polynomials onAn

K vanishing on any component of W form a proper Zariski closed subset of theaffine space of all polynomials of degree (at most) d, we have reduced the statementdown to the case that X itself is affine space over S. In this case, if we first supposethat x is k-rational, one checks directly (for instance by induction on the numberof variables, and using that the maximal ideal of R is necessarily infinite) that wecan find a lift to R which does not vanish for any particular polynomial over K.On the other hand, if x is defined over some finite extension k′ of k, we can find avaluation ring R′ with residue field k′ and fraction field K ′ an algebraic extensionof K, such that R′ dominates R, and taking the base change to R′ we reduce backto the rational case. �

4.3. Proofs of subadditivity and lifting. We now use the principal ideal the-orem to prove subaddivity of codimension under intersection and deduce a liftingtheorem for proper intersections, in smooth schemes over a valuation ring of rank1. Our proof follows the classical argument for smooth varieties, which is simplerthan Serre’s proof for regular schemes.

Remark 4.3.1. Codimension is not subadditive under intersection in smooth schemesover valuation rings of rank greater than 1, as shown by the following example. Nev-ertheless, this failure of subadditivity can be understood and controlled, and properintersections in special fibers still lift to more general fibers over arbitrary valuationrings. See Appendix B.

Example 4.3.2. Let A be a valuation ring of finite rank r greater than 1, andlet a be an element of A that generates an ideal of height r. Then A[x] has Krulldimension r + 1, by [Nag66, Theorem 2]. The ideals of A[x] generated by x andx − a each have height 1, but the ideal (x, x − a) = (x, a) has height r + 1, whichis greater than 2.

Our proof of Theorem 4.1.1 involves a reduction to the diagonal and requires thefollowing lemma on dimensions of fiber products over S.

Lemma 4.3.3. Let Y be irreducible and finite type over S, and let X and X ′ beirreducible closed subschemes. Then for every irreducible component Z of X ×S X ′,

codimY×SY Z ≤ codimY X + codimY X ′.

Furthermore, equality holds unless Y is dominant over S and X and X ′ are bothcontained in the special fiber.


Proof. If X and X ′ both meet the generic fiber then they are flat over S. In thiscase, Z must also meet the generic fiber, and the proposition holds with equality,by classical dimension theory over K. Similarly, if Y is not dominant over S, thenthe proposition holds with equality by dimension theory over k.

Suppose Y is dominant and X is contained in the special fiber. Then Z is alsocontained in the special fiber. If, furthermore, X ′ is contained in the special fiber,then Z has dimension dimX + dimX ′. Therefore, by Lemma 4.2.3,

codimY×SY Z = 2dimYK − dimX − dimX ′ + 1.

The right hand side is equal to codimY X +codimY X ′−1, so the inequality is strictby exactly one in this case. A similar argument shows that the proposition holdswith equality if X ′ meets the generic fiber. �

We will prove subadditivity of codimension by intersecting with diagonal in Y×S

Y, and applying the previous proposition and the following lemma.

Proposition 4.3.4. If Y is smooth over S then the diagonal ∆ in Y ×S Y is alocal complete intersection subscheme.

Here, by local complete intersection we mean that the number of local generatorsfor I∆ is equal to the codimension.

Proof. First note that the diagonal is locally of finite presentation, by [Gro64,Corollary 1.4.3.1]. The lemma follows, since a locally finite presentation immersionof a smooth scheme in another smooth scheme over an arbitrary base is a localcomplete intersection. See Proposition 7 of Section 2 in [BLR90]. �

We now proceed with the proof of subadditivity of codimension and lifting ofproper intersections in the special fiber, over valuation rings of rank 1.

Proof of Theorem 4.1.1. The intersection X ∩ X ′ can be realized as in the inter-section of X ×S X ′ with the diagonal in Y ×S Y ′. Now, the codimension of anycomponent Z of X ×S X ′ is at most

codimY×SY Z ≤ codimY X + codimY X ′,

by Lemma 4.3.3. Then the principal ideal theorem, Theorem 4.2.1 and Lemma 4.2.4together imply that codimension can only decrease when intersecting with a localcomplete intersection subscheme of Y×SY. Proposition 4.3.4 says that the diagonalis a local complete intersection, and the theorem follows. �

Proof of Theorem 4.1.3. Let Z be a component of X ∩ X ′ that contains x. ByTheorem 4.1.1,

codimY Z ≤ codimY X + codimY X ′.

The codimension of the special fiber Zk in Y is equal to codimY X +codimY X ′+1,by Lemma 4.2.3, so Z must meet the generic fiber. Therefore, the generic point ofZK is a point of XK ∩ X ′

K specializing to x. If x is closed in its fiber, we can thenfind a closed point of ZK specializing to x, by Theorem 4.2.5. �


4.4. Intersection multiplicities over valuation rings of rank 1. The principleof continuity says that intersection numbers are constant when cycles vary in flatfamilies. See [Ful98, Section 10.2] for a precise statement and proof when thebase is a smooth variety. For applications to tropical lifting with multiplicities,we need to apply a principle of continuity over the spectrum S of a possibly non-noetherian valuation ring of rank 1. Lacking a suitable reference, we include aproof. The argument generalizes immediately to valuation rings of higher rank,and the analogous statement over an arbitrary base scheme can be reduced to thiscase. See Remark 4.4.4.

Definition 4.4.1. Suppose Y is smooth over a field k, and let X and X ′ beclosed subschemes of Y whose intersection X ∩X ′ is finite. Then the intersectionnumber i(X ·X ′;Y ) of X and X ′ in Y is

dim Y∑i=0

(−1)i dimk ToriOY

(OX ,OX′).

In other words, the intersection number is the sum of the local intersection mul-tiplicities at the finitely many points of X ∩X ′, weighted by degrees of extensionfields,

i(X ·X ′;Y ) =∑

P∈X∩X′

[κ(P ) : k]i(P,X ·X ′;Y ).

While we are primarily interested in the case where X and X ′ have complementarydimension, so the finiteness of X ∩X ′ means that X and X ′ meet properly in Y ,this hypothesis is not technically necessary, since the local intersection multiplicitiesvanish when the intersection is not proper [Ser65, Theorem V.3].

Theorem 4.4.2. Let Y be smooth and quasiprojective over S, and let X and X ′

be closed subschemes of Y that are flat over S. If X ∩ X ′ is finite over S then

i(XK · X ′K ;YK) = i(Xk · X ′

k;Yk).

In other words, intersection numbers are invariant under specialization over S. Wenow give a direct proof of this equality, by first establishing coherence properties forthe structures sheaves of these schemes and then using free resolutions to computethe Tor groups. An alternative approach, using noetherian approximation andspecialization properties of intersection theory over a noetherian base, is sketchedin Remark 4.4.5.

In general, the structure sheaf of a non-noetherian scheme is not necessarilycoherent. The following proposition asserts that such pathologies do not occur onthe schemes we are considering.

Proposition 4.4.3. Let X be of finite presentation over S. Then OX is coherent.

Proof. We first claim that OX is coherent in the special case where X is the affinespace Spec R[t1, . . . , tn]. Since the question is local, it is enough to show that forany affine subset U of X , and any homomorphism f : Om

X |U → OX |U , the kernelof f over U is finitely generated. Now, the image of f is finitely generated, andis the quasicoherent ideal sheaf I associated to some ideal I in OX (U). The idealI is torsion-free and hence flat over S, so it is finitely presented. Therefore, by[Mat89, Theorem 2.6], the kernel of the given presentation OX (U)m → I is finitelygenerated, which proves the claim.


For the general case, since coherence is local, we may again assume that X isaffine, so X = Spec A[t1, . . . , tn]/I for some finitely generated ideal I. The sameargument as in the special case above shows that I is finitely presented. ThereforeI is coherent on Spec R[t1, . . . , tn], and hence OX is coherent [Gro60, 5.3.10]. �

Proof of Theorem 4.4.2. Since Y is quasiprojective and X∩X ′ is finite over S, thereis an affine open subset of Y containing the intersection, and hence we may assumeY is affine. Now OY is coherent, by Proposition 4.4.3, and OX and OX ′ , being flatand finitely generated, are finitely presented and hence coherent as OY -modules.Let P• be a resolution of OX by free OY -modules of finite rank. Then, since OXis flat over S, the restrictions P• ⊗K and P• ⊗ k to the generic and special fibersremain exact, giving resolutions of OXK

and OXk, respectively. Therefore, not only

does the homology ofQ• = P• ⊗OX ′

compute TorY(X ,X ′), but also the homology of the base changes Q•⊗K and Q•⊗kcomputes TorYK

(XK ,X ′K) and TorYk

(Xk,X ′k), respectively.

Now, ToriY(X ,X ′) is coherent, and supported on X ∩ X ′ [Gro63, 6.5.1]. Since

π : X ∩ X ′ → S is finite and hence projective, it follows that the push forwardπ∗ Tori

Y(X ,X ′) is coherent [BGI71, Expose III, Theorem 2.2.2], and since π isaffine, this push forward can be computed as the homology of the complex

L• = π∗Q•.

As above, since OX is flat over S, the homology of L• ⊗ K and L• ⊗ k computeTori

YK(XK ,X ′

K) and ToriYk

(Xk,X ′k), respectively. Note also that since OX ′ is flat

over S, the terms of Q• and hence L• are flat OS-modules.Since the homology of L• is coherent, there is a quasi-isomorphic bounded below

complex M• of free OS-modules of finite rank, and since L• is flat over S, therestrictions M• ⊗ K and M• ⊗ k are quasi-isomorphic to L• ⊗ K and L• ⊗ k,respectively [Gro61b, Chapter 0, Remark 11.9.3]. The homology of M• vanishes inhigh degree, so we can truncate the complex, replacing some Mi by the image ofMi+1, which is coherent and torsion-free and hence free of finite rank. In particular,we may assume that M• is also bounded above. It is then clear that

i(Xs · X ′s,Ys) =

∑i

(−1)i rkMi,

for s equal to either the generic or closed point of S, and the theorem follows. �

Remark 4.4.4. The proofs of Proposition 4.4.3 and Theorem 4.4.2 go through un-changed over an arbitrary valuation ring. It is also easy to see that intersectionnumbers are invariant under extension of the base field. A standard reduction tothe case of valuation rings, using [Gro61a, Proposition 7.1.4], then shows that inter-section numbers are preserved under specialization over an arbitrary base scheme.

Remark 4.4.5. Theorem 4.4.2 can also be proved over an arbitrary base schemeby first passing to the closure of the more general point, and then taking an affineneighborhood around the special point, to reduce to the case where the base is affineand irreducible. This ensures that X and X ′ are finitely presented over the baseS, because they are flat and finite type, and then we can proceed by noetherianapproximation. Say the affine base is Spec A. Then there is a finitely generatedsubalgebra A0 of A over which X, X ′, and Y and all of the relevant morphisms


are defined, and all can be chosen so that the relevant geometric properties arepreserved, including that the models of X and X ′ are flat [Gro66, Sections 11 and12], and the model of Y is smooth [Gro67, Section 17.7]. In particular, we mayassume the base is noetherian. Then, since the base is noetherian, we can reduce tothe case of a DVR, instead of an arbitrary valuation ring, and the theorem followsfrom the fact that intersection numbers are preserved by specialization over a DVR[Ful98, Section 20.3].

In the geometric case that we are interested in, where the fraction field K isalgebraically closed, and hence the residue field k is algebraically closed as well,we apply Theorem 4.4.2 on intersection numbers to get the following result onindividual intersection multiplicities, along components of the intersection of thespecial fibers. Note that this result holds even when the intersections are positivedimensional.

Theorem 4.4.6. Assume K is algebraically closed. Let Y be smooth and quasipro-jective over S, and let X and X ′ be closed subschemes that are flat over S. SupposeXk and X ′

k meet properly along a component Z of their intersection. Then

i(Z,Xk · X ′k;Yk) =

∑eZ

m(Z, Z) · i(Z,XK · X ′K ;YK),

where the sum is over irreducible components Z of XK ∩X ′K whose closures contain

Z, and m(Z, Z) is the multiplicity of Z in the special fiber of the closure of Z insideY.

An important lemma is the following flatness result, complementing Proposition4.2.2.

Lemma 4.4.7. Let X be finite type and flat over S. Suppose that D is a locallyprincipal closed subscheme in X that does not contain any associated point of thespecial fiber Xk. Then D is flat over S.

Note that in the proof which follows, S could be any integral scheme.

Proof. We first use noetherian approximation to reduce to the case that S is thespectrum of a DVR. Observe that the condition that D does not contain any asso-ciated point of Xk is invariant under base change, by [Gro65, Proposition 4.2.7(i)].Also, the statement is local, so we may assume that X = Spec A is affine, andD = V (f) for some f ∈ A. Now, X is necessarily of finite presentation, so by[Gro66, Corollary 11.2.7], we have a noetherian approximation for X over S whichis still flat at any chosen point, and which can be enlarged so that f is defined overit. Thus, we may assume S (and therefore X ) is noetherian, at the cost of allowingit to be an arbitrary integral affine scheme. However, by [Gro66, Theorem 11.8.1],to check flatness of D it is then enough to check flatness after arbitrary base changeto the spectrum of a DVR, so we have reduced to the case that R is a DVR, asdesired.

Now, flatness of D is equivalent to the assertion that no non-zero t ∈ R is a zerodivisor in A/fA. This is turn is equivalent to showing that f is not a zero divisorin A/tA. Given g ∈ A r tA, suppose that fg ∈ tA. Since the valuation on R isdiscrete, there exists some s ∈ R such that s|g, and g′ := g/s is not in mA. Thenν(s) < ν(t), so fg = sfg′ ∈ tA together with flatness of A over R implies that


fg′ ∈ (t/s)A ⊆ mA, and we conclude that f is a zero divisor modulo mA. But by[Gro65, Corollary 3.1.9], this contradicts the hypothesis that f is not contained inany associated prime of A. �

Proof of Theorem 4.4.6. We first prove the special case where X and X ′ have com-plementary dimension. Suppose Z is a point. We claim that there is a Zariskiopen neighborhood U of Z such that X ∩ X ′ ∩ U is finite over S and has specialfiber exactly Z. By upper semicontinuity of fiber dimension, the union W1 of thepositive dimensional components of XK ∩ X ′

K and Xk ∩ X ′k is closed in Y. The

intersection of XK and X ′K in the complement of W1 is a finite set of K-points. Let

W2 be the closure of those points in this set that do not specialize to Z. We claimthat U = Y r (W1 ∪ W2) is the desired neighborhood. Now X ∩ X ′ ∩ U is sepa-rated, quasifinite over S and has special fiber Z, by construction, and it is locallyof finite presentation, since X and X ′ are finite type and flat, and hence of finitepresentation. By [Gro66, Theorem 8.11.1], to show that this intersection is finite itonly remains to check that it is proper over S. The generic fiber consists of finitelymany K-points, each of which extends to an R-point, and one can check that thisimplies the valuative criterion for universal closedness [Gro61a, Theorem 7.3.8],which proves the claim. Now, note that m(Z, Z) is 1 for every Z, since each Z is aK-point whose closure is a section. The required equality of intersection numbersthen follows immediately from Theorem 4.4.2.

We now prove the general case, where the intersection may have positive dimen-sion, by reducing to the case of a zero-dimensional intersection. Since the statementis local on Y, we may assume Y is affine, that Z is the only component of Xk ∩X ′

k,and that every component of XK ∩X ′

K contains Z in its closure. Let Z1, . . . , Zm bethe irreducible components of XK ∩ X ′

K . Localizing further, we may also assumethat Z and the Zi are all smooth (although we cannot assume that the closures ofthe Zi are smooth, as Z may appear with multiplicity greater than 1).

Now, let Lk be a general linear subspace of complementary dimension in thespecial fiber, so Lk meets Z transversely at finitely many points. Choose a generallift L of Lk to S; by the Zariski density of Theorem 4.2.5 applied to the Grassman-nian over S, the generic fiber LK meets each component XK ∩ X ′

K transversely, atisolated points. After further localizing, we may assume Lk meets Z at a singlepoint z, and every point z of XK ∩X ′

K ∩LK specializes to z. Since L is a completeintersection, applying [Ful98, Example 8.1.10] in the special fiber, we have

i(Z,Xk · X ′k;Yk) = i(z, (Xk ∩ Lk) · (X ′

k ∩ Lk);Lk).

Now, note that induction application of Lemma 4.4.7 implies that X ∩L and X ′∩Lremain flat over S, so by the zero-dimensional case treated above, the right handside is equal to ∑

ez i(z, (XK ∩ LK) · (X ′K ∩ LK);LK).

Now, if Zi is the component of XK ∩ X ′K containing z, then a similar argument in

the generic fiber gives

i(z, (Xk ∩ Lk) · (X ′k ∩ Lk);Lk) = i(Zi,XK · X ′

k;YK).

The theorem follows, since again using the 0-dimensional case we have that themultiplicity m(Zi, Z) is the number of points z in Zi ∩ LK specializing to z. �


5. Lifting tropical intersections with multiplicities

We now use the results of Sections 3 and 4 to prove the main tropical liftingtheorems, both as stated in the introduction and in refined forms with multiplicities.We also prove generalizations to intersections of three or more subvarieties anddiscuss weaker lifting results when the tropicalizations do not meet properly.

Without further comment, we make repeated use of Proposition A.1 and The-orem A.4. The first says that the integral model Xw is of finite type over Spec Rwhen w is in NG, and the second is a version of invariance of initial degenerationsunder extensions of valued fields that allows many questions to be reduced to thiscase.

5.1. Intersections of two subvarieties. Let Y be a subvariety of T , and let Xand X ′ be subvarieties of codimension j and j′, respectively, in Y .

Proof of Theorem 1.3. Suppose Trop(X) meets Trop(X ′) properly at a simple pointw of Trop(Y ). After extending the valued field, we may assume w ∈ NG.

The initial degeneration Yw is a torus containing Xw and X ′w, and the tropical-

izations Trop(Xw) and Trop(X ′w) with respect to the trivial valuation are the stars

of w in Trop(X) and Trop(X ′), respectively, and hence meet properly in the vectorspace Trop(Yw). Theorem 3.3.1 then says that

Trop(Xw ∩X ′w) = Trop(Xw) ∩ Trop(X ′

w).

In particular, Xw∩X ′w is nonempty of codimension j + j′, and the theorem follows,

since Yw is smooth. �

Proof of Theorem 1.4. Suppose Xw meets X ′w properly at a smooth point x of Yw.

After extending the valued field we may assume w ∈ NG. In this case, the integralmodel Yw is of finite type over Spec R, by Proposition A.1. Theorem 4.1.3 thensays that there is a point of X ∩X ′ specializing to x, and hence x is contained in(X ∩X ′)w. �

As noted in the introduction, Theorem 1.2 follows immediately from Theo-rems 1.3 and 4.1.3, and Theorem 1.1 is the special case where Y is the torus T . Sothis concludes the proof of the theorems stated in the introduction.

We now state and prove a refined version of Theorem 1.2 with multiplicities. IfTrop(X) and Trop(X ′) meet properly at a simple point w of Trop(Y ), and τ isa facet of Trop(X) ∩ Trop(X ′) containing w, then the tropical intersection multi-plicity i(τ,Trop(X) · Trop(X ′); Trop(Y )) along τ in Trop(Y ) is defined by a localdisplacement rule, as in Section 2.6, but the displacement vector is required to bein the subspace of NR parallel to the affine span of the facet of Trop(Y ) containingw.

Theorem 5.1.1. Suppose Trop(X) meets Trop(X ′) properly along a face τ ofcodimension j + j′ that contains a simple point of Trop(Y ). Then τ is contained inTrop(X∩X ′) with multiplicity bounded below by the tropical intersection multiplicity

mX∩X′(τ) ≥ i(τ,Trop(X) · Trop(X ′); Trop(Y )),

and both are strictly positive. Furthermore, the tropical intersection multiplicity isequal to the weighted sum of algebraic intersection multiplicities

i(τ,Trop(X) · Trop(X ′); Trop(Y )) =∑Z

i(Z,X ·X ′;Y ) mZ(τ),


where the sum is over components Z of X ∩X ′ whose components contain τ .

Proof. First, by Theorem 1.2, the face τ is contained in Trop(X ∩X ′). We provethe identity between the tropical intersection multiplicity and the weighted sum ofalgebraic intersection multiplicities and then deduce the inequality for mX∩X′(τ),as follows. Let w be a point in τ that is a simple point of Trop(Y ). After extendingthe valued field, we may assume w ∈ NG. Then the tropical intersection multiplicityalong τ is equal to the local tropical intersection multiplicity at w

i(τ,Trop(X) · Trop(X ′); Trop(Y )) = i(τw,Trop(Xw) · Trop(X ′w); Trop(Yw)).

Then, since Yw is a torus, Theorem 3.3.1 says that this local tropical intersectionmultiplicity is given by

i(τw,Trop(Xw) · Trop(X ′w); Trop(Yw)) =

∑Z0

i(Z0, Xw ·X ′w;Yw)mZ0(τw),

where the sum is over components Z0 of Xw ∩X ′w whose tropicalizations contain

τw. By Theorem 4.4.6, for each such Z0,

i(Z0, Xw ·X ′w;Yw) =

∑Z

m(Z,Z0) i(Z,X ·X ′;Y ),

where the sum is over components Z of X ∩ X ′ and m(Z,Z0) is the multiplicityof Z0 in the special fiber of the closure of Z. Combining these identities gives therequired identity, since mZ(τ) =

∑Z0

m(Z,Z0)mZ0(τ), where the sum ranges overcomponents Z0 of special fiber of the closure of Z.

The inequality follows immediately from this identity, since

mX∩X′(τ) =∑Z

length(OZ,X∩X′)mZ(τ),

and length(OZ,X∩X′) ≥ i(Z,X ·X ′;Y ), for each Z. �

If Trop(X) meets Trop(X ′) properly in NR, then part (2) of Theorem 5.1.1 has aparticularly simple statement equating the tropicalization of the refined intersectioncycle with the stable tropical intersection.

Corollary 5.1.2. Suppose Y is smooth, Trop(X) meets Trop(X ′) properly inTrop(Y ), and simple points of Trop(Y ) are dense in their intersection. Then

Trop(X ·X ′) = Trop(X) · Trop(X ′).

In particular, if the ambient space Y is just the dense torus T , then this identityholds whenever Trop(X) and Trop(X ′) meet properly in NR.

We also note that the inequality in Theorem 5.1.1 can be replaced by an equalityif X and X ′ are smooth, or mildly singular.

Corollary 5.1.3. Under the hypotheses of Theorem 5.1.1, if X and X ′ are Cohen-Macaulay then

mX∩X′(τ) = i(τ,Trop(X) · Trop(X ′); Trop(Y )).

Proof. If X and X ′ are Cohen-Macaulay, then the Tor groups TorjOY

(OX ,OX′)vanish, for j > 0. Therefore, the length length(OZ,X∩X′) of the scheme-theoretic


intersection is equal to the intersection multiplicity i(Z,X ·X ′;Y ) for each compo-nent Z of X ∩X ′ of expected dimension. The result then follows from the equalityin Theorem 5.1.1, since

mX∩X′(τ) =∑Z

length(OZ,X∩X′)mZ(τ),

where the sum is over components Z of X ∩X ′ such that Trop(Z) contains τ . �

5.2. Intersections of three or more subvarieties. In applications, and partic-ularly in the context of enumerative geometry, one frequently wants to intersectmore than two subschemes. Once intersection theory is developed suitably, theintersection product of several subvarieties X1 · · ·Xr may be treated either by in-duction from results on intersection of pairs or by reducing to the intersection ofa product X1 × · · · ×Xr with the small diagonal in Y r. The corresponding state-ment for stable tropical intersections follows from the compatibility of algebraic andstable tropical intersections, discussed in Section 2.7. Because the intersection oftwo tropicalizations may not be the tropicalization of any variety, for the followinglifting theorem we do not argue by induction, but instead by a reduction to thediagonal.

Theorem 5.2.1. Let X1, . . . , Xr be subvarieties of codimension j1, . . . , jr in Y ,respectively. Suppose τ is a facet of Trop(X1) ∩ · · · ∩ Trop(Xr) of codimensionj1 + · · · + jr in Trop(Y ) that contains a simple point of Trop(Y ). Then the trop-icalization Trop(X1 ∩ · · · ∩ Xr) contains τ with multiplicity bounded below by thetropical intersection multiplicity

mX1∩···∩Xr(τ) ≥ i(τ,Trop(X1) · · ·Trop(Xr); Trop(Y )),

and both are stricly positive. Furthermore, the tropical intersection multiplicity isequal to the weighted sum of algebraic intersection multiplicities

i(τ,Trop(X1) · · ·Trop(Xr); Trop(Y )) =∑Z

i(Z,X1 · · ·Xr;Y )mZ(τ),

where the sum is over components of X1 ∩ · · · ∩Xr such that Trop(Z) contains τ .

Proof. The multiple intersection X1 ∩ · · · ∩ Xr in Yw is naturally identified withthe pairwise intersection of the product X1 × · · · ×Xr with the small diagonal ∆in Y r. Similarly, there is a natural identification

Trop(X1) ∩ · · · ∩ Trop(Xr) ∼= Trop(X1 × · · · ×Xr) ∩ Trop(∆).

It follows from the compatibility of algebraic and tropical intersection multiplici-ties, discussed in Section 2.7, that each of the tropical intersection multiplicitiesi(τ,Trop(X1) · · ·Trop(Xr); Trop(Y )) is equal to the tropical intersection multiplic-ity of Trop(X1 × · · · ×Xr) and Trop(∆) along the corresponding face of their in-tersection. The theorem then follows immediately from Theorem 5.1.1 for pairwiseintersections. �

The following application of Theorem 5.2.1, giving a formula for counting pointsin zero-dimensional complete intersections, has been used by Rabinoff in a morearithmetic setting [REF]. Let fi =

∑ai(u)xu be a defining equation for Xi, with

Pi = conv{u|ai is nonzero} its Newton polytope. Projecting the lower faces of thelifted Newton polytope conv{(u, ν(ai(u))} in MR ×R gives the Newton subdivisionof Pi. This Newton subdivision is dual to Trop(Xi), in the following sense. The


faces of the unique minimal polyhedral structure on Trop(Xi) are in canonical order-reversing bijection with the positive dimensional faces of the Newton subdivision.A face τ of Trop(Xi) corresponds to the convex hull of the lattice points u suchthat ai(u)xu is a monomial of minimal w-weight, for w in the relative interior of τ .

Our formula is phrased in terms of mixed volumes of faces of the Newton subdi-vision. Recall that, for lattice polytopes Q1, . . . , Qn in MR, the euclidean volume ofthe Minkowski sum b1Q1 + · · ·+ bnQn is a polynomial of degree n in b1, . . . , bn, andthe mixed volume V (Q1, . . . , Qn) is the coefficient of b1 · · · bn. If Σ is the smallestcommon refinement of the inner normal fans of the Qi, then each Qi correspondsto a nef line bundle Li on the toric variety Y (Σ), and the mixed volume is equal tothe intersection number

V (Q1, . . . , Qn) = (c1(L1) · · · c1(Ln)).

See [Ful93, p. 116] for further details on mixed volumes and their relation to toricintersection theory.

Theorem 5.2.2. Let X1, . . . , Xn be hypersurfaces in T , and suppose w is an iso-lated point in Trop(X1)∩· · ·∩Trop(Xn). Let Qi be the face of the Newton subdivisioncorresponding to the minimal face of Trop(Xi) that contains w. Then the numberof points in X1 ∩ · · · ∩ Xn with tropicalization w, counted with multiplicities, isexactly the mixed volume V (Q1, . . . , Qn).

Proof. Fix a complete unimodular fan Σ in NR such that, for i = 1, ..., n, every faceof Starw Trop(Xi) is a union of faces of Σ. Since Starw Trop(Xi) is the codimension1 skeleton of the inner normal fan of Qi, it follows that Qi corresponds to a nefline bundle Li on Y (Σ). Furthermore, the Minkowski weight of codimension 1on Σ given by the tropical multiplicities on Starw Trop(Xi) corresponds to c1(Li).It follows, by the compatibility of toric and stable tropical intersections that thetropical intersection multiplicity

i(w,Trop(X1) · · ·Trop(Xn)) = (c1(L1) · · · c1(Ln)),

and the latter is the mixed volume V (Q1, . . . , Qn). By Theorem 5.2.1, this tropicalintersection multiplicity is strictly positive, and equal to the number of points inX1 ∩ · · · ∩Xn with tropicalization w, counted with multiplicities, as required. �

Remark 5.2.3. In Theorem 5.2.2, the multiplicities on the points x in X1∩ · · ·∩Xn

with tropicalization w are equal to the lengths of the scheme-theoretic intersectionof X1, . . . , Xn along x, as in Corollary 5.1.3, since complete intersections are Cohen-Macaulay.

5.3. Non-proper intersections. Our main tropical lifting results depend heavilyon the hypothesis that Trop(X) meets Trop(X ′) properly. However, we still obtainsome nontrivial consequences under weaker hypotheses. For instance, we have

Corollary 5.3.1. Suppose that Yw is smooth for every w ∈ Trop(X) ∩ Trop(X ′).Suppose further that for every w ∈ Trop(X) ∩ Trop(X ′), we have that Xw meetsX ′

w properly in Yw if the intersection is non-empty. Then the set of w such thatXw ∩X ′

w 6= ∅ is a polyhedral complex of pure codimension j + j′ in Trop(Y ).

Proof. Given x ∈ X∩X ′, let w = Trop(x); then x specializes to a point of Xw∩X ′w

inside Yw, which is smooth by hypothesis. It follows that Y is smooth at x, andthus the codimension of X ∩X ′ in Y at x is at most j + j′. On the other hand, by


hypothesis Xw∩X ′w has codimension j+j′ inside Yw, so it follows by semicontinuity

of fiber dimension that X ∩X ′ has codimension exactly j + j′ in Y at x. Finally,it follows from Theorem 1.4 that under our hypotheses, the set of w with Xw ∩X ′

w

non-empty is precisely Trop(X ∩X ′), so we conclude the desired result. �

As a further application of these methods, we extend Proposition 2.7.4 to thecase where the valuation may be nontrivial, identifying stable tropical intersectionswith tropicalizations of intersections with general translates.

If t ∈ T (K), then the tropicalization of the translate tX is the translate ofTrop(X) by the vector Trop(t) in in NG. In particular, the tropicalization is invari-ant under translation by points t such that Trop(t) is zero. The set of such pointsis Zariski dense in T (K) [Pay09b], but not Zariski open. Nevertheless, we say thata property holds for a general point t such that Trop(t) is zero if there is a Zariskiopen subset U0 of the initial degeneration T0 such that the property holds for all twith t0 ∈ U0. This set is again Zariski dense in T (K).

Theorem 5.3.2. Let X and X ′ be subvarieties of T . Then, for a general point tsuch that Trop(t) is zero,

Trop(X ∩ tX ′) = Trop(X) · Trop(X ′).

Proof. We assume that the valuation is nontrivial, since we have already handledthe case of trivial valuation. Let τ be a face of Trop(X)∩Trop(X ′), and let w ∈ NG

be a point in τ . The initial degeneration (tX ′)w is the translate of X ′w by t0. Since

t0 is general in T0, Proposition 2.7.4 says that

Trop(Xw ∩ (tX ′)w) = Trop(Xw) · Trop(X ′w).

In particular, if w is not in the stable tropical intersection then the initial degen-erations Xw and (tX ′)w are disjoint, so w is not in Trop(X ∩ tX ′). On the otherhand, if w is in the stable intersection then Xw meets (tX ′)w properly in Tw, andall points of Xw ∩ (tX ′)w lift to the general fiber X ∩ tX ′, by Theorem 4.1.3. Notethat for w,w′ in the relative interior of τ , because the isomorphisms Xw

∼= Xw′ andtX ′

w∼= tX ′

w′ can be simultaneously induced by a single isomorphism Tw∼= Tw′ , if

Xw meets tX ′w then Xw′ likewise meets tX ′

w′ properly. Since there are only finitelymany faces of Trop(X) ∩ Trop(X ′), the point t can be chosen sufficiently generalso that this holds in every face.

This shows that the underlying sets of Trop(X ∩ tX ′) and Trop(X) · Trop(X ′)are equal. The fact that the multiplicities agree is a straightforward applicationof Proposition 2.7.4 and Theorem 4.4.6, with the latter used as in the proof ofTheorem 5.1.1. �

6. Examples

Here we give a number of examples illustrating our tropical lifting theorems andthe necessity of their hypotheses. The first example involves tropicalizations thatmeet properly, but not necessarily in the interiors of maximal faces.

Example 6.1. Inside Y = (K∗)2, let X be the line y = x + 1, and let X ′ be theparabola y = ax2, for some a ∈ K∗. We consider three cases, according to whetheror not ν(a) is zero, and its sign if it is nonzero.


Trop(X)

ν(a) < 0

ν(a) = 0

ν(a) > 0

Trop(X ′)

��

��

��

��

��

��

If ν(a) is positive then Trop(X) meets Trop(X ′) at two points, each with multi-plicity 1. In this case, Theorem 5.1.1 says that X meets X ′ at two points, eachwith multiplicity 1, and one of these intersection points lies over each of the pointsof Trop(X) ∩ Trop(X ′). It is also straightforward to check this directly. If ν(a) isnegative, then Trop(X) meets Trop(X ′) at a single point, but with tropical inter-section multiplicity 2. In this case, Theorem 5.1.1 says that X meets X ′ at eithera single point with multiplicity 2, or at two points of multiplicity 1. One can checkthat both possibilities do occur. In both of the previous cases, the nonemptinessof X ∩X ′ also follows from the transverse tropical lifting result of [BJS+07], sincethe tropicalizations meet properly in the interiors of maximal faces.

Suppose ν(a) is zero. Then Trop(X) and Trop(X ′) meet at a single point withtropical multiplicity 2, but the intersection is in a nonmaximal face of Trop(X),so transverse lifting results do not apply. Nevertheless, Theorem 5.1.1 still saysthat X and X ′ meet at either a single point with multiplicity 2, or at two pointsof multiplicity 1. Either possibility can occur; the algebraic intersection is a singlepoint of multiplicity 2 when the characteristic is not 2 and a is equal to −1/4, andtwo points of multiplicity 1 otherwise.

In the following example, the tropicalizations meet nonproperly along a positivedimensional set that does not contain the the tropicalization of any curve.

Example 6.2. Inside Y = (K∗)2, let X and X ′ be given by y = x+1 and y = ax+b,respectively. Assume a and b are not both 1, so the closures of X and X ′ are distinctlines in K2. In particular, X and X ′ intersect in at most one point.

Suppose ν(a) is zero and ν(b) is positive. Then Trop(X) and Trop(X ′) intersectnonproperly, along the ray R≤0(1, 1).


(0, 0)

Trop(X)

Trop(X ′)

��

��

��

��

��

��

��

If a is 1, then the closures of X and X ′ are parallel lines in K2, so none of thetropical intersection points lift.

Suppose a is not 1. Then the unique algebraic intersection point is

X ∩X ′ = ((1− b)/(a− 1), (a− b)/(a− 1)),

and the unique point of Trop(X ∩ Trop(X ′) that lifts is (−ν(a − 1),−ν(a − 1)).In particular, for a suitable choice of a congruent to 1 modulo m, any nonzeroG-rational point in the tropical intersection can lift. For such a, the initial degen-erations Xw and X ′

w coincide for all w in Trop(X)∩Trop(X ′). In particular, noneof the initial degenerations meet properly.

On the other hand, if a is not congruent to 1 modulo m, then the initial degen-erations Xw and X ′

w meet transversely at a single point for w = (0, 0), and aredisjoint otherwise. In this case (0, 0) is the unique tropical intersection point thatlifts, as it must be by Theorem 1.4.

The remaining examples involve tropicalizations of skew lines inside a smoothquadric surface. These examples demonstrate the necessity of requiring the pointof proper intersection to be a simple point of Trop(Y ) or a smooth point of Yw inour tropical lifting results for intersections inside an ambient subvariety, such asTheorems 1.2, 1.3, and 1.4.

Example 6.3. Let Y be the surface in (K∗)3 given by

z2 + xy + x + y = 0.

Then Y contains the curves X and X ′ given by (x + 1, z − 1) and (y + 1, z + 1),respectively. Now, Trop(X) and Trop(X ′) still meet properly at w = (0, 0, 0), asshown.


Trop(Y )

Trop(X ′)

Trop(X)

w

but this tropical intersection point does not lift because the closures of X andX ′ are skew lines in K3. This is consistent with Theorems 1.2 and 1.3 becausew = (0, 0, 0) is not a simple point of Trop(Y ), and the initial degenerations Xw andX ′

w are disjoint. The closures of Xw and X ′w in k3 are skew lines in the closure of

Yw, which is a smooth quadric surface.

Example 6.4. Let a ∈ K∗ be an element of positive valuation, and let Y ′ be thesurface given by

z2 − 1 + a(xy + x + y + 1) = 0,

where a ∈ K∗ has valuation 1, and let X and X ′ be as in Example 6.3. Then X andX ′ are contained in Y ′, and their tropicalizations still meet properly in Trop(Y ) atw = (0, 0, 0). In this case, w does lie in the relative interior of a facet σ of Trop(Y ′),but that facet has multiplicity m(σ) = 2.

Trop(X)

Trop(X ′)

Trop(Y ′)

wm(σ) = 2

The initial degeneration Y ′w has two disjoint components,


Xw

X ′w

Y ′w

and each component contains the initial degeneration of one of the curves.

Example 6.5. Assume the characteristic of K is not 2. Let a ∈ K∗ be an elementof positive valuation, with Y ′′ be the surface in (K∗)3 given by

2xy + xz + yz + z2 + x + y + 1 + a(x + z) = 0.

Let X again be as in Example 6.3, and let X ′′ be the curve given by (y+1, z−1+a).Once again, Trop(X) and Trop(X ′′) meet properly at w = (0, 0, 0). Furthermore,the initial degenerations Xw and X ′′

w meet properly at a single point in Y ′′w .

Y ′′w

Xw

X ′′w

This intersection point does not lift, because the closures of X and X ′′ are skewlines in K3. This is still consistent with Theorem 1.4 because Y ′′

w is a cone, andXw meets X ′′

w at the singular point.


Appendix A. Initial degenerations, value groups, and base change

Here we study how initial degenerations behave with respect to arbitrary exten-sions of valued fields. These basic results are used throughout the paper to reduceour main lifting theorems to the case where w is NG, by extending the ground field.

Let MG,w be the maximal sublattice of M on which w is G-rational. In otherwords,

MG,w = {u ∈ M | 〈u, w〉 is in G}.Since K is algebraically closed, the value group G is divisible, and hence MG,w issaturated in M . If the weight of a monomial axu is zero, then 〈u, w〉 = −ν(a), andhence u is in MG,w. In particular, MG,w contains all exponents of monomials thatrestrict to nonzero functions on Tw, and Tw is a torsor over the torus associated toMG,w.

Proposition A.1. Suppose the valuation is nontrivial. Then the integral modelT w is of finite type over Spec R if and only if w is in NG. Furthermore, if w is inNG, then T w is of finite presentation over Spec R.

Proof. Suppose w is in NG. Let u1, . . . , ur be a basis for M , and choose a1, . . . , ar

in K∗ such that ν(ai) = −〈ui, w〉. Then R[M ]w is generated over R by{a1x

u1 , (a1xu1)−1, . . . , arx

ur , (arxur )−1

},

and hence is of finite type. The ring R[M ]w is torsion free and hence flat overR. By Raynaud-Gruson [RG71, Theorem 3.4.6], we conclude that it is of finitepresentation.

For the converse, suppose w is not in NG. However, MG,w is a proper sublatticeof M , and hence the special fiber Tw has dimension strictly less than the dimensionof T . Since fiber dimension is semicontinuous in families of finite type, and T w isirreducible, it follows that T w is not of finite type. �

Remark A.2. Suppose the valuation is trivial, and let Sw be the set of latticepoints u ∈ M such that 〈u, w〉 ≥ 0. Then Sw is a subsemigroup of M , and R[M ]w

is naturally identified with the semigroup ring K[Sw]. The semigroup Sw is finitelygenerated if and only if the ray spanned by w has rational slope, and it follows thatT w is of finite type over K if and only if w is a scalar multiple of a lattice point.

However, note that unlike the case of nontrivial valuation, whether or not T w isof finite type is unrelated to whether or not the dimension of Tw is less than thatof T , because Tw is not obtained as a fiber over R. Indeed, if w 6= 0, the dimensionof Tw is always less than that of T .

When the schemes T w and Xw are not of finite type, there are many technicaldifficulties in handling them directly. These technical difficulties can be overcomeby extending scalars; the models T w become finite type after a suitable base change,as follows.

Suppose the value group G is a proper subgroup of R, and b is a real number thatis not in G. Then there is a unique valuation ν on the function field K(t) such thatν(t) = b. This valuation extends to the algebraic closure K of K(t). Iterating thisprocedure finitely many times, we can ensure that an arbitrary w is rational overthe value group of a suitable extension of K. In particular, for a suitable choice ofextension K|K, the integral model of the base change Tw is of finite type over thevaluation ring R.


Let K|K be an arbitrary extension of valued fields, and let G be the value groupof K, with R the valuation ring in K. For any subvariety X of T over K, thetropicalization of the base change X is equal to the tropicalization of X [Pay09a,Proposition 6.1], so the initial degeneration Xw is nonempty if and only if Xw isnonempty. Here we give a more precise geometric relationship between these initialdegenerations. First we treat the associated integral models over the valuationrings. Let

ϕ : Tw → Tw

be the natural map of integral models induced by the inclusion of tilted grouprings, which is equivariant over the projection of tori induced by the inclusionsMG,w ↪→ M eG,w and R ↪→ R.

Theorem A.3. The integral model Xw is the preimage of Xw under ϕ.

Proof. It is clear that ϕ maps Xw into Xw. We must show that any Laurentpolynomial in the tilted group ring R[M ]w that vanishes on X is in the ideal ofthat ring generated by IX ∩R[M ]w.

Let f =∑

u α(u)xu be a Laurent polynomial over K in I eX . Then f can bewritten as

f = α1f1 + · · ·+ αnfn,

with f1, . . . , fn in IX and linearly independent over K, and αi in K. Say ai(u) isthe coefficient of xu in fi, so the coefficient of xu in f is

α(u) = α1a1(u) + · · ·+ αnan(u).

Now, suppose f is in R[M ]w. If each αifi is in R[M ]w, then it is easy to seethat f is in the ideal generated by IX ∩ R[M ]w, by applying the case n = 1,below. The difficulty is that there may be some cancellation of leading terms in theabove expression for f , by which we mean that the valuation of some coefficientα(u) may be strictly larger than mini{ν(αiai(u))}. When n is greater than 1, weproceed by carefully eliminating one term in the summation, writing f as a K-linearcombination of f1, ..., fn−1 plus a single element of K times a K-linear combinationof f1, ..., fn. A suitable choice in the elimination procedure ensures that each ofthese two terms is in R[M ]w, and we deduce that they are in the ideal generatedby IX ∩R[M ]w, by induction on n.

Suppose n = 1. Let m = a1(u)xu be a monomial of lowest weight in f1, and letg = f1/m. Then f can be expressed as

f = α1mg,

with g in IX ∩R[M ]w, and α1m in R[M ]w, as required.We proceed by induction on n. Given u1, . . . , un−1 in M , consider the matrix

whose (i, j)th entry is ai(uj), and let δi be the ith maximal minor

δi =

∣∣∣∣∣∣∣∣∣∣∣∣

a1(u1) · · · a1(un−1)...

...ai(u1) · · · ai(un−1)

......

an(u1) · · · an(un−1)

∣∣∣∣∣∣∣∣∣∣∣∣.


Since f1, . . . , fn−1 are linearly independent over K, we may choose u1, . . . , un−1 sothat δn is nonzero and

ν(α(u1)) + · · ·+ ν(α(un−1))− ν(δn)

is as large as possible. This choice is essential in the proof of the following claim.We claim that

h =αn

δn

n∑i=1

(−1)n−iδifi

is in R[M ]w. The claim implies that h is in the ideal generated by IX ∩ R[M ]w,by the case n = 1, above, and also that the difference f − h is in R[M ]w. Thecoefficient of fn in the above expression is equal to αn, by construction, so thedifference f−h can be written as a K-linear combination of f1, . . . , fn−1. It followsby induction that f − h is also in the ideal generated by IX ∩ R[M ]w, and thisproves the theorem.

It therefore remains to show that h =∑

β(u)xu is in R[M ]w, which means thatthe w-weight of each monomial β(u)xu is nonnegative. Fix one such monomial,write un = u for its exponent, and let A be the square n× n matrix whose (i, j)thentry is ai(uj),

A =

a1(u1) · · · a1(un)...

. . ....

an(u1) · · · an(un)

.

Let Aij be the (i, j)th minor of A, the determinant of the submatrix obtained bydeleting the ith row and jth column. So δi = Ain. Expanding detA in the lastcolumn shows that

β(un) =(−1)nαn

Anndet A.

Therefore, the valuation of β(un) is

ν(β(un)) = ν(detA)− ν(Ann) + ν(αn).

Since f is in R[M ]w by hypothesis, it will be enough to show that ν(β(un)) is atleast as large as ν(α(un)). To compare β(un) with α(un), we consider the matrix

A′ =

α1a1(u1) · · · α1a1(un)

.... . .

...αn−1an−1(u1) · · · αn−1an−1(un)

α(u1) · · · α(un)

.

Recall that the coefficient α(uj) in the bottom row is α1a1(uj) + · · · + αnan(uj),so the determinant of A′ is α1 · · ·αn det A. Expanding detA′ in the last row givesalso

det A′ = α1 · · ·αn−1

( n∑i=1

(−1)n−iα(ui)Ani

),

and comparing these two expressions for detA′ yields

det A =1

αn

( n∑i=1

(−1)n−iα(ui)Ani

),


Therefore,ν(detA) ≥ min

i{ν(α(ui) + ν(Ani)} − ν(αn).

Now u1, . . . , un−1 were chosen so that this minimum occurs at i = n. Substitutingthe resulting inequality for ν(detA) into the expression for ν(β(un)) above showsthat ν(β(un)) is greater than or equal to ν(α(un)). This proves the claim, and thetheorem follows. �

We now pass from the integral models to the initial degenerations. Let

φ : Tw → Tw

be the natural projection of torus torsors induced by the inclusions of tilted grouprings, modulo monomials of strictly positive w-weight.

Theorem A.4. The initial degeneration Xw is the preimage of Xw under φ.

Proof. It is clear that φ maps Xw into Xw. Any element fw of the ideal of Xw isthe residue of a Laurent polynomial f in I eX ∩ R[M ]w, which is then in the idealgenerated by IX ∩R[M ]w, by Theorem A.3. Taking residues shows that fw in theideal generated by the pullback of IXw , and it follows that Xw is the full preimageof Xw. �

Appendix B. Topology of finite type morphisms

Because the results may be of independent interest, we explain how Theo-rem 4.2.5 on the existence of closed points in fibers, and Theorem 4.1.3 on liftingpoints of intersection, can both be extended to an arbitrary base scheme. The proofof the former is by reduction to the case of valuation rings, while the latter is adirect application of the techniques of Section 4, despite the fact that subadditivityno longer holds.

The proof of Proposition 4.2.2 does not use that the base scheme is the spectrumof a valuation ring of rank 1, and in fact yields the following result.

Proposition B.1. Let X → S be a flat morphism of finite type of irreducibleschemes, and suppose that D is a locally principal closed subscheme of X that doesnot meet the generic fiber. Then, for every s ∈ S, every irreducible component ofthe fiber Ds is an irreducible component of Xs.

Remark B.2. Some hypothesis such as flatness is necessary for such results onlocally principal subschemes over general base schemes, as shown by the followingexample. Suppose X → S is the blowup of the affine plane at the origin. Thenthe strict transform D of a line through the origin is locally principal and does notmeet the generic fiber, but its intersection with the exceptional divisor is a singlepoint.

We can now prove the first stated result.

Theorem B.3. Let X → S be a morphism locally of finite type, and let s be aspecialization of s′ in S. Suppose x′ is a point in Xs′ specializing to a closed pointx in Xs. Then there is a closed point x′′ in Xs′ which specializes to x, and suchthat x′ specializes to x′′. Moreover, the set of such x′′ is Zariski dense in the closureof x′ inside Xs′ .


Proof. First, in light of the generalized Proposition B.1, the argument of Theorem4.2.5 goes through to prove the desired result in the case that S is an arbitraryvaluation ring, with s′ the generic point of S. We thus reduce to this case. ReplacingX by the closure of x′, we may assume X is integral and x′ is its generic point. By[Gro61a, Proposition 7.1.4(ii)], there is a valuation ring A in k(x′) with a dominantmorphism

Spec A → X

mapping the closed point to x. Let R be the valuation ring in the residue fieldk(y′) given by intersecting with A in k(x′), and let X ′ = X ×Y Spec R. Then thefollowing diagram is commutative.

X ′ //

��

X

��Spec R // Y

By construction the map from Spec A to X factors through X ′. So, by the casewe have already handled, there is a closed point x′′ in the generic fiber of X ′ overSpec R specializing to the image in X ′ of the closed point of Spec A; moreover,such points are Zariski dense in the generic fiber. Since k(y′) is identified with thefraction field of R, by construction, the generic fiber of X ′ maps isomorphically tothe generic fiber of X, and we conclude the desired statement for X. �

Naive statements about global codimension and subadditivity do not extendfrom valuation rings of rank 1 to valuation rings of higher rank, as shown by Ex-ample 4.3.2. Nevertheless, hypotheses on codimension of intersection can still yieldlifting results even when subadditivity of codimension fails. Indeed, the follow-ing theorem can be reduced immediately to the case that S is integral with s′ itsgeneric point, and then follows from a reduction to the diagonal and an inductiveapplication of Proposition B.1.

Theorem B.4. Let Y → S be a smooth morphism, and let X and X ′ be closedsubschemes of Y , flat over S, such that the codimension in Y of every componentof X and every component of X ′ is less than or equal to j and j′, respectively.Suppose that, for some s ∈ S, the fibers Xs and X ′

s meet in codimension j + j′ ata point x in Ys. Then for any s′ ∈ S specializing to s

(1) There is a point x′ in Xs′ ∩X ′s′ specializing to x.

(2) If x is closed in Ys, then x′ may be chosen to be closed in Ys′ .

Remark B.5. In Example 4.3.2, the special fibers of the two subschemes coincide,and the generic fibers are disjoint. This does not contradict Theorem B.4 becausethe special fibers do not meet properly, and the failure of subadditivity is for simplenumerical reasons. The intersection that does not lift has dimension one larger thanexpected, but is supported in a fiber of codimension r, which is greater than 1.

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Department of Mathematics, University of California, Davis, CA 95616E-mail address: [email protected]

Department of Mathematics, Stanford University, Stanford, CA 94305E-mail address: [email protected]

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