aa r X i v : . [ m a t h . AG ] J u l LOGARITHMIC DONALDSON–THOMAS THEORY
DAVESH MAULIK & DHRUV RANGANATHANA
BSTRACT . Let X be a smooth threefold with a simple normal crossings divisor D . We construct theDonaldson–Thomas theory of the pair ( X | D ) enumerating ideal sheaves on X relative to D . Thesemoduli spaces are compactified by studying subschemes in expansions of the target geometry, andthe moduli space carries a virtual fundamental class leading to numerical invariants with expectedproperties. We formulate punctual evaluation, rationality and wall-crossing conjectures, in parallelwith the standard theory. Our formalism specializes to the Li–Wu theory of relative ideal sheaveswhen the divisor is smooth, and is parallel to recent work on logarithmic Gromov–Witten theorywith expansions. C ONTENTS
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21. Flavours of tropicalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62. Tropical degenerations and strong transversality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123. Moduli of target expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194. Moduli of ideal sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275. Logarithmic Donaldson–Thomas theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356. First examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407. Transverse limits and generically expanded targets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 & DHRUV RANGANATHAN I NTRODUCTION
A basic technique in enumerative geometry is to degenerate from a smooth variety to a singularone, whose irreducible components may be easier to analyze. In Gromov–Witten theory, thisis an essential tool in the subject, with both symplectic and algebraic incarnations [28, 35, 36,37]. Associated to a smooth pair ( X, D ) , relative Gromov–Witten theory studies maps with fixedtangency relative to D , and uses these to study degenerations of a smooth variety to two smoothcomponents meeting transversely. The key geometric idea is that the target is dynamic, expandingalong D to prevent components mapping to the divisor.Logarithmic Gromov–Witten theory was developed [2, 15, 25] to handle more general logarith-mically smooth pairs ( X, D ) , for instance, where D is a normal crossings boundary divisor; theseoccur as components of degenerations with more complicated combinatorics. In this approach,the target is no longer dynamic and, instead, logarithmic structures are used to maintain a formof transversality in limits.When X is a threefold, there are sheaf-theoretic approaches to enumerating curves, developedby Donaldson–Thomas and Pandharipande–Thomas [19, 60, 53]. In the case of a smooth pair ( X, D ) , there is again a powerful relative theory, developed by Li and Wu [38], which is one of thecentral tools of the subject. The goal of this paper is to construct the Donaldson–Thomas theory ofa pair ( X, D ) where D is a simple normal crossings divisor, referred to here as the logarithmic DTtheory of ( X | D ) .One challenge in doing so is there is no clear analogue of the unexpanded formalism of [25]. Themoduli space of prestable curves and its universal family are themselves simple normal crossingspairs, so the logarithmic mapping space provides a starting point. The Hilbert scheme exhibitsno such structure, and there is not, as yet, a good definition of a logarithmic structure on an idealsheaf. Instead, our strategy is closer to the original expanded formalism of [38]. In subsequentwork, we plan to complete the parallel to logarithmic Gromov–Witten theory, with its associateddegeneration formalism [3, 4, 56].0.1. Main results.
We work over the complex numbers. Let X be a smooth threefold equippedwith a simple normal crossings divisor D . We further assume the intersection of any collectionof components of D is connected. Our main results are described below. Precise definitions andstatements can be found in the main text. Fix a curve class β in H ( X ) and an integer for theholomorphic Euler characteristic χ . Theorem A.
There exists a proper Deligne–Mumford moduli stack DT β,χ ( X | D ) compactifying the modulispaces of ideal sheaves on X relative to D with numerical invariants β and χ , equipped with universaldiagram Z Y X DT β,χ ( X | D ) . The fibers of DT β,χ ( X | D ) parameterize relative ideal sheaves on expansions of X along D . The space DT β,χ ( X | D ) carries a perfect obstruction theory and a virtual fundamental class with expected properties. The structure of our result mirrors the structure of relative Donaldson–Thomas theory. Namely,we first construct a moduli stack parametrizing allowable target expansions; the moduli space ofstable relative ideal sheaves will then consist of ideal sheaves on fibers of the universal family,
OGARITHMIC DT THEORY 3 satisfying a certain stability condition. However, in our setting, both the classifying stack and thecorrect notion of stability are more subtle, and ideas from tropical geometry are crucial to findingthe correct formulation. The approach is outlined in the final sections of this introduction.Numerical invariants are obtained by integration against the virtual fundamental class. Primaryand descendant fields are given by the Chern characters of the universal ideal sheaf, as in thestandard theory [42]. The Hilbert scheme of points on the divisor components, relative to theinduced boundary divisor, can be analogously compactified.
Theorem B.
Each Donaldson–Thomas moduli space DT β,χ ( X | D ) is equipped with an evaluation mor-phism ev : DT β,χ ( X | D ) → Ev ( D ) . The space Ev ( D ) is a compactification of the Hilbert scheme of points on the smooth locus of D . Donaldson–Thomas invariants defined by pairing the virtual class with cohomology classes pulled back from Ev ( D ) andprimary and descendant fields of the universal subscheme. Once the numerical invariants are defined, we formulate the basic conjectures of the subject inour setting, namely the precise evaluation of the punctual series and the rationality of the normal-ized generating function for primary insertions. These are collected in Section 5.4.Formal similarities between the map and sheaf sides give a natural generalization of the rela-tive GW/DT correspondence to normal crossings geometries, and motivate a study of the corre-spondence via normal crossings degenerations [43, 44]. A prototype version of this theory playsa central role in the study of the GW/Pairs correspondence of Pandharipande and Pixton [52].These ideas form a central motivation for our study, and will be pursued in future work.The results complete an exact parallel to [56, Theorem A], where an expanded version of loga-rithmic Gromov–Witten theory is constructed. As in that paper, a notable feature of our modulispaces is that they are not unique. Rather, there is a combinatorial choice in constructing the stackof expansions, which leads to an infinite collection of moduli spaces with the desired proper-ties, naturally organized into an inverse system. There are natural compatibilies between virtualclasses, so that the numerical invariants are independent of this choice. This is very similar tothe combinatorial choice in studying familes of degenerating abelian varieties and constructingtoroidal compactifications. While the geometry is a little different, the non-uniqueness arises in anidentical fashion, via a system of polyhedral structures on a fan-like object.The approach of [56] begins with the mapping stack of logarithmic maps to the static targetand obtains transverse maps to expansions by applying logarithmic modifications. In our case,since no unexpanded space exists, we are forced to construct our space directly and check thedesired properties. We first provide a tropical algorithm for finding transverse limits for familiesof subschemes of X and use this to guess the correct moduli problem. In the following subsections,we give a detailed outline of our approach.0.2. Transversality.
Let Z ⊂ X be a subscheme with ideal sheaf I Z . We are interested in sub-schemes that intersect D in its smooth locus, with the property no one-dimensional componentsor embedded points of Z lie in D . Algebraically, this is the condition that the map I Z ⊗ O D → O D is injective. We refer to subschemes satisfying this condition as strongly transverse subschemes . Thecondition forces Z to be dimensionally transverse to the strata of X . DAVESH MAULIK & DHRUV RANGANATHAN
The locus of strongly transverse subschemes is a non-proper open subscheme of the Hilbertscheme of X . We aim to find a compactification of this moduli problem with the prescription thatthe universal subscheme continues to be strongly transverse.0.3. Limits from tropicalization, after Tevelev.
In order to achieve transversality for limits offamilies, the scheme X must be allowed to break. The degenerations here are built from tropicalgeometry, using an elegant argument due to Tevelev, based on Kapranov’s visible contours [29, 58].The cone over the dual complex of D is denoted Σ and can be identified with R k > . Let Σ + beproduct of the cone R > with Σ . We view these as the fans associated to the toric stacks A k and A k × A respectively. Given an injection of cone complexes ∆ ֒ → Σ + , the toric dictionary gives riseto a modification Y → X × A → A . Geometrically, the expansion is obtained from the constant family by performing toric birationalmodifications to the strata of D in the special fiber and passing to an open subscheme. When D isa smooth divisor, this recovers the class of expansions considered by Li and Li–Wu [36, 38]. In thegeneral case, they recover the class of targets in [56].We can now apply Tevelev’s approach. Given a strongly transverse family of subschemes Z η over a C ⋆ , the tropicalization of Z η is a subset of Σ + . This subset can be given the choice of a fan ∆ contained in Σ + . For an appropriate choice of fan structure, this produces a degeneration Y of X over A . The flat limit of Z η in this degeneration is strongly transverse to the strata of Y . Thiswas proved by Tevelev when ( X | D ) is toric. These limits have strong uniqueness and functorialityproperties, making them appropriate for constructing moduli.Precedent for building moduli spaces by using tropical limit algorithms is provided by work ofHacking–Keel–Tevelev, and the method has been used in Gromov–Witten theory, see [27, 55].0.4. The universal tropical expansion.
By axiomitizing the output of Tevelev’s argument, wepropose a class of ideal sheaves on X relative to D for the Donaldson–Thomas moduli problem. Theseare subschemes of expansions of X along D that are transverse to the strata. In order to constructa global moduli problem, we construct an Artin stack that encodes the possible expansions of X that could arise from Tevelev’s procedure.The discussion above predicts the one-parameter degenerations of the target, but subtletiesarise in extending them over higher dimensional bases, having to do with flattening the universaldegeneration. We tackle this by first studying an appropriate tropical moduli problem, usingrecent work that identifies a category of certain locally toric Artin stacks with purely combinatorialobjects [14].The outcome of the tropical study is a system of moduli stacks of universal expansions, relatedto each other by birational transformations, and organized into an inverse system. The system de-pends only on the combinatorics of the boundary divisor. Both the moduli space and its universalfamily have this structure and compatible choices give rise to a universal degeneration.0.5. Moduli space of stable relative ideal sheaves.
After fixing a stack of expansions, we definea notion of DT stability for ideal sheaves on X relative to D . An important subtlety appears whenconsidering tube subschemes , namely subschemes in a component of an expansion that are pulledback from a surface in its boundary. In relative DT theory, these are ruled out by stability; in oursetting, they are forced on us by the combinatorial algorithms in an analogous fashion to how trivial bubbles arise in the stable maps geometry. In any fiber of the universal family of the stack OGARITHMIC DT THEORY 5 of expansions, there are distinguished irreducible components, denoted tube components; the DTstability condition we impose is that these are precisely the components which host tube sub-schemes. We show this defines a moduli problem with the expected properties. The transversalityhypotheses guarantee that the morphism DT β,χ ( X | D ) → Exp ( X | D ) has a perfect obstruction theory and consequently a virtual class. This establishes an appropriateDonaldson–Thomas theory for the pair ( X | D ) .0.6. Pairs, etc.
We have chosen to focus on the moduli theory of ideal sheaves in this paper, but themethods appear to be adaptable to other settings. In particular, one can define logarithmic stablepair invariants by replacing the Hilbert scheme with the stable pair moduli spaces of Pandhari-pande and Thomas. The stable pair adaptations are recorded in Remark 4.5.2. It seems reasonableto hope for further applications. For instance, it may be possible to rederive logarithmic Gromov–Witten theory, relying on target expansions from the very beginning. In another direction, thelogarithmic theory of quasimaps has only been treated in the smooth pair case [11].0.7.
Toroidal embeddings.
The expectation is that the theory set up in this paper can be extendedto any logarithmically smooth target, without either the simple normal crossing or connectivityrestrictions placed on D . The arguments in the present paper carry over with cosmetic changesto treat generalized Deligne–Faltings logarithmic structures [2]. This includes all singular toricvarieties. A more delicate modification of the combinatorics can likely be used to treat divisorgeometries with disconnected intersections. These two variants, together with the case where D has a self-intersecting component, will be addressed elsewhere. Logarithmically ´etale descent andvirtual birational invariance techniques are likely to play a role, see [6, 9].0.8. Further directions.
We mention briefly some natural directions to pursue with the theoryconstructed here. In the sequel to this paper, we plan to develop a degeneration formalism, gener-alizing that of [38], and parallel to [56]. One can also ask for a logarithmic version of the GW/Pairscorrespondence, in a form that is compatible with normal crossings degenerations. With this inplace, it seems plausible to extend the inductive strategy of Pandharipande–Pixton and prove theGW/Pairs correspondence for a broader class of threefolds, i.e. varieties which are not easily stud-ied by double-point degenerations. For instance, a natural class of examples comes from takingzero loci of sections of toric vector bundles.In another direction, the formalism of relative DT theory (in cohomology and K -theory) interactswell with the representation-theoretic structure on the Hilbert scheme of points on a surface [45],and we expect that our logarithmic theory will extend this circle of ideas. In a similarly speculativevein, logarithmic Gromov–Witten invariants in genus are related to the symplectic cohomologyof the open variety X \ D , which is the natural replacement for quantum cohomology for opengeometries. It would be interesting to examine whether logarithmic Donaldson–Thomas theorycan be related to the symplectic cohomology of Hilbert schemes of points on open surfaces. Outline of paper.
We briefly outline the sections of this paper. In Section 1, we review some basicconstructions and results from tropical geometry. In Section 2, we give the tropical algorithm forconstructing strongly transverse flat limits of subschemes. In Section 3, we construct the tropicalmoduli spaces of expansions as well as their geometric counterparts. In Section 4, we define stablerelative ideal sheaves and show their moduli functor is represented by a proper Deligne–Mumfordstack. In Section 5, we study the virtual structure on the moduli space, define logarithmic DT
DAVESH MAULIK & DHRUV RANGANATHAN invariants, and state the basic conjectures. In Section 6, we give a handful of simple examples,demonstrating the basic theory. In Section 7, we complete the proof of the valuative criterion ofproperness, initiated in Section 2, to deal with the case where the generic fiber is expanded.
Background and conventions.
We have put some effort into minimizing the amount of logarith-mic geometry that is explicitly used in this paper, and there is nothing that we use beyond [5,Sections 1–5]. We do use the combinatorics of cone complexes and cone spaces heavily, and referthe reader to [14, Sections 2 & underlying . Acknowledgements.
We are grateful to Dan Abramovich, Dori Bejleri, Qile Chen, Johan de Jong,Mark Gross, Eric Katz, Max Lieblich, Hannah Markwig, Andrei Okounkov, Sam Payne, MattiaTalpo, Richard Thomas, Martin Ulirsch, Jeremy Usatine, and Jonathan Wise for numerous con-versations over the years on related topics. The first author would like to thank Daniel Tiger forproviding extra childcare during the final stages of this project. The second author was a MooreInstructor at MIT and a visitor at the Chennai Mathematical Institute during work on this project,and is grateful to both institutions for excellent working conditions.1. F
LAVOURS OF TROPICALIZATION
We require some elementary notions from logarithmic geometry, and a reference that is wellsuited to our point of view is [5, Sections 3–5]. There are a few different ways in which tropical-izations arise in logarithmic geometry and we recall these for the reader.1.1.
Cone complexes and their morphisms.
We first recall the basic building blocks of toric ge-ometry. A polyhedral cone with integral structure ( σ, M ) is a topological space σ equipped with afinitely generated abelian group M of continuous real-valued functions, such that the evaluation σ → Hom ( M, R ) is a homeomorphism onto a strongly convex polyhedral cone. If this cone is rational with respectto the dual lattice of M , then we say that ( σ, M ) is rational. We define the lattice of integral pointsof σ by taking the preimage of the dual lattice of M under the evaluation map. The set of elementsof M that are non-negative on σ form a monoid S σ referred to as the dual monoid . The cone σ isrecovered as the space of monoid homomorphisms Hom ( S σ , R > ) .Henceforth, a rational polyhedral cone with integral structure will be referred to as a cone . Definition 1.1.1 (Cone complexes) . A rational polyhedral cone complex is a topological space that ispresented as a colimit of a partially ordered set of cones, where all arrows are given by isomor-phisms onto proper faces. A morphism of cone complexes is a continuous map Σ ′ → Σ such that the image of every cone in Σ ′ is contained in a cone of Σ and that is given by an integrallinear map upon restriction to each cone of Σ ′ .Cone complexes are nearly identical to the notion of a fan considered in toric geometry [22]. Thekey differences are that they do not come equipped with a global embedding into a vector space,and two cones can intersect along a union of faces. Our hypotheses will typically be geared toforce an intersection of cones to be a face of each. OGARITHMIC DT THEORY 7
A cone complex is said to be smooth if every cone is isomorphic to a standard orthant withits canonical integral structure. We record two combinatorial notions associated to morphismsbetween cone complexes.
Definition 1.1.2 (Flat and reduced maps) . Let Σ be a smooth cone complex and let π : Σ ′ → Σ bea morphism of cone complexes. Then π is flat if the image of every cone of Σ ′ is a cone of Σ . A flatmap is said to have reduced fibers if for every cone σ ′ of Σ ′ with image σ , the image of the lattice of σ ′ is equal to the lattice of σ .The terminology is compatible with the analogous geometric notions for toric maps.1.2. Tropicalization for the target.
Every fine and saturated logarithmically smooth scheme givesrise to a cone complex. Let ( X | D ) be a smooth scheme of finite type over C equipped with a simplenormal crossings divisor D . Assume that the intersections of irreducible components of D arealways connected. The presence of D gives X the structure of a logarithmically smooth scheme,and we let X ◦ be the complement of the divisor, where the logarithmic structure is trivial.We unwind the definition of the logarithmic structure in this case for the benefit of the reader.The components of D give rise to a distinguished class of functions in the structure sheaf. Foran open set U ⊂ X , we record the values of the logarithmic structure sheaf and the characteristicmonoid sheaf : M X ( U ) = { f ∈ O X ( U ) : f is invertible on U r D } and M X ( U ) = M X ( U ) / O ⋆ X ( U ) . Locally at each point x ∈ X , there are functions, canonical up to multiplication by a unit, cuttingout the irreducible components of D passing through x . In particular, the stalk of the characteristicmonoid sheaf at x is naturally identified with N e , where e is the number the such components.The dual cones of the stalks of the characteristic monoids of X give rise to a collection of cones,one for each point of X . The generization maps naturally give rise to gluing morphisms, and thesecones form a finite type cone complex. We denote it by Σ X , refer to it as the tropicalization of X , andregard it as a cone complex equipped with an integral structure. In the case of a toric variety thisconstruction recovers the fan. The standard references are [5, 14, 30, 34].This construction extends to the case of a toroidal pair [1, 61]. What distinguishes the simplenormal crossings pairs among logarithmically smooth schemes is that their cones are smooth.1.3. Subdivisions.
Let ( X | D ) be a simple normal crossings pair with tropicalization Σ X . For sim-plicity, we assume that the non-empty intersections of components of D are connected. We willproduce target expansions by using subdivisions of the tropicalization. Definition 1.3.1. A subdivision is a cone complex ∆ and a morphism of cone complexes ∆ ֒ → Σ X that is injective on the support of ∆ and further, such that the integral points of the image of eachcone τ ∈ ∆ are exactly the intersection of the integral points of Σ X with τ .Note that this more flexible than the standard definition; the underlying map of sets need not bea bijection because we wish to have the flexibility to discard closed strata after blowing up. Whenthe map on integral points is a bijection we call it a proper subdivision . These may regarded as sheaves in either the Zariski or ´etale topology.
DAVESH MAULIK & DHRUV RANGANATHAN
In toric geometry, subdivisions give rise to possibly non-proper birational models of X . Thesame is true for ( X | D ) . The cone complex Σ associated to D is smooth, and the intersection of anytwo cones is a face of each. If there are d rays in Σ , we may embed Σ X as Σ X ֒ → R d > by mapping each ray in Σ X isomorphically onto the positive ray on the corresponding axis. Thepositive orthant in this vector space is the fan associated to the toric variety A d with dense torus G dm . The subdivision ∆ ֒ → Σ X ֒ → R d > defines a non-complete fan with associated toric variety A ∆ , equipped with an G dm -equivariantbirational map A ∆ → A d . By passing to quotients, we have a morphism of stacks [ A ∆ / G dm ] → [ A d / G dm ] . The presence of D gives rise to a tautological morphism ( X | D ) → [ A d / G dm ] Definition 1.3.2.
The birational model of X associated to the subdivision ∆ ֒ → Σ is given by X ∆ := X × [ A d / G dm ] [ A ∆ / G dm ] . Remark 1.3.3.
In practice, we apply this construction to X × A with the divisor X × { } ∪ D × A .The deformation to the normal cone of the strata of D are special cases of the construction. Remark 1.3.4.
It is natural to formulate subdivisions in terms of subfunctors of the functor onlogarithmic schemes defined by X and by its cone complex. The birational model associated to asubdivision is defined by pulling back the subdivision along the tropicalization map, see [14, 30].1.4. Tropicalization for subschemes via valuations.
We assume that X is proper. The divisor D locally provides a system of functions on X and the orders of vanishing of these functions onsubschemes of X is the content of tropicalization.Let ( K, ν ) be a valued field extending C and let ν be the valuation. Consider a morphism Spec K → X ◦ . Since X is proper, this morphism extends to the valuation ring Spec R → X. Let x be the image of the closed point. The locally closed stratum of D containing x is the inter-section of a (possibly empty) subset of irreducible D i , . . . , D i r . The associated equations generatethe stalk of the characteristic sheaf N r . For each element f ∈ N r , we may lift it to a function f , pullback to Spec R along the map above, and compose with the valuation. Since any two such liftsdiffer by a unit, this gives rise to an element [ N r → R > ] ∈ Hom ( N r , R > ) ֒ → Σ X . For any valued field K , we have a well-defined morphism trop : X ◦ ( K ) → Σ X . OGARITHMIC DT THEORY 9
Let K be a valued field whose associated valuation map K → R is surjective. Let Z ◦ ⊂ X ◦ be asubscheme. Let trop ( Z ◦ ) be the subset of Σ X obtained by restricting trop to Z ◦ ( K ) ; it is independentof the choice of valued field K and its functoriality properties are outlined in [26, 62]. Remark 1.4.1.
When X is a toric variety, Tevelev has shown that there exists a choice of X ′ suchthat the closure is strongly transverse and we will require and refine this result in the course of ourmain result [58, Theorem 1.2]; a simple proof is given in [39, Theorem 6.4.17].1.5. Properties of the tropicalization.
The shapes of tropicalizations are governed by the Bieri–Groves theorem [12, 61].
Theorem 1.5.1.
Let Z ◦ ⊂ X ◦ be a closed subscheme. Then the set trop ( Z ◦ ) is the support of a rationalpolyhedral cone complex of Σ X . The topological dimension of trop ( Z ◦ ) is bounded above by the algebraicdimension of Z ◦ . If X ◦ is a closed subvariety of an algebraic torus, the topological dimension of trop ( Z ◦ ) isequal to the algebraic dimension of X . The set trop ( Z ◦ ) has no distinguished polyhedral structure in general [39, Example 3.5.4].The role of tropicalization in degeneration and compactification problems has its origin in thefollowing theorems, proved by Tevelev for toric varieties and Ulirsch for logarithmic schemes,see [58, Theorem 1.2] and [61, Theorem 1.2].The first concerns the properness of closures in partial compactifications of X ◦ . Let Z ◦ be asubscheme of X ◦ and let X ′ be a simple normal crossings compactification. Let X ′′ ⊂ X ′ be thecomplement of a union of closed strata of X ′ and let Σ X ′′ be the subfan of Σ X ′ obtained by deletingthe corresponding union of open cones. Theorem 1.5.2.
The closure Z of Z ◦ in the partial compactification X ′′ is proper if and only if trop ( Z ◦ ) isset theoretically contained in Σ X ′′ . The second concerns transversality. The closure Z of Z ◦ in X ′ intersects strata in the expecteddimension if dim E ∩ Z = dim Z − codim X ′ E. Theorem 1.5.3.
The closure Z of Z ◦ in the compactification X ′ of X ◦ intersects strata in the expecteddimension if and only if trop ( Z ◦ ) is a union of cones in Σ X ′ . Remark 1.5.4. If X is not proper, there is no longer a tropicalization map defined on the set X ◦ ( K ) ,because limits need not exist. However, the above relationship to the cone complex persists. Foreach valued field K with valuation ring R as above, let X i ( K ) be the subset X ◦ ( K ) consisting ofthose K -points that extend to R -points. There is a morphism X i ( K ) → Σ X defined exactly as defined above. Similarly, given a subscheme Z ⊂ X ◦ , we can define its tropical-ization as the image of Z i ( K ) in Σ X , see [62, Section 5.2]. Comprehensive treatments of tropicalization regularly make use of Berkovich’s analytification. We omit this hereto avoid additional technical machinery, although the proofs of these basic results use them. See [5] for a survey of therelationship between tropical, logarithmic, and non-archimedean geometry. & DHRUV RANGANATHAN
Tropicalization via compactifications.
In the previous section, tropicalizations were seen todetermine a family of compactifications in which the closure of a subvariety meets each stratumin the expected codimension. There is a partial converse.Let X ′ be a simple normal crossings compactification of X ◦ . Let Z be a subscheme of X ′ whoseintersection with the strata of X ′ have the expected codimension, with Z ◦ = X ◦ ∩ Z. The following perspective is due to Hacking–Keel–Tevelev, extended by Ulirsch, and is sometimesreferred to as geometric tropicalization , see [27, 61].
Theorem 1.6.1.
The tropicalization of Z ◦ is equal to the union of cones σ in Σ X ′ such Z nontriviallyintersects the locally closed strata dual to σ . That is, there is an equality of subsets of Σ X ′ given by trop ( Z ◦ ) = [ σ ∈ Σ X : V ( σ ) ∩ Z = ∅ σ, where V ( σ ) is the locally closed stratum of X ′ corresponding to the cone σ . The definition of trop ( Z ◦ ) via coordinatewise valuation described in the previous subsectiondepends on a choice of compactification of X ◦ , but is insensitive to strata blowups. The resultabove can therefore be viewed as a computational tool; it describes the tropicalization using asingle compactification in which Z is dimensionally transverse. We return to this in Section 7. Remark 1.6.2 (Asymptotics and stars) . Let Z ◦ ⊂ X ◦ be a subscheme with tropicalization trop ( Z ◦ ) in Σ X . Assume that the tropicalization is a union of cones in Σ X and let Z denote the closure of Z ◦ .If D i ⊂ X is an irreducible component, we can view it as a simple normal crossings pair in its ownright with interior D ◦ i and divisor equal to the intersection of D i with the remaining componentsof D . It contains the subscheme Z ◦ i = Z ∩ D ◦ i . The tropicalization trop ( Z ◦ i ) can be read off from trop ( Z ◦ ) as follows. The divisor D i determines a ray ρ i in Σ X . The star of ρ i is the union of conesthat contain ρ i and can be identified with Σ D i . The tropicalization of trop ( Z ◦ i ) is the union ofthe cones under this identification where trop ( Z ◦ ) itself is supported. In practice, it visible as thecollection of asymptotic directions of trop ( Z ◦ ) parallel to ρ i .1.7. Tropicalization for a family of subschemes.
The constructions extend to flat families of sub-schemes of a logarithmic scheme. We will consider subschemes in X over valued fields K thatextend the trivially valued ground field C . In order to avoid foundational issues, we will assumethat all subschemes are defined over the localization of a smooth algebraic curve of finite type.The relevant moduli spaces will be shown to be of finite type, so this is a harmless assumption.1.7.1. Tropicalization over a valued field.
Let ( X | D ) = X ◦ ֒ → X be a simple normal crossings compact-ification as above and let K be a valued field extending C as above. Consider a strongly transversesubscheme Z η ֒ → X × C K, and let Z ◦ η be the open subscheme contained in X ◦ . By the strong transversality hypothesis, thisopen subscheme is dense. After passing from K to a valued extension L with real surjective valu-ation, we may once again consider the tropicalization map trop : Z ◦ η ( L ) → Σ X . The tropicalization is independent of the choice of L , and we denote the set trop ( Z ◦ ) . OGARITHMIC DT THEORY 11
The basic structure result extends to this setting with a small twist. The toric case of this resultwas established by Gubler without restriction on the base field [26]. The statement for simplenormal crossings targets is a consequence of [13, Theorem 5.1].
Theorem 1.7.1.
The set trop ( Z ◦ η ) is the support of a rational polyhedral complex of Σ X . The topologicaldimension of trop ( Z ◦ η ) is bounded above by the algebraic dimension of Z ◦ . If X ◦ is a closed subvariety of analgebraic torus, then the topological dimension of the tropicalization is equal to the algebraic dimension of Z ◦ . To emphasize the point, the tropicalization of a variety that is defined over a nontrivial valuedextension of C is polyhedral but not necessarily conical . In other words, it can have bounded cells.The target X itself is still defined over C , not merely over K , so its tropicalization remains conical.1.7.2. Families over a punctured curve.
Let C be a smooth algebraic curve of finite type with a dis-tinguished point ∈ C , let C ◦ be the complement of this point. The pair ( C, 0 ) has a cone complex Σ C , canonically identified with the positive real line R > .Let ( X | D ) be a pair as before and X ◦ its interior. Consider a flat family of -dimensional sub-schemes over C ◦ Z ◦ ⊂ C ◦ × X ◦ . There are two tropicalizations associated to this family. For the first, we consider Z ◦ as a -dimensional subscheme of C ◦ × X ◦ . The partial compactification C × X determines a tropicalization,as explained in Remark 1.5.4: trop ( Z ◦ ) ⊂ R > × Σ X . For the second, note that the function field of C is equipped with a discrete valuation arising fromorder of vanishing at . Let K be an extension of this field with real surjective valuation and notethat there is a canonical inclusion of Spec K to C ◦ . Consider the tropicalization of the K -valuedpoints of the base change Z η = Z ◦ × C ◦ K ⊂ X ( K ) , where the morphism Spec K → C ◦ is determined by the fraction field of the local ring of C at .Denote the result by trop ( Z η ) – note that the subscript η indicates that we are considering it as ageneric fiber.These two procedures are related. When considering the total space of Z ◦ as a surface over C ◦ ,there is a map υ : trop ( Z ◦ ) → R > . The fiber υ − ( ) coincides with the second tropicalization trop ( Z η ) . This follows from a tracingof definitions for the tropicalization of fibers of maps of Berkovich spaces and the functorialityresults in [61].1.8. Transversality for families.
The appropriate generalizations of the properness and transver-sality statements earlier in this section are as follows. Let R be the valuation ring of K and equip Spec R with the divisorial logarithmic structure at the closed point. The scheme X × Spec R isequipped with the simple normal crossings divisor given by D × Spec R ∪ X × . Once equippedwith this divisor, it has tropicalization Σ X × R > .Let Y ′ → X × Spec R be a toroidal modification of the constant family and let Y ′′ ⊂ Y ′ bethe complement of a union of closed strata. These have tropicalizations Σ Y ′ and Σ Y ′′ that are, & DHRUV RANGANATHAN respectively, a proper subdivision and a subdivision of Σ X × R > . There is a morphism of conecomplexes by composition: Σ Y ′′ → R > and we denote the fiber over by Σ Y ′′ ( ) . We view this as a polyhedral complex.Let Z ◦ be a subscheme of X ◦ × Spec K . We examine the question of when the closure is proper. Theorem 1.8.1.
The closure Z of Z ◦ in the degeneration Y ′′ of X is proper over Spec R if and only if trop ( Z ◦ ) is set theoretically contained in Σ Y ′′ ( ) .Proof. This is well-known to experts, but in the form stated, we have been unable to locate asuitable reference. We explain how it can be deduced from results that do appear in the literature.First, we note that the closure of Z ◦ in the larger degeneration Y ′ is certainly proper over Spec R because X is proper and Y ′ → X × Spec R is a proper and birational morphism. We now usethe hypothesis on K . Specifically, since K is a localization of the function field of a smooth curve C with the valuation associated to a closed point with complement C ◦ . Moreover, the givensubscheme Z ◦ , as well as the degenerations Y ′ and Y ′′ can be assumed to arise via base changefrom corresponding families over C , along the inclusion Spec R → C of the local ring at . Rather than overburdening the notation, we replace the families over Spec R with the corresponding families over C . We now view C , X × C , Y ′ and Y ′′ as logarithmic schemesover C . Note that due to the potential non-properness of C , Remark 1.5.4 is in effect. The readeris also advised to keep in mind the relationship between the two tropicalizations of Z ◦ describedabove.We are now in a position to apply the Tevelev–Ulirsch Lemma for logarithmic schemes as statedin [61, Lemma 4.1]. From it, we deduce that the closure of Z ◦ in Y ′ coincides with the closure in Y ′′ precisely under the hypotheses stated in the theorem. We conclude the result. (cid:3) We keep the notation above, and now deal with the corresponding flatness statement.
Theorem 1.8.2.
The closure Z of Z ◦ in the degeneration Y ′′ of X intersects the strata of Y ′ in the expecteddimension if and only if trop ( Z ◦ ) is a union of polyhedra in Σ Y ′′ ( ) .Proof. Proceed as in the proof of the theorem above and spread out the family until it is definedover a curve. To calculate the dimension of the intersections of Z with the strata of Y , the map to Spec R is not relevant, so we directly apply Theorem 1.5.3 in the previous section and conclude. (cid:3)
2. T
ROPICAL DEGENERATIONS AND STRONG TRANSVERSALITY
Let ( X | D ) be a simple normal crossing pair such that the all intersections of irreducible compo-nents of D are connected. We further assume that X is proper. In this section, we first introducethe precise notion of expansion of X that we consider in this paper. We then examine how to usetropical data to construct strongly transverse flat limits for families of subschemes of X , along thelines of work of Tevelev [58] and Ulrisch [61]. This will motivate our construction of the stackof expansions in the next section, and provide the main ingredient in our proof of properness.Throughout this section, we specialize our discussion to subschemes of dimension at most . OGARITHMIC DT THEORY 13
Graphical preliminaries.
Let G be a finite graph, possibly disconnected but without loops orparallel edges. We enhance it with two additional pieces of data. The first is a finite set of rays ,formally given by a finite set R ( G ) equipped with a map to the vertex set r : R ( G ) ֒ → V ( G ) . The second is the metrization of the edge set, given by the edge length function ℓ : E ( G ) → R >0 . These data give rise simultaneously to a metric space and a polyhedral complex enhancing thetopological realization of G , as follows. The topological realization of G is a CW complex and weendow an edge E with a metric by identifying it with an interval in R of length ℓ ( E ) . For eachelement h ∈ R ( G ) we glue on a copy of the metric space R > to the point r ( h ) . As we are now freeto think of each edge or half edge as being either a polyhedron or a metric space, the result is aspace G that is simultaneously a metric space and a polyhedral complex of dimension at most .In particular, it makes sense to talk about real valued continuous piecewise affine functions on G . We refer to the metric space G obtained above, together with its choice of polyhedral structure and its groupof piecewise affine functions, as a -complex. Let Σ be a smooth cone complex such that the intersection of any two cones is a face of each.Note that Σ embeds canonically as a subcomplex of a standard orthant via Σ ֒ → R Σ ( ) > . Given apiecewise affine map of polyhedral complexes F : G → Σ, each non-contracted edge E maps to a cone σ , and thus maps onto a line L E in σ gp . We refer to theexpansion factor of the induced map F : E → L E as the slope of F along the edge E . We consider anembedding G ֒ → Σ in the category of rational polyhedral complexes – that is, an embedding given by a collection ofpiecewise affine functions on G that have slope on each edge.2.2. Target geometry.
Let Σ X be the cone complex of ( X | D ) . An embedded -complex G ⊂ Σ X gives rise to a class of target geometries as follows. Place the embedded -complex G at height inside Σ X × R > and let C ( G ) be the cone over it. Lemma 2.2.1.
The cone C ( G ) over G is a cone complex embedded in Σ X × R > Proof.
The cone over each face in G certainly forms a cone, so we check that this collection of conesmeet along faces of each. Following the argument in [24, Theorem 3.4] this check is nontrivial inthe fiber over in Σ X × R > . Since G has dimension either or , the cones in the fiber are eitherrays starting from the origin or the origin itself. Since two such rays either coincide or intersectonly at the origin, the check is immediate. (cid:3) By applying the construction in Section 1.3 and Remark 1.3.3 to the subdivision C ( G ) ֒ → Σ X × R > , we obtain a target expansion Y G → X. One might be inclined to call this object an embedded tropical curve, but we reserve this terminology for use in con-texts where it is natural to decorate this -complex with data such as Hilbert polynomials or intersection multiplicities. & DHRUV RANGANATHAN
It is typically not proper.
Remark 2.2.2.
The construction of such target expansions using subdivisions goes back at least toMumford’s work on degenerations of abelian varieties [49]. Its first appearance in enumerativegeometry is in work of Nishinou and Siebert [50].Every affine toric variety carries a canonical logarithmic structure. Consider a toric monoid P with associated toric variety U P and a closed point u in the closed torus orbit. By pulling backthe toric logarithmic structure to u , we obtain the P -logarithmic point . It is denoted Spec C P . The standard logarithmic point is Spec C N and is the natural logarithmic structure at in A . Definition 2.2.3 ( Expansions and families) . A rough expansion of X over Spec C N is a logarithmicscheme Y → X × Spec C N which is the fiber over in the modification of ( X | D ) × A induced by asubdivision ∆ ֒ → Σ X × R > . A rough expansion is called simply an expansion if, in addition, the following two conditions aresatisfied(Exp 1) The scheme Y is reduced.(Exp 2) The subdivision is given by the cone over an embedded -complex in Σ X . Equivalently,the relative logarithmic rank of Y → Spec C N is at most .If S is a fine and saturated logarithmic scheme, a morphism Y /S → X × S/S is called an expansionof X over S if all pullbacks Spec C N → S are expansions and if the morphism Y → S is flat andlogarithmically smooth. Remark 2.2.4.
The definition has a peculiar feature when we consider X itself. Specifically, X is arough expansion of itself, but is typically not an expansion of itself because of the codimension logarithmic strata. The complement of these strata in X is an expansion of X , as is the interior of X .The reason for imposing this condition is that we will later require our families of subschemes tohave nonempty intersection with all logarithmic strata, which is not satisfied for a -dimensionaltransverse subscheme unless there are no codimension strata. This definition yields a cleaneruniversal property when we consider the valuative criterion of properness.2.3. Flat limit algorithms: existence.
We let X ◦ be the complement of D in X . Let C be a smoothcurve with a distinguished point , whose complement is denoted C ◦ . Proposition 2.3.1.
Consider a flat family of subschemes Z ◦ ⊂ X ◦ × C ◦ over C ◦ . There exists a roughexpansion Y → X × C over C such that the closure Z of Z ◦ in Y has the following properties:(E1) The scheme Z is proper and flat over C ,(E2) The scheme Z has non-empty intersection with each stratum of Y .(E3) The scheme Z intersects all the strata of Y in the expected dimension.Moreover, after replacing C with a ramified base change, the degeneration of X can be guaranteed to havereduced special fiber, i.e. the rough expansion can be chosen to be an expansion. We will refer to a family satisfying these properties as as dimensionally transverse family.
Proof.
Let
Spec C [[ t ]] → C OGARITHMIC DT THEORY 15 be the spectrum of the completed local ring of C at . Pull back the subscheme family Z ◦ to thegeneric point of this valuation ring to obtain a subscheme Z ◦ η of X ◦ over C (( t )) . After passing to avalued field extension with value group R , apply the tropicalization map to obtain trop ( Z ◦ ) ⊂ Σ X . By the structure theorem for tropicalizations, this is the support of a polyhedral complex of di-mension at most in Σ . Choose a polyhedral structure on this set and call it G . Let C ( G ) be thecone over G in Σ X × R > . We obtain an associated target family Y G → C. By construction, the tropicalization of Z ◦ when viewed as a single subscheme of the total space X ◦ × C ◦ is equal to C ( G ) . The theorems in Section 1.4 imply that the closure is proper. Since thebase is a smooth curve, the closure is also flat over C .We examine the strata intersections. Given a stratum W of Y G dual to a cone σ in C ( G ) , choosea point v ∈ σ . By definition, this is the image of a K -valued point of Z ◦ under the tropicalizationmap. Since the family of subschemes is proper this extends to an R -valued point where R is thevaluation ring of K . Since the tropicalization of this K -valued point lies in the interior of σ , theclosed point maps to W , so it follows that Z intersects all strata.Finally, we show that we can obtain a degeneration with reduced special fiber after a basechange. By using the toric dictionary, we observe that the special fiber of Y G → C is reduced if andonly if the vertices of G are lattice points in Σ X . This can be engineered by using Kawamata’s cycliccovering trick, explained in [7]. The toroidal procedure is carried out by replacing the integrallattice in Σ C with a finite index sublattice, thereby ensuring that the fiber over the new primitivegenerator of Σ C is a dilation of G which has integral vertices. The main result of [7, Section 5] is thatthis produces the requisite ramified base change. The statements concerning strata intersectionsare unaffected by the base change, and we conclude the result. (cid:3) Flat limit algorithms: uniqueness.
The tropical limit algorithm in the previous section inher-its a uniqueness property that should be thought of as a separatedness result for the forthcomingmoduli problem. Let K be the discretely valued field associated to the valuation at in C and let R be its valuation ring. Proposition 2.4.1.
Let Z ◦ η be a flat family of subschemes of X ◦ over Spec K . Assume that the closure of Z ◦ η in X × Spec K is dimensionally transverse. Then there exists a canonical triple ( R ′ , Y ′ , Z ′ ) comprising of aramified base change R ⊂ R ′ with fraction field K ′ and expansion of X Y ′ → Spec R ′ such that the closure Z ′ of Z ◦ η ⊗ K K ′ in Y is dimensionally transverse. Moreover, the triple satisfies thefollowing uniqueness property: ( ⋆ ) For any other choice ( R ′′ , Y ′′ , Z ′′ ) satisfying these requirements, there exists a unique toroidal bira-tional morphism Y ′′ → Y ′′ over Spec R ′′ with subscheme Z ′′ , and a ramified covering Spec R ′′ → Spec R , such that Z ′′ ⊂ Y ′′ isobtained from Z ′ ⊂ Y ′ by base change. We deduce the uniqueness results from two simple combinatorial observations. & DHRUV RANGANATHAN
Lemma 2.4.2.
Let ι : G ֒ → Σ X be an embedded -complex. The underlying set | G | of G carries a uniqueminimal polyhedral structure G such that the map ι descends to an embedding G ֒ → Σ X . The term minimal here is used in the sense that any other polyhedral structure is obtainedfrom the putatutive unique minimal one by subdividing along edges. Note that a morphism ofpolyhedral complexes is required to map vertices and edges of G to into cones of Σ X . Proof.
Recall that G comes with a distinguished vertex set and | G | has the structure of a metricspace independent of the chosen vertex set. The non-bivalent vertices of G are necessarily con-tained in the vertex set of any polyhedral structure on | G | . Let x ∈ G be a -valent vertex. Call v inessential if both of the following conditions hold. (1) There exists an open neighborhood U x of x that is completely contained in the relative interior σ ◦ of a cone σ ∈ Σ X .(2) Every point in this neighborhood lies on the same line in the vector space σ gp .A vertex that is not inessential is essential. Let G be the polyhedral complex obtained from themetric space | G | by declaring the vertices of G to be the essential vertices of G . The morphism ι descends to an embedding G → Σ X . Conversely, any polyhedral structure must contain the pointsof | G | that are not inessential to ensure that ι is a morphism of polyhedral complexes. It followsthat ι : G → Σ X is minimal. (cid:3) Passing to the cone over G , we obtain a uniqueness property concerning the dilations of G obtained by uniformly scaling the edge lengths. Lemma 2.4.3.
Let G ֒ → Σ X be an embedded -complex. Let C ( G ) ֒ → Σ X × R > be the cone over theminimal polyhedral structure G of G . There exists a minimum positive integer b such that all vertices inthe fiber of C ( G ) over b lie in the lattice of Σ X .Proof. The proof is an exercise in least common multiples. (cid:3)
Proof of Proposition.
The uniqueness is a translation of the combinatorial observations above, whichwe now undertake. Let Z ◦ η be a subscheme as in the proposition. The algorithm for finding limitshas the following steps. We calculate the tropicalization of Z ◦ η and obtain a canonical -complex G ֒ → Σ X . We then pass to the cone over this complex C ( G ) in Σ X × R > . Finally, we find theminimum positive b integer in R > over which the fiber in C ( G ) has integral vertices. We thenperform the order b cyclic ramified base change to obtain the requisite triple ( R ′ , Y ′ , Z ′ ) .Now consider another set ( R ′′ , Y ′′ , Z ′′ ) satisfying these properties. Let C ( G ′′ ) → R > be thefan associated to this expansion. Tropicalization is invariant under taking valued field extensions;since the subscheme Z ′′ nontrivially intersects all the strata of the expansion, we can concludeusing Theorem 1.6.1 that the support of the tropicalization is equal to C ( G ′′ ) where G ′′ is theheight slice of the cone. If we replace G ′′ with its minimal polyhedral structure G ′′ , we obtain arefinement of cone complexes C ( G ′′ ) → C ( G ′′ ) . The latter cone complex gives rise to an expansionof X in which the closure of Z ◦ η is still dimensionally transverse. Intuitively, x is inessential if the map ι doesn’t bend at x ; thus, the map ι descends to a polyhedral map aftercoarsening the polyhedral structure to exclude x from the vertex set. OGARITHMIC DT THEORY 17
For the base change, by using the preceding lemma we let G ′ be the minimal dilation of G whose vertices are all integral. The cones over G ′ and G ′′ coincide, so G ′′ must be a dilation of G ′ .It follows that the morphism C ( G ′′ ) → R > is obtained from C ( G ′ ) → R > by passing to a finite index sublattice in the integral structure of the base. The result follows. (cid:3) Dimensional transversality to strong transversality.
We have constructed limits of trans-verse subschemes as dimensionally transverse subschemes in expansions of X along D . We requirea stronger form of transversality. Definition 2.5.1 (Strong transversality) . Let Y be an expansion of X and let Z ⊂ Y be a subschemewith ideal sheaf I Z . Then Z is said to be strongly transverse if it is dimensionally transverse and forevery divisor stratum S ⊂ Y , the induced map I Z ⊗ O Y O S → O Y ⊗ O Y O S is injective.Equivalently, the subscheme intersects every stratum non-trivially, it has no embedded pointsor components on divisorial strata, and for every double divisor S contained in components Y i and Y j of Y , the restrictions of the subscheme to Y i and Y j have the same restriction to S .In their study of ideal sheaves relative to a smooth divisor, Li and Wu prove that, starting with astrongly transverse family over the generic point of a discrete valuation ring, a strongly transverselimit can always be found, but it requires further expansion of the target. Using the dimensionaltransversality guaranteed by the previous section, we deduce strong transversality in the normalcrossings setting by reducing to a situation where we can use their results.Our main result is the following variation on Proposition 2.4.1. Recall that K is the discretelyvalued field associated to in the curve C and R is its valuation ring. Proposition 2.5.2.
Let Z ◦ η be a flat family of subschemes of X ◦ over Spec K whose closure in X is stronglytransverse. Then there exists a canonical triple ( R ′ , Y ′ , Z ′ ) comprising of a ramified base change R ⊂ R ′ with fraction field K ′ and expansion of X Y ′ → Spec R ′ such that the closure Z ′ of Z ◦ η ⊗ K K ′ in Y is strongly transverse, satisfying the following uniqueness prop-erty: ( ⋆ ) For any other choice ( R ′′ , Y ′′ , Z ′′ ) satisfying these requirements, there exists a unique toroidal bira-tional morphism Y ′′ → Y ′′ over Spec R ′′ with subscheme Z ′′ , and a ramified covering Spec R ′′ → Spec R , such that Z ′′ ⊂ Y ′′ isobtained from Z ′ ⊂ Y ′ by base change.Proof. The proposition is established in the following sequence of lemmas. (cid:3)
The following lemma is an immediate consequence of Proposition 2.4.1. & DHRUV RANGANATHAN
Lemma 2.5.3.
Let K be a discretely valued field with valuation ring R and let Z η be a strongly transversesubscheme of X over Spec K . Let Z ′ be the canonical dimensionally transverse limit over a ramified exten-sion Spec R ′ , in a subscheme Y ′ . Let Z ′′ ⊂ Y ′′ be a strongly transverse limit over a ramified extension Spec R ′′ . Then Z ′′ is obtained from Z ′ by a ramified base change and a sequence of weighted blowups of thecodimension strata of the special fiber degeneration. In combinatorial language, given the graph G ′ whose cone yields the minimal dimensionallytransverse extension Z ′ ⊂ Y ′ , if a strongly transverse extension exists, it is obtained by introducinga finite set of new -valent vertices along G ′ to form G ′′ , forming the cone C ( G ′′ ) → R > and thenpassing to a finite index sublattice on the base to make the new vertices of G ′′ integral. Lemma 2.5.4.
Let K be a discretely valued field with valuation ring R and let Z η be a strongly transversesubscheme of X over Spec K . After passing to an extension R ⊂ R ′′ , a strongly transverse limit exists.Proof. We first form the dimensionally transverse limit Z ′ ⊂ Y ′ over Spec R ′ and let G ′ be theassociated embedded polyhedral complex. If such a limit fails to be strongly transverse, it is be-cause there exists a codimension stratum that hosts an embedded point of Z ′ . Call this stratum W , and observe that it is dual to a possibly unbounded edge of e of G ′ and let C ( e ) be the corre-sponding cone in C ( G ′ ) . If we pass to the completion of Z ′ and Y ′ along W , we are in preciselythe Li-Wu situation, specifically [38, Section 5], where we have a double-point degeneration of asmooth scheme. It follows from their argument that, after a ramified base change and a sequenceof blowups over W , the closure of the generic fiber family is strongly transverse.We can repeat this procedure for every edge in G ′ , taking the least common multiple of the basechanges for each edge, to obtain a refined polyhedral structure G ′′ such that the closure of Z η inthe associated degeneration Y ′′ is strongly transverse. (cid:3) Lemma 2.5.5.
There exists a strongly transverse limit which is universal in the sense of Proposition 2.5.2.Proof.
To see this, we start with the family which is universal with respect to having a dimension-ally transverse limit. As before, we then look in a neighborhood of each codimension stratum W and apply the corresponding uniqueness statement in the Li-Wu situation. More precisely, in theirproof of separatedness, they show (in the setting of double-point degenerations), the existence ofa unique expansion such that the limit of Z ′ η is strongly transverse and stable. Here, stability intheir sense means that the restriction to every irreducible component lying over W has finite stabi-lizer with respect to the natural G m -action. Moreover, it follows from their construction that anystrongly transverse limit can be stabilized to this unique stable limit, by contracting all destabiliz-ing components. In our setting, we apply this to each edge of G ′ to construct a strongly transverselimit with the desired property. (cid:3) Remark 2.5.6.
We have the following consequence of the algorithm for producing strongly trans-verse limits. If G ֒ → Σ X denotes the -complex associated to the minimal strongly transverse limit, G may involve refining the minimal polyhedral structure G by adding bivalent vertices to subdi-vide edges. However, if we look at the irreducible components Y i corresponding to these bivalentvertices, the embedded subscheme Z i ⊂ Y i is stable, i.e. not fixed by the natural G m -action on Y i .With the results of this section as motivation, we define a subscheme of X relative to D . OGARITHMIC DT THEORY 19
Definition 2.5.7 (Relative subschemes) . Let S be a logarithmic scheme. A subscheme of X relativeto D over S is an expansion Y /S of X over S together with a flat family of strongly transversesubschemes Z ⊂ Y .Proposition 2.5.2 suggests that strongly transverse subschemes will satisfy the valuative crite-rion for properness. Our goal is now to construct moduli for these objects – first for the expansionsthemselves which happens in the next section, and then for the strongly transverse subschemeswhich happens in the one after.3. M ODULI OF TARGET EXPANSIONS
The purpose of this section is to construct moduli spaces for the expansions in the previous sec-tion, in analogy with the stack of expansions of a smooth pair; see [5, Section 6.1] for an expositionof the latter. Our approach is to use the dictionary between combinatorial moduli spaces and Artinfans, which we review in the first subsection. Using this, we then study the combinatorial modulispace parametrizing embedded -complexes inside Σ X , and show it can be given the structure ofa cone space, in the sense of [14].Two important subtleties arise in this process. First, as discussed in the introduction, the choiceof cone decomposition is not unique, and requires an auxiliary combinatorial choice. Second,in order to endow the universal family with the structure of a cone space, further subdivisionis required, which translates into allowing additional codimension- bubbling in the geometricexpansion. See Remarks 3.4.2 and 3.5.1.3.1. Artin fans and cone stacks.
The stack
Exp ( X | D ) is similar in nature to Olsson’s stack LOG oflogarithmic structures and are instances of the Artin fans that are extracted from
LOG , see [9, 51].An
Artin cone is a global toric quotient stack: A P = Spec C [ P ] / Spec C [ P gp ] , where P is a toric monoid. These stacks are endowed with a logarithmic structure in the smoothtopology, from the toric logarithmic structure on Spec C [ P ] . Observe that A P recovers P from itslogarithmic structure, so these data naturally determine each other. By varying the monoids P , thecanonical morphisms from Artin cones A P → LOG form a representable ´etale cover of the latter,by work of Olsson [51, Section 5]. This stack is logarithmically ´etale over a point.
Definition 3.1.1. An Artin fan is a logarithmic algebraic stack A that has a strict ´etale cover by adisjoint union of Artin cones.Under mild assumptions, every logarithmic structure on a scheme arises from a morphism toan Artin fan [5, Proposition 3.2.1]. In practice, it is convenient pass to an equivalent but explicitlycombinatorial -category, for which we follow [14, Sections 2.1–2.2]. Definition 3.1.2 (Cone spaces) . A cone space is a collection C = { σ α } of rational polyhedral conestogether with a collection of face morphisms F such that the collection F is closed under compo-sition, the identity map of every cone σ α lies in F , and every face of a cone is the image of exactlyone face morphism in F .A cone complex is obtained when the further assumption is placed that there is at most onemorphism between any two cones. A cone stack can be defined similarly, as a fibered categoryover the category of rational polyhedral cones satisfying natural conditions. We will not need to & DHRUV RANGANATHAN work with cone stacks in a serious way, so we refer to the reader to [14, Section 2] for a definition.Their results allow us to systematically work with Artin fans using combinatorics.
Theorem 3.1.3 ([14, Theorem 3]) . The -categories of Artin fans and of cone stacks are equivalent. Under this equivalence, an Artin cone A P is carried to the cone Hom ( P, R > ) . If Σ is a fan embed-ded in a vector space and W Σ is the resulting toric variety with dense torus G Σ , the equivalencecarries the global quotient Artin fan [ W Σ / G Σ ] to the cone complex Σ .3.2. Embedded complexes.
Section 2 constructed expansions of X along D that accommodatelimits of transverse subschemes. The construction was presented over valuation rings with loga-rithmic structure. In order to globalize this we construct moduli for the tropical data over higherdimensional cones, and then glue these cones to form a cone space. By the categorical equivalencein Theorem 3.1.3, we will have produced Exp ( X | D ) as an Artin stack.Recall that Σ is a cone complex with unimodular cones such that the intersection of any twocones is a face of each, and we view it as a subcomplex of the standard orthant by the embedding Σ ֒ → R Σ ( ) > . We will make use of the Euclidean geometry of this ambient vector space.In Section 2.1 we defined a -complex to be a possibly disconnected metric graph G togetherwith polyhedral complex structure, and that an embedding of -complexes was an injective mapof polyhedral complexes G ֒ → Σ given by piecewise affine functions of slope . Let | T ( Σ ) | be the set of isomorphism classes of -complexes G equipped with an embedding G ֒ → Σ. For each embedded -complex, we can dilate the embedded complex by any positive real scalar,giving rise to an isometric copy of R > in | T ( Σ ) | . We will endow this set with the structure of acone space, whose rays will be a subset of the rays obtained in this fashion.3.3. Construction of cones of embedded graphs.
By the conventions in this text, a graph refers toa possibly disconnected finite graph without loop or parallel edges, together with a finite collec-tion of rays placed at the vertices. As is standard in Gromov–Witten theory, we visualize the raysas “legs”.In order to put a cone space structure on | T ( Σ ) | , we first study spaces of maps from graphs to Σ and then identify maps with the same image. For nw, we will suppress discussion of the integralstructure, and fix the correct integral lattice afterwards. Definition 3.3.1. A combinatorial -complex G in Σ is a graph G with vertex, ray, and edge sets V ( G ) , R ( G ) , and E ( G ) , equipped with the following labeling data:(C1) For each vertex V ∈ V ( G ) a cone σ V in Σ .(C2) For each edge or ray E ∈ E ( G ) ∪ R ( G ) , a cone σ E together with a primitive integral vector inthe associated lattice σ E ( N ) (referred to as a direction vector )subject to the compatibility condition that if V is an incident vertex of an edge or ray E , the cone σ V is a face of σ E . OGARITHMIC DT THEORY 21
We define a linear path in G to be a path (consisting of edges and/or rays) where every edge/rayhas the same direction vector. Let P ( G ) denote the set of linear paths in G .An isomorphism of combinatorial -complexes is an isomorphism of the underlying graphsthat is compatible with the labels. Fix a representative for each isomorphism class. We considerembeddings of a combinatorial -complex into Σ . Let | Σ | denote the support of the cone complex.Define the set X G to be the set of functions f : V ( G ) → | Σ | such that(1) For each V ∈ V ( G ) , the image f ( V ) lies in σ V .(2) For each E ∈ E ( G ) with adjacent vertices V and W , the line segment between f ( V ) and f ( W ) has direction vector equal to the one labeling E .Each such function f ∈ X G determines a polyhedral complex in | Σ | . It can be constructed asfollows. Take the collection of points obtained as images vertices of G under f . Given an edge E between V and W introduce a segment between f ( V ) and f ( W ) , whose edge direction is nec-essarily the one given by E . For each ray, glue an unbounded ray with the corresponding basepoint and edge direction dictated by G . This topological space has a minimal polyhedral complexstructure whose vertices are of two types: (1) images of vertices of G under f and (2) points whereedges or rays intersect.We refer to this embedded complex as the image, and denote it by im ( f ) . We restrict to those G such that there exists at least one function f ∈ X G such that im ( f ) is isomorphic to the complex | G | ,the latter with its canonical polyhedral structure. Lemma 3.3.2.
The set X G has the structure of a cone.Proof. The set of all functions f : V ( G ) → | Σ | satisfying only the first of the two conditions aboveis given by the product of σ V ranging over all vertices V . Inside this product, we consider thelocus where the second condition is imposed. Consider an edge E with incident vertices V and W . Consider the line containing f ( V ) with edge direction given by E . The intersection point ofthis line with the cone σ W is linear in the coordinates of f ( V ) . It follows that the locus where thesecond condition above is satisfied for a given edge E is linear, and that X G is cut out from theproduct of all cones σ V by a finite collection of linear equations. This gives us a cone structure on X G . (cid:3) As mentioned above, each point f ∈ X G determines an embedded -complex im ( f ) . Our pri-mary interest is in embedded complexes, so we would like to replace the space X G with the cor-responding set of images. However, the topological type of the image is not constant even forpoints in the relative interior of a cone in X G . A key example is provided by two skew edges ina -dimensional cone that intersect in their interiors. The following definition is meant to capturethis phenomenon of varying topological type. Let G and H be combinatorial -complexes. Definition 3.3.3. A surjection from G to H is given by the a pair of maps τ : V ( G ) → V ( H ) υ : E ( G ) ⊔ R ( G ) → P ( H ) subject to the following conditions for:(1) If E is an edge of G , then the edge directions for each edge in υ ( E ) coincide with that of E . & DHRUV RANGANATHAN (2) If V and W are adjacent vertices with E , then τ ( V ) and τ ( W ) are the endpoints of the path υ ( E ) .(3) For each vertex of G the cone σ τ ( V ) is equal to a face of σ V .(4) Each edge or ray in H is contained in a path which is in the image of υ .Each point f ∈ X G determines a surjection of G , as follows. Given a point f ∈ X G , we considergraph associated to the image im ( f ) , considered with its natural structure as a combinatorial -complex. A basic finiteness result holds. Lemma 3.3.4.
Let G and X G be as above. The set of surjections of G associated to points of X G is finite.Proof. We may view the cone X G as a cone of tropical maps with type G to Σ , in the sense of [25,Remark 1.21]. It comes equipped with a moduli diagram G Σ × X G X G . The fibers of the vertical map are the metrizations of G given by endowing each edge E with lengthequal to the distance in Σ between the images of its endpoints. Since the horizontal map is linearon cones, the image of G in Σ × X G can be given the structure of a cone complex. Choose anysuch cone complex structure im ( G ) . After subdividing this image further, and replacing X G by asubdivision, the induced map e G → e X G is flat in the combinatorial sense – every cone of the source surjects onto a cone of the image. Thisfollows from [7, Section 4]. After this subdivision, the combinatorial type of the -complex in thefibers in the relative interior of any cell is constant. It follows that there are only finitely manysurjections that appear from taking image, as claimed. (cid:3) Let G be as above, and let H be a surjection of G associated to a point in X G . Let X H be the coneassociated to the combinatorial -complex H . There is an associated inclusion X H ֒ → X G . Specifically, a point of X H in particular determines a function on V ( G ) by using the map on vertexsets V ( G ) → V ( H ) . The definitions have been made in order to guarantee that this induces a pointof X G . Lemma 3.3.5.
Let G → H and G → K be two surjections of G obtained from points of X G by the imageconstruction. Consider a point f ∈ X H ∩ X K ⊂ X G . The surjection G → J determined by f is a common surjection of H and K .Proof. The lemma follows immediately from the definitions by taking J to be the image of f . (cid:3) Fix a combinatorial -complex G and let Aut G be the automorphism group. There is a naturalaction of Aut G on the cone X G . Our goal will be construct an automorphism-equivariant subdivi-sion of X G such that the subsets X H for surjections H of G become subcomplexes. The followinglemma will be of use to us. OGARITHMIC DT THEORY 23
Lemma 3.3.6.
Let C be a cone and let F ⊂ C be an embedded cone complex. Let Γ be a finite group actingon C and such that F is Γ -stable. There exists a Γ -equivariant proper subdivision e C of C such that F is aunion of faces of e C . If F is a smooth cone complex, then e C can be chosen to be smooth.Proof. Let N R denote the associated vector space of the cone C . We consider F and C as subsetsof this vector space. The existence of equivariant completions for toric varieties guarantees thatthere exists a complete fan ∆ in N R that contains F as a subcomplex [21]. Consider the commonrefinement of ∆ and C : C ′ = { σ ∩ σ : σ ∈ ∆ σ ∈ C } . This is a cone complex structure on the intersection of the supports of the two fans, that is on | C | ,see [39, Section 2.3]. It clearly contains F as a subcomplex. The subdivision is not Γ equivariant.We fix this by averaging over the group. Given γ ∈ Γ , the set of translates of cones in C ′ form acone complex with support | C | . The common refinement over all group elements gives rise to thesubdivision e C as claimed in the lemma. Since F is Γ -stable, this last common refinement step doesnot subdivide it. The final statement on smoothness follows from the fact that toric resolution ofsingularities for any fan can be performed without subdividing smooth cones [18, Theorem 11.1.9]and the fact that these subdivisions can be made Γ equivariant [8, Section 2]. (cid:3) Note that this procedure typically involves choices, and is not canonical. Our application is tothe cones associated to combinatorial one complexes.
Lemma 3.3.7.
For each combinatorial -complex G , there exists an Aut G -equivariant subdivision Y G → X G such that for each surjection G → H induced by a point of X G , the induced map Y H ֒ → Y G is an inclusion of a subcomplex. If Y H is smooth, then Y G can be chosen to be smooth.Proof. Given G , the set of all surjections of G forms a partially ordered set, since the notion is stableunder composition. Choose any total order extending this partial order. At the minimal elementsof this partial order, there is no subdivision necessary, and the union of the cones associated tothese types forms a cone complex in X G . Proceeding in the chosen order, we repeatedly applyLemma 3.3.6. Given H and its surjections J , . . . J k , we observe that by Lemma 3.3.5, the union ofthe inductively chosen Y J i is a cone complex. We may therefore apply the preceding lemma toobtain an Aut H -equivariant subdivision Y H of X H such that each previously constructed Y J i is aunion of faces. The lemma follows. The smoothness statement follows from the correspondingstatement in the preceding lemma. (cid:3) Tropical moduli of expansions.
In order to pass from our constructions in the previous sec-tion to the moduli of expansions, we need to forget the labellings of the graph G and then pass tothe colimit over all comibnatorial graphs.We fix a set of subdivisions Y G as guaranteed by the lemma above. For a fixed combinatorial -complex G , we consider the “coarse” quotient space Y G / Aut G . This quotient exists in the categoryof generalized cone complexes defined by Abramovich–Caporaso–Payne [1, Section 2.6]. It isexplained there that the quotient also carries a natural cone complex structure, obtained from thebarycentric subdivision of Y G . We equip the quotient with this structure and denote it Y G // Aut G . & DHRUV RANGANATHAN
Definition 3.4.1. A moduli space of tropical expansions T is defined to be the colimit of the diagram ofcone complexes { Y G // Aut G } G where G ranges over all combinatorial -complexes in Σ , and witharrows given by the morphisms Y H // Aut H ֒ → Y G // Aut G where G → H is a surjection.By construction, the morphisms in this system satisfy the conditions required for the limit to bea cone space, see Definition 3.1.2.By equivariant resolution of singularities, it can be guaranteed that the cones of T are smooth.This can be done either by applying equivariant resolution in the construction of the complexes Y G above, or by applying resolution to the output of the construction. From this point forth, weonly worth with those T that are smooth. Remark 3.4.2.
Note that the cone spaces T constructed above are non-unique. The ambiguityarises from the inductive construction of the subdivision Y G as a subdivision of X G , for each graph G . The construction is analogous to, and derived from, the result on equivariant completions oftoric varieties and on completions of fans [21]. In this sense, the root cause of the non-uniquenessof our spaces is the same as the non-uniqueness seen in the compactification problems for tori.3.5. Universal family and integral structure.
Given a moduli space T of tropical expansions, wecan equip it with a universal expansion. More precisely, we construct a diagram of cone spaces Υ ⊂ Σ × T Σ T , where Υ ⊂ Σ × T is a subdivision and the map Υ → T is flat.In order to construct this diagram, we observe that each X G comes equipped with a universalgraph G and a map to Σ × X G . In order to construct Υ we simply replicate every step in theconstruction of T above, modified as follows.For each G , we consider the augmented graph G ∗ consisting of the disjoint union of G with adistinguished vertex ∗ . If we denote s a choice of vertex, edge, or ray of G , we consider the set U G ( s ) consisting of a function from G ∗ to Σ whose restriction to G is in X G and such that f ( ∗ ) lieson the image of f ( s ) .Each element of U G ( s ) determines a -complex in Σ equipped with distinguished basepoint . Theset U G ( s ) forms a cone, since the choice for the basepoint is a cone, namely the cone formed bythe edge or vertex s , as f varies in X G . As we vary s , we obtain a cone complex U G equipped withan action of Aut G and a morphism ι G : U G → X G × Σ. Given a surjection G → H , an edge s of G , and an edge t of H which is contained in the path υ ( s ) ,we have U H ( t ) ֒ → U G ( s ) . We have similar maps when s or t are rays or vertices. Notice given a surjection, there could bemany distinct embeddings of U H ( t ) into U G . OGARITHMIC DT THEORY 25
Following our previous approach, the lemmas again yield a system of subdivisions g U G of U G which are compatible with the action of Aut G , the pullback of the subdivision Y G , and the inclu-sions of ^U H ( t ) . If we apply the map ι G to this subdivision, it is injective on every cone, and theimages of any two cones intersect along a unique face of each. Therefore, there is a natural conestructure on its image V G ⊂ Y G × Σ , along with natural maps V H → V G associated to surjections.We pass to Aut G -quotients and declare the colimit to be Υ .Finally, we fix the integral structure on T and Υ as follows. Given a rational point P of T , we sayit is integral if every vertex of the embedded -complex Υ P ⊂ Σ lies on an integral lattice point of Σ. Similarly, a rational point Q ∈ Υ is integral if its image in T × Σ is integral. One can check thatthis definition of integral points in each cone σ of T is compatible with the usual notion of integralstructure, i.e. it is the restriction of a lattice in the ambient rational vector space. It follows fromour definition that the map Υ → T is flat and reduced. Remark 3.5.1.
Given any point P of T , we obtain two -complexes associated to this point. First,since we can identify points of T and | T | , we have the -complex G P associated to this point.Second, we can take the fiber Υ P of the map Υ → T . It follows from the construction that there isa map Υ P → G P which is a refinement of polyhedral sets, obtained by adding -valent vertices along edges. Wewill refer to the vertices of Υ P that are not present in G P as tube vertices , and these will play animportant role in our definition of stability.As with our construction of T , a combinatorial choice is required in the construction of Υ . Con-sequently, given P , while the -complex G P is canonically given, the subdivision Υ P depends onthis choice. Moreover, the integral structure on T also depends on this additional choice. Remark 3.5.2 (Relative compactification) . Fix a moduli space of tropical expansions T and a fi-nite type subspace T β,χ . We abuse notation mildly, allowing Υ denote the universal embedded -complex in T β,χ × Σ X . The subdivisions we have constructed for the universal target expansionsare typically non-proper. For appropriate choices in the construction of T it is possible to ensurethat Υ can be completed to proper subdivision of T β,χ × Σ X . Indeed, after itreated barycentricsubdivision T β,χ can be embedded in a vector space [6, Section 4.6]. Results on equivariant com-pletions for fans of toric varieties give rise to the completion [21]. Toroidal semistable reductionguarantees flatness after a further sequence of blowups of T β,χ .3.6. Common refinements.
The different choices involved in constructing T above lead to differ-ent cone spaces, but they only differ by subdivision. We record this for future use. Let Λ denotethe set of all conical structures on the set | T | that arise from the construction above. Given λ ∈ Λ ,let T λ denote the associated moduli space of tropical expansions. Proposition 3.6.1.
Any two conical structures λ, µ ∈ Λ share a common refinement. Precisely, thereexists ϕ ∈ Λ and morphisms T ϕ T λ T µ such that each of the two arrows is a composition of a proper subdivision. & DHRUV RANGANATHAN
Proof.
We reinspect the procedure described above. For each combinatorial -complex G and sur-jection G → H , we choose a subdivision of X G such that the cone complex Y H is a union of faces ofthis subdivision. In turn, there are choices involved in constructing Y H from X H , given by study-ing surjections H → K and so on. The moduli spaces T are then formed by taking the colimit of thequotients of these cones by the appropriate automorphism groups. However, we note that twocone complexes on the same underlying space always have a common refinement, whose conesare given by intersections of the cones in the two complexes. Given cone complexes Y λG and Y µG ,as their support is the cone X G we can construct their common refinement Y ϕG . The common re-finement yields a compatible system of subdivisions for the different combinatorial -complexesin the following sense. Given a graph surjection G → H and an inclusion X H ֒ → X G we may take the common refinement of X H with each of these three cone complex structures on X G to obtain Y ϕG to obtain Y λH , Y µH , and Y ϕH . A definition chase reveals that Y ϕH is the commonrefinement of Y λH and Y µH . Further, the common refinement of a Aut G -equivariant subdivisionsof X G remains equivariant. It follows that the preceding colimit construction can be run with thequotients Y ϕG // Aut G . By construction, this cone space refines both T λ and T µ . (cid:3) Remark 3.6.2.
There is a broader context for the construction here that we have skirted around inorder to keep the discussion concrete. The functor | T | has been defined above on the monoid N , butit has a natural extension to all valuative monoids . Functors on valuative monoids (resp. valuativelogarithmic schemes) often arise as inverse limits of cone complexes (resp. logarithmic schemes)along subdivisions (resp. logarithmic modifications). However, a functor defined directly on thevaluative category needs to satisfy additional properties in order to arise in this fashion. Oncethese are satisfied, the functor in question, in our case | T | is determined as a polyhedral complexup to refinement. Rather than working with the inverse limit directly, we work with the entireinverse system, making compatible choices to produce morphisms where necessary. There are,at this point, relatively few genuinely valuative approaches to moduli problems. The logarithmicPicard group is a notable exception [48].3.7. Geometric moduli.
We now pass to Artin fans to produce our moduli stack of target expan-sions. Consider Σ X the cone complex for ( X | D ) .Given a moduli space of tropical expansions T for Σ X and its universal family Υ , we set Exp ( X | D ) to be the Artin fan associated to the cone space T via Theorem 3.1.3. It carries a natural logarithmicstructure and is smooth if we choose T to be smooth. The points of Exp ( X | D ) correspond to conesof T .To produce a universal expansion over Exp ( X | D ) , we apply the Artin fan construction to theinclusion Υ ֒ → Σ × T to obtain A Υ → A Σ × Exp ( X | D ) . Using the natural map X → A Σ , we set Y := A Υ × A Σ × Exp ( X | D ) ( X × Exp ( X | D )) . and consider the diagram Y X Exp ( X | D ) . OGARITHMIC DT THEORY 27
Since we arranged for the morphism of cone complexes Υ → T to be flat and reduced, Y is a familyof expansions of X over Exp ( X | D ) .Recall that, for every point P of T , we have a distinguished set of tube vertices which are -valent vertices that appear in the subdivision Υ P → G P . Given a cone σ of T , the combinatoricsof the universal complex does not change in the interior of σ , so the tube vertices of Υ are well-defined over σ by choosing any point P in its interior. If Y σ denotes the geometric expansioncorresponding to this point of Exp ( X | D ) , the irreducible components corresponding to these tubevertices will be called tube components . These are components that are not needed for applying thevaluative criterion, but appear when producing families over the entire stack of degenerations.4. M ODULI OF IDEAL SHEAVES
In this section, we define stable ideal sheaves on X relative to D and show that, for fixed numer-ical invariants, their families are parameterized by a proper Deligne-Mumford stack DT ( X | D ) .4.1. Stable ideal sheaves.
Let
Exp ( X | D ) be a stack of target expansions of X along D and let T ( Σ X ) the associated tropical moduli stack. We will denote it by T when there is no chance ofconfusion. Recall from the last section and Remark 3.5.1 that fibers of the universal expansion Y → Exp ( X | D ) have distinguished irreducible components, denoted tube components , associated tocertain -valent vertices of fibers of Υ . Each such component is a P -bundle P over a surface S .Given such a bundle P , a subscheme of P is a tube if it is the schematic preimage of a zero-dimensional subscheme in S . These are the analogues of “trivial bubbles” in relative and expandedGromov–Witten theory. We can now state our main definition: Definition 4.1.1. A stable ideal sheaf on X relative D over a closed point p is a morphism p → Exp ( X | D ) with associated expansion Y p and a strongly transverse subscheme Z p ⊂ Y p , which satisfies thefollowing stability condition: DT stability : the subscheme Z is a tube precisely along the tube components of Y P .A family of stable ideal sheaves on X relative to D over a scheme S is a morphism S → Exp ( X | D ) withassociated expansion Y S and a flat family of strong transverse subschemes Z S ֒ → Y S such that DTstability is satisfied in each geometric fiber.Recall that strongly transverse implies that Z meets every stratum of Y p . Also, notice that afamily of stable ideal sheaves over a scheme S induces a logarithmic structure on S by pullingback the natural structure on the Artin fan Exp ( X | D ) .4.2. Geometry of the space of relative ideal sheaves.
Let DT ( X | D ) denote the fibered categoryover schemes of families of stable ideal sheaves on X relative to D . We prove that the moduliproblem is algebraic with finite stabilizers. Theorem 4.2.1.
For each fixed stack
Exp ( X | D ) , the fibered category DT ( X | D ) is an algebraic stack. The stack comes with a map
Exp ( X | D ) , and we equip it with the pullback logarithmic structure. We suppress the choice from the notation for brevity. & DHRUV RANGANATHAN
Proof.
The proof is broken up into four steps; we prove algebraicity, first for the stack of expan-sions, then for the relative Hilbert scheme on the universal family of the stack of expansions, thenfor the substack of strongly transverse maps, and finally for the substack of transverse maps sat-isfying the DT stability condition.First, the stack
Exp ( X | D ) is an Artin fan, and therefore is algebraic. Second, since the map Y → Exp ( X | D )) is separated, it follows from [59, Tag 0D01] that the Hilbert functor Hilb ( Y / Exp ( X | D )) isrepresentable by an algebraic space over Exp ( X | D ) .We next consider strong transversality. Suppose we are given a family of expansions Y S → S and a family of subschemes Z S ⊂ S . Since transversality is clearly a constructible condition, itsuffices to show it is stable under generization, so we assume S is a DVR. Given any expansion,the closure of any stratum of the generic fiber contains strata of the closed fiber. Therefore, byflatness, every stratum of the generic fiber must intersect Z non-trivially. Similarly, by upper semi-continuity, the Tor-vanishing condition for divisor strata of the generic fiber follows by the samecondition on the closed fiber (by choosing a divisor stratum of the closed fiber in the closure).Finally, we deduce openness of the DT stability condition from the following lemmas. (cid:3) Lemma 4.2.2.
The DT stability condition is open and therefore DT ( X | D ) ⊂ Hilb ( Y / Exp ( X | D )) is an openalgebraic substack.Proof. As before, we need to check that the condition of DT stability is preserved by generization.Let S be the spectrum of a discrete valuation ring with generic point η and closed point andconsider a morphism S → Hilb † ( Y / Exp ( X | D )) . Assume that DT stability is satisfied at . We willprove that it is satisfied at the generic point.Examine the map B → Exp ( X | D ) . The generic and closed point each determine strata of thelatter space. Let F and F η be the corresponding cones in T ; note that F η is a face of F . Eachcomponent determines a -complex embedded in Σ X and by slight abuse of notation, we let G η and G be the combinatorial types of the associated graphs and note that there is a specializationmap from G to G η . To see this specialization, we may choose particular metric structures on G and G η such that there is a morphism from a polyhedral complex G → R > , with G ⊂ Σ X × R > whose fibers over and are G and G respectively. The specialization map is obtained by takingthe cone over a face in G and intersecting with the fiber.Let Z be the family of subschemes over B with generic and special fibers Z η and Z . By theDT stability condition we see that at the closed point , the subscheme Z is a tube along the tubevertices of G . We wish to show that the same is true at η .We claim that, if v is a tube vertex of G η , then all vertices w of G that specialize to v are alsotube vertices. Indeed, tube vertices arise by adding -valent vertices to subdivide the minimalgraph structure on G and then specializing these vertices to obtain a subdivision of the minimalstructure on G η . It follows that non-tube vertices always specialize to non-tube vertices, so the setof vertices specializing to tube must all be tube vertices.Let v be a tube vertex. The star around v in G can be identified with the fan of P . Let w , . . . , w k be the vertices of G η specializing to v . Since tube vertices are always -valent, thestar of v in the total space G can be identified with the cone over a polyhedral subdivision of R with k vertices. The toric dictionary provides us with the follow structure for the component Y v dual to v over B . We have morphisms Y v → S → B, OGARITHMIC DT THEORY 29 where S is a smooth surface over B and Y v → S is a proper family of semistable genus curves thatis generically a P bundle – that is, X η → S η is a P -bundle. The surface S is obtained as a locallyclosed stratum in a logarithmic modification of X , and in particular, is typically non-proper. Let Y be the union of the irreducible components corresponding to the vertices w i .After replacing our degeneration with this local model, we are left to check that if Z is a tubesubscheme on every component of Y , then Z η is also a tube subscheme. This is established by thelemma below, which then concludes the proof. (cid:3) Lemma 4.2.3.
As above let Y v → S → B, be a generically smooth semistable genus curve fibration over a family of smooth surfaces over B . Let Z and Z η be the special and generic subschemes obtained by restriction to the local model as above. If Z ⊂ Y is the preimage of a zero-dimensional subscheme W ⊂ S , then there exists a subscheme W ⊂ S such that Z is the preimage of W .Proof. By applying the criteria for properness and transversality from Theorems 1.5.2 and 1.5.3,we see that the family of subschemes obtained by the construction above to the component aboveis flat and proper. Properness and upper-semicontinuity of dimension guarantees that the imageof Z η in S η is zero-dimensional. Since the special fiber Z is the preimage of a zero dimensionalsubscheme along a nodal curve fibration, the special fiber has no embedded points. The Cohen-Macaulay condition is open, so the generic fiber also has no embedded points [59, Tag 0E0H].We examine the Hilbert polynomial of the subscheme. By flatness of the family of subschemes,the fiber degree of the subscheme is constant. We therefore conclude that the subscheme Z η isa Cohen–Macaulay thickening of a union of fibers in a P -bundle over S η . For each such fiber,the thickening gives rise to a morphism from P to the Hilbert scheme of points Hilb ( S η , d ) onthe surface S η . The minimal possible holomorphic Euler characteristic of this thickening occurs ifand only if the subscheme is a tube. Indeed, if the subscheme is not a tube, then the morphism P → Hilb ( S η , d ) is non-constant. The difference of the Euler characteristic and the degree of thismoduli map is constant, from which the claim follows. (cid:3) Remark 4.2.4.
Consider the functor of morphisms between two fixed logarithmic schemes. Un-der natural hypotheses, Wise has proved that these are always algebraic spaces equipped withlogarithmic structure [63, Theorem 1.1]. Each of the objects in DT ( X | D ) give rise to a logarithmicmonomorphism to X and in particular, a morphism to Wise’s mapping space. This is the closestavailable analogue on the sheaf theory side to the morphism from the moduli spaces of maps toexpansions to the logarithmic stable map spaces [56, Theorem A]. In particular, this morphism tothe mapping space will typically be a logarithmic modification near the image.4.3. Boundedness.
Fix a curve class β and holomorphic Euler characteristic χ and consider themoduli space DT β,χ ( X | D ) of ideal sheaves on X relative to D with these numerical data . Specifi-cally, we consider subschemes for which the pushforward of the fundamental cycle to X is β . Weestablish boundedness of the moduli problem. Theorem 4.3.1.
There exists a finite type scheme S and a map S → DT β,χ ( X | D ) which exhausts the geometric points of the target. & DHRUV RANGANATHAN
Proof.
We prove this in two lemmas. First, in Lemma 4.3.2, we bound the number of tropical typesthat can arise by examining points of DT β,χ ( X | D ) and is achieved in . Second, in Lemma 4.3.3, weexhaust the locus of transverse subschemes in a fixed degenerate target that give a fixed tropical-ization and is achieved (cid:3) Let G and G ′ be embedded -complexes in Σ X . We say that G and G ′ have the same type if theylie in the interior of the same cone of T ( Σ X ) . Lemma 4.3.2.
There are finitely many combinatorial types of embedded -complexes in Σ X that arise fromideal sheaves in X relative to D with fixed curve class β and Euler characteristic χ . That is, the moduli map DT β,χ ( X | D ) → Exp ( X | D ) factors through a finite type substack of the target.Proof. The blueprint for this result on the Gromov–Witten side is [25, Section 3.1]. The proofbounds the possible slopes of edges of -complexes embedded in Σ X . This is possible by means ofthe balancing condition, which we formulate below, before appealing to known finiteness resultsfor embedded tropical curves. I: Embedded tropical curves.
Let Z ⊂ Y be a strongly transverse subscheme of an expansion of X along D over a point. This point is equipped with a logarithmic structure pulled back from Exp ( X | D ) . It has a monoid P . After pulling back the family to a standard logarithmic point via amap P → N , we obtain a -complex G Z embedded in Σ X . Let v be a vertex of G Z and e an edgeincident to v . The edge e is dual to a codimension stratum of Y . The intersection of Z with thisstratum is a zero dimensional subscheme in this component. Define w e to be the length of thissubscheme. Since the subschemes on either side of a double divisor share the same intersection,the number w e depends only on e and not on v . Decorate the vertex v with the homology class ofthe subscheme Z v in the homology of X under the pushforward H ( Y ) → H ( X ) .As discussed previously, Σ X can be embedded as a subcomplex of R k > – the latter with itsstandard cone structure – for k equal to the number of irreducible components in D . We obtaina -complex with vertex and edge decorations in R k > . We refer to this object as an embeddedtropical curve. We abuse notation slightly and continue to denote it by G Z . II: Piecewise linear functions.
The group of positive integral linear functions on R k > is naturallyidentified with N k . Every linear function in N k restricts to a function on Σ X that is piecewise linearand linear. The toric dictionary gives rise to a line bundle on X with a canonical section. If α is apiecewise linear function, we let ( O X (− α ) , s α ) be the associated line bundle and section.Let v be a vertex of the G Z . Restrict α to the star G Z ( v ) of v in G Z . The result is a piecewise affine function at the star of v ; subtract the constant α ( v ) to obtain a piecewise linear function onthe (zero or) one-dimensional fan G Z ( v ) . III: The balancing condition.
Fix a linear function α ∈ N k . By the procedure above, this de-termines a line bundle ( O Y v (− α ) , s v,α ) on the component Y v of the expansion, together with acanonical section. Let Z v be the subscheme in the component Y v . The length of the vanishinglocus of the canonical section along the subscheme gives a monoid homomorphism ϕ v : N k → N α length ( Z v ∩ V ( s v,α )) . OGARITHMIC DT THEORY 31
The ambient space of the embedded tropical curve is canonically identified with the real positivedual of this copy of N k . In other words, we have G Z ֒ → R k > = Hom
Mon ( N k , R > ) . This homomorphism is an integral element in the ambient cone R k > ; we use this to decorate thingsa little more. The element ϕ v is an integral element in the cone R k > . Its negative is thereforean element in the ambient vector space R k . To each vertex v , attach a ray e v starting at v in thedirection of − ϕ v . Moreover, ϕ v is a positive multiple of a primitive integral element in R k . Declarethis multiple to be the weight of the new edge e v . Call the resulting object the balanced tropical curveassociated to Z . We view it as a new graph b G Z equipped with an immersion into R k that restrictsto an embedding on the -complex G Z into the standard quadrant.The term balanced is justified as follows. When equipped with these edge decorations, the sumof the outgoing vectors at each vertex, weighted by the edge decoration, is zero. Balancing holdsdue to geometric considerations. Each piecewise linear function α gives rise to a Cartier divisoron a component of Y v , which is represented by a linear combination of boundary divisors on thatcomponent. The intersection number of Z v with this Cartier divisor can be computed in two ways– schematically and tropically. The first is by taking the length of the schematic intersection of thesubscheme with the boundary divisor. The second is by summing the edge decorations for theedges incident to v , with weights given by the slopes of α . These two computations agree for alllinear functions α , and this is what we have encoded as the balancing balancing condition. IV: Bounding the balanced curves.
With the balancing condition at hand, we are essentially re-duced to the toric case. We refer to the unbounded rays of the balanced curve b G Z as the asymptoticdirections . For each of the k directions corresponding to divisors of X , the number of asymptoticrays with this direction in b G Z is at most the intersection number of β with the corresponding divi-sor. Additionally, the edge weights along rays in a fixed direction must add up to this intersectionnumber.We can control the rays introduced in the previous step. There is a balancing condition at eachvertex, and by summing over the vertices, we obtain a balancing condition that depends only onthe asymptotic rays, and therefore only on the geometry of X and the degree of the subschemesunder consideration.The number of asymptotic rays and their weights have been bounded. We apply a result of TonyYue Yu [65, Proposition 4.1]. Momentarily ignoring -valent vertices, from this result, we concludethat the number of vertices of a balanced, embedded, tropical curve, with at least trivalent verticesand these asymptotics, is bounded. The image of b G Z is such a curve, after removing -valentvertices. Since the number of vertices and the unbounded directions are bounded, the genus isalso bounded, and in turn, the genus of the immersed tropical curve b G Z as well. Viewing b G Z as a parameterized tropical curve, we apply [50, Proposition 2.1] to conclude finiteness of thecombinatorial types, after removing -valent vertices.The number of isolated points in the tropicalization is bounded because we have fixed the de-gree and Euler characteristic. Moreover, by the DT stability condition, non-tube -valent verticescannot contain a tube subscheme, so contribute to either the degree or increase the Euler charac-teristic. Therefore there are only finitely many such -valent vertices. The number of tube vertices For readers wishing to see a more “logarithmic” point of view on the proof, we point out that the remainder of theproof runs parallel to the stable maps geometry [25, Theorem 3.8], and also closely to [15, Section 5]. & DHRUV RANGANATHAN only depends on the combinatorial type determined in the previous paragraph. We concludethat the number of combinatorial types of embedded -complexes arising from ideal sheaves isbounded after fixing the numerical data. (cid:3) To conclude boundedness we claim that, after fixing the Hilbert polynomial for the pullbackof a polarization from X , the space of realizations of given contact data is parameterized by opensubschemes in finitely many Hilbert schemes.Since the polarization is pulled back from X , it is not ample on the expansion, and we may havesubschemes in these components that do not contribute to the Hilbert polynomial. The extremecase occurs where the stratum in question is a toric variety Y that is collapsed to a point on projec-tion to X . In this case, we must examine the family of subschemes of Y with fixed contact ordersalong the toric boundary. The result is already known in this case; Katz and Payne proved in greatgenerality that realization schemes of tropical fans have finite type [33, Theorem 1.2]. See also [32].We use a modest generalization. Lemma 4.3.3.
The stack of ideal sheaves on X relative to D with fixed combinatorial type and fixed numer-ical data is bounded.Proof. Fix an expansion Y and examine the strata. The top dimensional strata of Y have the follow-ing forms. They are either: a toroidal modification of X , a compactified G m -bundle over a locallyclosed boundary surface in X , a compactified G -bundle over a boundary curve in X , or a toricthreefold. We examine the first. The toroidal modification X ′ → X introduces finitely many curvesthat are contracted in X . The Hilbert polynomial with respect to the pullback polarization has beenfixed, and since there are no embedded points on the higher codimension strata of X ′ , there arefinitely many choices for the Hilbert polynomial of a subscheme appearing in X ′ and therefore thesubschemes in X ′ are exhausted by a finite list of finite type Hilbert schemes.We come to the top dimensional expanded strata. As stated above, for strata that are toricthreefolds we may simply apply the main result from [33, Theorem 1.2]. For the general case, weargue similarly. The locally closed top dimensional strata are all compactified torus bundles overlocally closed strata of X . By the strong transversality hypothesis, the boundedness is insensitiveto further logarithmic modifications or partial compactifications of Y . In particular, after replacing Y with a further logarithmic modification, any component V has a logarithmically ´etale map V → W → B where B is a closed stratum in X and W a P r –bundle B with fiber dimension r = − dim V .Moreover, morphism can be taken to be a composition of an open immersion and sequence ofblowups.Let Z ⊂ V be a subscheme. We claim that there are finitely many choices for the homology classof Z in V . We argue as follows. Since the Hilbert polynomial in X is fixed, there are finitely manychoices for the Chow homology class of the pushforward of [ Z ] to B . On the other hand, since wehave fixed the contact orders of Z , there are finitely many choices for the fiber degree of the class [ Z ] after pushforward to the toric bundle W . Moreover, the cohomology of V can be obtained as amodule over that of W by the blowup formula, and is generated by classes of exceptional divisorsand exceptional curves in the blowup. However, as the contact orders are fixed, the class of [ Z ] is a strict transform of its pushforward to P r × B . It is therefore is determined by its intersectionmultiplicity with the strata, which are in turn fully determined by the contact order. OGARITHMIC DT THEORY 33
Choose a projective embedding of V into some projective space and observe that its degree isbounded. We may now directly the argument of Katz and Payne, specifically [33, Para. 3 &
4, Pf.of Thm. 3.2]. That is, since the degree is bounded, the Castelnuovo–Mumford regularity of Z isalso bounded, so the Hilbert polynomial of Z is determined by finitely many values of the Hilbertfunction depending only on the degree. Finally, noting that the holomorphic Euler characteristicis fixed and that the Hilbert function of a subvariety of projective space is bounded in terms of thedegree, the dimension Z , and the ambient dimension it follows that there are only finitely manypossible values for the Hilbert polynomial of Z . Since contact orders are locally constant, thereis an open subscheme of a finite disjoint union of finite type Hilbert schemes which exhaust thestack of ideal sheaves. The result follows. (cid:3) Finiteness of automorphisms.
We continue with the notation from the previous sections. Let DT β,χ ( X | D ) be the moduli space of ideal sheaves on X relative to D , with a fixed choice of targetexpansion moduli. There is a structure map DT β,χ ( X | D ) → Exp ( X | D ) . We use this to deduce finiteness of automorphisms the space of relative ideal sheaves.
Theorem 4.4.1.
The isotropy groups of the moduli space DT β,χ ( X | D ) are finite.Proof. We show that the objects of DT β,χ ( X | D ) have finite automorphism group. Recall that apoint of this stack is given by a morphism Spec C → Exp ( X | D ) together with an ideal sheaf I Z on the expansion Y of ( X | D ) ; here Z denotes the associated sub-scheme. Moreover, we have imposed the condition that the ideal sheaf gives rise to a subschemethat is a tube precisely along the tube components. In order to analyze the stabilizer group of theideal sheaf I Z , we first note that the expansion Y maps to a substack B G rm of Exp ( X | D ) . Specifically,the map above determines a locally closed stratum of Exp ( X | D ) corresponding to a cone σ of theassociated cone space. The isotropy group at this point is naturally identified with the torus whoselattice of one-parameter subgroups is σ gp ( N ) . We claim that I Z is not stabilized by any positivedimensional subgroup of this isotropy group.To see this, we unwind how the isotropy group acts on Y . The components of the expansion areeither modifications of X , or an equivariantly compactified G km bundle over S ◦ where S ◦ is a locallyclosed stratum in X of codimension k . Each of these bundles therefore possesses a translationaction by the torus. Upon restriction to each component, the isotropy group acts as by scaling thetorus directions described above (however, not all such translation actions come from the actionof isotropy).Assume that there is a one-parameter subgroup of G rm that stabilizes I Z . We may pullbackthe expansion Y along the inclusion B G m → B G rm , and note that by our hypothesis on the tubescomponents of the expansion, we have a contraction Y Y B G m such that the subscheme Z is the inverse image, along the contraction, of a subscheme Z ⊂ Y .By the definition of strong transversality, every component of Y has nonempty intersection with & DHRUV RANGANATHAN the subscheme. The only possibility for a G m to stabilize a subscheme is at the tube components.However, by hypothesis, the subscheme Z has no tube components. Since the G m acts on thecontraction as well as Z itself, it follows that in fact no subtorus can stabilize the given point. Itfollows that the stabilizer must be finite, as required. (cid:3) Properness.
We establish the following result, delaying one technical part of the analysis toSection 7.
Theorem 4.5.1.
The moduli space DT β,χ ( X | D ) is proper.Proof. The stack
Exp ( X | D ) is locally of finite presentation, see [14, Corollary 6.7] so the space DT β,χ ( X | D ) is also quasiseparated; it is also of finite type. Therefore we need only check theweak valuative criterion for properness. Let K be a valued field and consider a morphism Spec K → DT β,χ ( X | D ) . Since DT β,χ ( X | D ) is of finite type, we may assume that K is the function field of a smooth curveand the valuation is given by order of vanishing at a closed point. After pulling back the universalexpansion and universal subscheme, we must check that, after replacing K with an extension, theresulting family extends to yield a map Spec R → DT β,χ ( X | D ) . We prove the result here in the case where the fiber over K is unexpanded. In the unexpanded case,this follows directly from Proposition 2.5.2; indeed, our construction has been rigged to arrangethis, and we spell out the details as follows. The general case is handled in Section 7.Suppose we have a cone space T with Artin fan A T . The data of a morphism Spec R → A T whichsends the generic point to the open stratum is equivalent to giving an integral point in T . Indeed,such a statement holds for an Artin cone by a direct analysis, and the general statement follows.If we pass to a base change of R , the corresponding integral point of the composed morphism isobtained by scaling the original point.To prove the existence of a stable limit, apply Proposition 2.5.2; possibly after a base change,there exists a -complex G ⊂ Σ X such that the corresponding expansion Y G → Spec R has astrongly transverse limit. Furthermore, by Remark 2.5.6, there are no tube subschemes containedin the components associated to bivalent vertices of G .The -complex G defines a rational point of T . After a possible scaling to ensure integrality,along with the corresponding base change of R , we have a point P ∈ T and an associated mor-phism Spec R → Exp ( X | D ) . By construction, the corresponding expansion Y R → Spec R factorsthrough Y G , and the contracted components are precisely the tube vertices of Υ P . As a result theflat limit of Z K in Y R will be strongly transverse and stable.For uniqueness of the limit, given an expansion Y → Spec R associated to an integral point P ′ of T , the universal property of G implies that there is a subdivision Υ P ′ → G and correspondingmodification Y → Y G . There are no tube subschemes for Y G , so the new vertices of the subdivisionare precisely the tube vertices of this family, so P ′ and P coincide and the stable limit is unique. (cid:3) Remark 4.5.2 (Stable pair moduli) . In this remark, we sketch how to modify the definitions andconstructions to handle stable pairs instead of ideal sheaves. A stable pair consists of a pair ( F, s ) consisting of a sheaf F of fixed Hilbert polynomial with purely -dimensional support and a section O X s −→ F whose cokernel is -dimensional [53]. A stable pair ( F , s ) on an expansion Y of X isstrongly transverse if the scheme theoretic support of F is strongly transverse and the support of OGARITHMIC DT THEORY 35 the cokernel is disjoint from the divisorial strata of of Y . The existence and uniqueness of stronglytransverse limits is analogous. Specifically, given a family of strong transverse stable pairs overa valuation ring, we find a dimensionally transverse limit for the schematic support using thetropicalization, and then apply the stable pairs arguments used by Li and Wu [38] to producea strongly transverse limit. A tube stable pair is a pair that is pulled back from a -dimensionalsubscheme on a surface along a P -bundle, with the condition that the cokernel of the sectionvanishes. The moduli space PT ( X | D ) can be constructed as a relative stable pair space on theuniversal expansion over Exp ( X | D ) . Properness follows from the tropical limit algorithms, andthe virtual class follows from [46, Proposition 10] as in the ideal sheaf case.5. L OGARITHMIC D ONALDSON –T HOMAS THEORY
We now specialize to the case where X is a threefold and construct logarithmic Donaldson–Thomas invariants for ( X | D ) by showing that the moduli spaces constructed in the previous sec-tion come equipped with virtual classes and evaluation morphisms to the relative divisors. Wealso show that the virtual class, and hence the numerical theory, does not depend on the choicesthat have been in constructing the moduli stack of target expansions. We define DT generatingfunctions and state the basic rationality conjecture, analogous to the non-relative setting.5.1. Virtual classes.
Each moduli space of ideal sheaves comes equipped with a structure map t : DT β,χ ( X | D ) → Exp ( X | D ) . Let I be an ideal sheaf. The traceless Ext groups Ext ( I Z , I Z ) Ext ( I Z , I Z ) govern the deformation and obstruction spaces of I Z . Proposition 5.1.1.
The structure map t : DT β,χ ( X | D ) → Exp ( X | D ) . is equipped with a perfect obstruction theory, with deformation and obstruction spaces as above. If β is thecurve class, then the there is an associated virtual fundamental class [ DT β,χ ( X | D )] vir in Chow homologyof degree vdim DT β,χ ( X | D ) = Z β c ( T X ) . Proof.
The existence of the relative obstruction theory as described is established in the requisitegenerality in [46, Proposition 10]. Since the base
Exp ( X | D ) of the obstruction theory is a connectedArtin fan, it is irreducible and equidimensional. Virtual pullback of the fundamental class givesrise to the virtual fundamental class in the usual fashion [40]. (cid:3) Virtual birational models.
The spaces constructed in the paper to this point have all requireda choice of tropical target moduli. We show that different choices lead to compatible theories.We supplement the notation in this section to keep track of the different theories arising fromthe tropical choices. Recall that the construction of T in Section 3 produces an infinite collectionof spaces, depending on the choices made in the algorithmic steps of Section 3.3. Moreover, thereare additional choices that are made in the construction of the universal family. & DHRUV RANGANATHAN
We denote by Λ the set of outputs of that construction. More precisely, each element in Λ is amorphism of polyhedral complexes π λ : Υ λ → T λ where T λ is a moduli space of tropical expansions and π λ is the projection from the universalfamily. Let Exp ( X | D ) λ be the associated moduli space of expansions and let Y λ → Exp ( X | D ) λ be the projection from the universal family. We fix the discrete data and establish the following bi-rational invariance statement, parallel to [56, Section 3.6], which is itself modeled on [9, Section 6]. Theorem 5.2.1 (Independence of choices) . For any λ, µ ∈ Λ there exists ϕ ∈ Λ and logarithmicmodifications DT ϕβ,χ ( X | D ) DT λβ,χ ( X | D ) DT µβ,χ ( X | D ) π λ π µ such that π λ ⋆ [ DT ϕβ,χ ( X | D )] vir = [ DT λβ,χ ( X | D )] vir and π µ ⋆ [ DT ϕβ,χ ( X | D )] vir = [ DT µβ,χ ( X | D )] vir . Proof.
We drop the discrete data to avoid overcrowding the notation. By applying Proposition 3.6.1in conjunction with the categorical equivalence between Artin fans and cone stacks, we can find acommon refinement
Exp ϕ ( X | D ) Exp λ ( X | D ) Exp µ ( X | D ) where both vertical arrows are birational. The definitions furnish a forgetful morphism DT ϕ ( X | D ) → DT λ ( X | D ) . We argue that pushforward identifies virtual classes. We have a universal diagram Z ϕ Y ϕ X DT ϕ ( X | D ) . The analogous diagram exists for λ . We claim that the induced morphism p : Y ϕ → Y λ × Exp λ ( X | D ) Exp ϕ ( X | D ) is obtained by adding tube components to the target geometry. Indeed, this is a tautology – bydefinition, tropicalizations of the expansions of this family of targets induce the same modulimap to | T | . Since the DT stability condition guarantees the tube components only contain tube OGARITHMIC DT THEORY 37 subschemes, the universal subscheme Z ϕ is simply obtained by pulling back the family Z λ alongthe modification. It follows that we have a commutative square DT λ ( X | D ) DT µ ( X | D ) Exp λ ( X | D ) Exp µ ( X | D ) As we have argued above, each of the two vertical arrows have perfect obstruction theories. Theintroduction of tube components has no effect on the obstruction theory. Indeed, the universalideal sheaf I ϕ is pullback of the universal ideal sheaf I λ along the induced map of targets, so theright vertical obstruction pulls back to the same obstruction theory as the one on the left. Sincethe lower horizontal is a subdivision, it is proper and birational; equality of virtual classes followsfrom [17, Theorem 5.0.1]. (cid:3) Remark 5.2.2.
There is a further birational invariance statement that one might expect for the target . Specifically, if X ′ → X is the blowup of X along a stratum of D with total transform D ′ , onemay hope that there is an associated map DT ( X ′ | D ′ ) → DT ( X | D ) that identifies virtual classes, indirect analogy with [9]. While this appears plausible, we delay further investigation to a sequel.5.3. Evaluations at the relative divisors.
Let ( X | D ) be as above. Let E , . . . E k be the irreduciblecomponents of D . These components are smooth by hypothesis. Each intersection of distinctdivisors E i ∩ E j is either empty or a connected divisor, again by hypothesis. Thus, each variety E i is equipped with a simple normal crossings divisor of its own. We abuse notation slightly anddenote these by ( E i | D ) .By applying the constructions of the preceding sections to E i , we obtain a stack of expansions Exp ( E i | D ) . For our purposes, it will suffice to examine with the substack parameterizing thoseexpansions whose associated -complex is a finite set of points. We pass to the subscheme of theuniversal expansion where the irreducible components of the expansion are disjoint.For each positive integer d i , we consider the Hilbert scheme of points in the universal expansion E → Exp λ ( E i | D ) of zero dimensional subschemes of length d i . As before, the symbol λ indicates that we have cho-sen some stack of target expansions. By studying the locus comprising subschemes of expansionssuch that no component of the expansion is nonempty, we obtain a proper and finite type stack Hilb d i λ ( E i | D ) as a special case of the construction above.As a special case of the virtual birational invariance of the Donaldson–Thomas moduli spaces,we obtain the parallel statement for the Hilbert schemes. However, since the Hilbert schemesof points on smooth curves and surfaces are unobstructed moduli schemes, the same argumentproduces an honest birationality statement.Given a curve class β , define d i to be the intersection number D i · β . We therefore have anevaluation space for relative insertions Ev β ( X | D ) = Hilb d ( E | D ) × · · · × Hilb d k ( E k | D ) . Lemma 5.3.1.
There exists compatible stacks
Exp ( X | D ) and Exp ( E i | D ) and an evaluation morphism ev : DT β,χ ( X | D ) → Ev β ( X | D ) of the associated moduli spaces of ideal sheaves and points. & DHRUV RANGANATHAN
Proof.
We first explain how the evaluation looks at the tropical level. The reader may first wishto reconsult with Remark 1.6.2. Fix a space of expansions for
Exp ( X | D ) with associated tropicalmoduli space T ( X | D ) . The latter is a particular conical structure on the space of embedded -complexes. Examine such a -complex G ֒ → Σ . Each divisor component E i determines a ray ρ i .The transversality hypothesis implies that outside a bounded set in Σ , the -complex is made upof a finite set of parallel to one of the rays ρ i . The star of ρ i in Σ is a cone complex of dimensionone lower than that of Σ . Following the toric dictionary, rays parallel to ρ i determine points in thestar Σ ( ρ i ) . This star is canonically identified with the tropicalization of ( E i | D ) . For each cone σ inthe cone space of T ( X | D ) , we therefore obtain a family over σ of points in Σ ( ρ i ) .We now apply the construction in Section 3.3 to the cone complex Σ ( ρ i ) to produce a tropicalmoduli space T ( E i | D ) . By the description above, given a cone of T ( X | D ) , we produce a map to T ( E i | D ) . The image of σ need not be contained in a single cone of T ( E i | D ) , but after replacing both T ( E i | D ) and σ with subdivisions T ′ ( E i | D ) and σ ′ , we obtain a morphism σ ′ → T ′ ( E i | D ) of cone spaces. For any choice of conical model, the morphism from DT β,χ ( X | D ) → Exp ( X | D ) factors through a finite type substack of the image, given by a finite collection of cones in T ( X | D ) .It follows that after a finite collection of subdivision operations, we obtain a morphism DT β,χ ( X | D ) → Exp ′ ( E i | D ) again for an appropriate system of choices. In order to lift this to an evaluation morphism, wepullback the universal expansion E over Exp ′ ( E i | D ) to DT β,χ ( X | D ) . There is a inclusion of expan-sions E × Exp ′ ( E i | D ) Exp ′ ( X | D ) ֒ → Y . Intersecting the universal subscheme Z with the expansion, the strong transversality conditionimplies that we obtain a subscheme of length d i , and thereby a point of Hilb d i ( E i | D ) . Performingthe construction for each divisor, we have constructed the claimed morphism. (cid:3) Remark 5.3.2.
When combined with the projection formula, the birationality statements for themoduli spaces of sheaves and of points arising from Theorem 5.2.1 guarantee that numerical in-variants are independent of the choices made in the construction of the spaces of expansions.
Remark 5.3.3.
This evaluation morphism imposes schematic tangency conditions via the Hilbertscheme of points on the boundary surfaces. A more refined evaluation is to the relative Hilbertscheme of points on the universal surfaces over
Exp ( X | D ) . This refined object is more natural fromthe point of view of the degeneration formula, and for the GW/DT correspondence. In particular,this is the way in which the gluing formula is proved on the Gromov–Witten side [56]. We willexamine this refined evaluation in future work, in its appropriate context.5.4. Logarithmic DT invariants.
We define descendent insertions for logarithmic DT invariantsas follows. Let ( π DT , π X ) : Y → DT β,χ ( X | D ) × X denote the universal expansion over the DT moduli space. While the map π D T is not proper,it is proper when restricted to the universal subscheme Z ֒ → Y . Because of flatness and strongtransversality, the ideal sheaf I Z is a perfect complex, and admits Chern classes ch k ( I ) ∈ H ∗ ( Y ) ,which are supported on Z for k > .Given a cohomology class γ ∈ H ∗ ( X ) , we define descendent operators for k > k ( γ ) : H ∗ ( DT β,χ ( X | D ) , Q ) → H ∗ ( DT β,χ ( X | D ) , Q ) OGARITHMIC DT THEORY 39 via the formula τ k ( γ ) := (− ) k + π DT , ∗ ( ch k + ( I ) · π ∗ X ( γ ) ∩ π ∗ DT . Here, the pullback on homology is well-defined since π DT is flat by [20], and the pushforward iswell-defined since π DT is proper on Z .Given a cohomology class µ ∈ H ∗ ( Ev β ( X | D )) , and cohomology classes γ , . . . , γ r ∈ H ∗ ( X ) andindices k , . . . , k r , we can define descendent invariants h τ k ( γ ) . . . τ k r ( γ r ) | µ i β,χ = deg r Y i = τ k i ( γ i ) ev ∗ µ ∩ [ DT ] vir ! where deg denotes the degree of the pushforward to a point. If we sum over χ we have thepartition function Z DT X, D ; q | r Y i = τ k i ( γ i ) | µ ! β = X χ h r Y i = τ k i ( γ i ) | µ i β,χ q χ which is a Laurent series since the logarithmic DT spaces are empty for χ ≪ .In analogy with absolute and relative DT invariants, we have the following basic conjectureregarding this series. Conjecture 5.4.1. (i) The DT series for zero-dimensional subschemes is given by Z DT ( X, D ; q )) β = = M (− q ) R X c ( T log X ⊗ K log X ) , where M ( q ) = Y n > ( − q n ) n is the McMahon function.(ii) For any curve class β , and insertions γ , . . . , γ r of degree > and relative insertion µ , the normal-ized DT series Z ′ DT X, D ; q | r Y i = τ k i ( γ i ) | µ ! β := Z DT ( X, D ; q | Q ri = τ k i ( γ i ) | µ ) β Z DT ( X, D ; q )) β = is the Laurent expansion of a rational function in q . We expect the first part of this conjecture can be proven once the degeneration formalism is inplace.If we consider the analogous PT generating function for stable pairs invariants, we have ratio-nality and wall-crossing conjectures.
Conjecture 5.4.2.
For any curve class β , relative insertion µ , and insertions γ , . . . , γ r , the PT series Z PT X, D ; q | r Y i = τ k i ( γ i ) | µ ! β is a rational function in q . If the degree of each γ i is at least , this series equals the normalized DT series Z ′ DT from the previous conjecture. These rationality statements are prerequisites for formulating the GW/DT and GW/PT corre-spondences. This correspondence will be addressed more systematically in future work. & DHRUV RANGANATHAN
6. F
IRST EXAMPLES
We sketch a handful of basic examples, focusing on the space of targets, where the main newcomplexity lies. The goal of this section is to show that in a reasonable range of test cases, thelogarithmic Donaldson–Thomas moduli spaces can be worked with fairly explicitly.6.1.
Target yoga.
There is a well-known procedure by which the strata of the expanded targetgeometry may be recognized from the tropical pictures, as a consequence of the toric dictionaryconcerning closed torus orbits and stars around cones [22, Section 3.1]. Let G ֒ → Σ be an embedded -complex. Let Y G be the associated expansion of X . Let V be a vertex of G that maps to a cone σ of Σ of dimension k . There is an associated irreducible component Y V ֒ → Y G → X. This map factors through the inclusion of the closed stratum X σ ֒ → X . The induced morphism Y V → X σ is a partially compactified torus bundle of rank k . For instance, if X is a threefold and σ has dimen-sion , then Y V is simply a toric threefold. The torus bundles are obtained from the line bundlesassociated to the divisorial logarithmic structure on X and its blowups. The partial compactifica-tion of Y V is obtained (fiberwise) as the toric variety associated to the star of V in G , which givesrise to a fan embedded in the star of σ .6.2. Dual plane.
Let ( X | D ) be the pair consisting of P and its toric boundary divisor. The mostbasic example is the moduli space of lines in P , with transverse contact orders. There is a canoni-cal space of ideal sheaves DT ( P | D ) , and it is identified with the blowup of the dual P at its threetorus fixed points. This example is worked out carefully in [55, Section 4.1] for stable maps, andthe situation is identical to the sheaf theory setup. The figures in op. cit. may be of particular use.The fibers of the expanded target family Y → DT ( P | D ) can be described as follows. The targetis either isomorphic to P r { p , p , p } or the complement of the codimension strata in thedeformation to the normal cone of a line in P . The latter arises as a transverse replacement in twoways, either when the subscheme limits to a coordinate line, or when the intersection point of thesubscheme with the coordinate line limits to a coordinate point.6.3. Plane curves.
The case of degree d curves in P allows us to make contact with the studyof discriminants, and in particular, the secondary fan of a toric fan, constructed by Gelfand–Kapranov–Zelivinsky [23]. The discussion here collapses to the dual plane for d = .Let P d be the d -fold dilation of the standard lattice simplex in R and let P d be the projectivizedlinear system, which coincides with the Hilbert scheme of degree d curves. An open subset of thisprojective space parameterizes curves that are strongly transverse to the coordinate lines in P .This subset is stable under the G action on P . Limits under the action of this torus capture therelevant degenerations to construct the Donaldson–Thomas moduli space, as we now explain.Any expansion P along its toric strata arising from our construction is necessarily toric. Let G ⊂ Σ P be a -complex expansion. Given a strongly transverse subscheme Z ⊂ Y G , of the associated expansion, we may proceed as in Section 4.3 and label each edge E of G with thelength of the intersection of Z with the divisor attached to E . Since the target is toric, the graph G OGARITHMIC DT THEORY 41 together with this weighting furnishes a balanced tropical curve. We abuse notation slightly andcontinue to denote this decorated object by the symbol G .The secondary fan T d of P d enters as a moduli space of tropical plane curves. Recall that thesecondary fan of P d parameterizes subdivisions of P d that are regular . i.e. they are induced bya strictly convex piecewise linear function [23]. The dual of such a subdivision gives rise to abalanced tropical curve in R , and moreover, a subdivision is regular precisely when it gives riseto a tropical curve [39, Remark 2.3.10]. The top dimensional cones correspond to unimodulartriangulations of P d , and the dual tropical curves have only trivalent vertices.Geometrically, the secondary fan is the toric fan associated to the Chow quotient P d // G , ofthe Hilbert scheme by the dilating action of the torus in P . The Chow quotient carries a universalflat family of broken toric surfaces, arising as limits of G -orbits in P d , see [10, Section 3].At the combinatorial level, these data are captured by the maps of polyhedral complexes Σ d R T d . The toric variety associated to Σ d is the universal family of the Chow quotient. It is a birationalmodification of the Hilbert scheme P d .In order to make a connection to the Donaldson–Thomas moduli space, a minor modificationis required. The expansions we propose are always equipped with a map to P , while the degen-erations arising from regular triangulations need not be modifications of the constant family. Theadaptation is as follows. The secondary fan comes equipped with a universal family of polyhedralsubdivisions of R : Σ d R T d . Let e Σ d be the fiber product Σ d × R Σ P , which is the common refinement of Σ d with the constantfamily T d × Σ P . The toric map induced by e Σ d → T d need not be flat with reduced fibers, but byuniversal weak semistable reduction, there is a canonical such family over a new fan T ′ d , see [47].Finally, let T ′′ d denote the quotient of T ′ d by the equivalence relation that identifies two embed-ded tropical curves if they have the same underlying -complex (i.e. we forget the weights). Itis straightforward to check that this is a cone space, and is a union of cones in a moduli spaceof tropical expansions T ( Σ ) constructed in Section 3.3. In fact, one can check that the interiors ofmaximal cells in T ′ d map isomorphically to interiors of cones in T ′′ d .Let Exp ( P | D ) and A d be the Artin fans associated to T ′′ d and T ′ d respectively. The constructionsin the previous section give rise to a Donaldson–Thomas moduli space; given a subscheme inan expansion, we can recover the weights on the edges using the schematic intersection numberswith the corresponding divisors. As a result, the structure morphism to Exp ( P | D ) factors through DT ( P | D ) → A d . The subdivision e Σ gives rise to a (non-representable) birational modification P ′ d of the universalorbit of the Chow quotient P d // G . The identification can either be made at the level of coarse & DHRUV RANGANATHAN moduli space, or the Chow quotient may be upgraded to a Deligne–Mumford stack using workof Ascher and Molcho [10]. This toric stack is naturally identified with DT ( P | D ) . Indeed, theyrepresent the same functor as families of curves in the universal expansion of P over Exp ( P | D ) .Analogous statements hold for any toric surface. We conclude with a series of remarks. Remark 6.3.1.
The stable pair moduli spaces in degree d on P but with arbitrary holomorphicEuler characteristic can be obtained as the relative Hilbert schemes of points on the universalsubscheme over the Donaldson–Thomas moduli space constructed above [54, Propostion B.10].At the combinatorial level, this amounts to allowing a fixed additional number of vertices on thebalanced -complexes considered above. The additional vertices allow for expansions along thedivisors to accommodate support for the cokernel of the section of the stable pair. Remark 6.3.2.
The subdivision step above can be avoided by working directly with the logarith-mic multiplicative group G log . The tropicalization is the R appearing above, with “fan” structuregiven by the single non-strictly convex cone R . We avoid the detour through the details of thelogarithmic multiplicative group, but a reader interested in the details of this perspective mayextract them using the arguments in [57]. Remark 6.3.3.
The moduli space above is naturally identified with the universal family over theChow quotient. The Chow quotient itself can also be interpreted as a Donaldson–Thomas modulispace for the rubber moduli space P , see [41]. The rubber moduli will be discussed elsewhere. Remark 6.3.4.
The relationship between the secondary fan and enumerative geometry shouldbe credited to Eric Katz [31, Section 9]. Katz suggested that the universal family of the Hilbertquotient of a toric variety should function as a moduli space of polarized target expansions forGromov–Witten theory relative to the toric boundary. In retrospect, it seems more natural to real-ize this connection within the context of ideal sheaves or stable pairs. A detailed study of the roleof the secondary fan in logarithmic curve counting theories seems worthwhile.6.4.
Subgroups in toric threefolds.
Let X be a smooth projective toric threefold with D the toricboundary divisor. Let Σ X be the fan in the cocharacter space N R . We make the further assumptionthat there exists a nonzero primitive vector v such that the ray generated by v and by − v are bothrays in Σ X . This guarantees that the associated compactified one-parameter subgroup ϕ v : P → X is transverse to D .The Donaldson–Thomas moduli space in the class [ im ( ϕ )] is again related to the geometry ofChow quotients. Consider the line ℓ v spanned by v . The relevant embedded -complexes in thisinstance are precisely the parallel translates of ℓ in N R . Given such a translate ℓ of ℓ v , we equip itwith a canonical polyhedral structure by taking the common refinement with Σ X inside N R .The moduli space of such embedded -complexes is canonically identified with the fan of theChow quotient X// G m by results of Chen–Satriano [16]. The identification is somewhat indirect,so we spell it out. In loc. cit. the authors identify the Chow quotient with the moduli space oflogarithmic stable maps to X in the class of ϕ v . The tropical moduli space of maps is preciselythe set of parallel translates ℓ v with the polyhedral structure described above, for instance fromthe arguments in [55]. However, it is straightforward to see that all tropical maps in this caseare forced to be embedded complexes, so we may identify the tropical moduli space with the setof embedded -complexes. The Donaldson–Thomas moduli space for this data is then equal to OGARITHMIC DT THEORY 43 the Chow quotient. The identification of coarse moduli spaces occurs can be upgraded using theChow quotient stack.
Remark 6.4.1.
Stable map, ideal sheaf, and stable pair spaces coincide in this case.6.5.
A fat point and a subgroup.
We maintain the notation from the previous example, and con-sider subschemes in the curve class ϕ v , but we increase the Euler characteristic by . Geometri-cally, we consider the class of a one-parameter subgroup together with a fat point. As the addi-tional point may wander the target, the space of relative ideal sheaves includes expansions causedby the presence of the fat point on the boundary D . The relevant -complexes are therefore trans-lates of the line ℓ together with an additional point, which can lie anywhere in N R , and whosepresence introduces a vertex.The space of expansions can be constructed as follows. Let S v be the toric fan associated to theChow quotient X// G m as above and let U be the universal orbit of the Chow quotient. The latteris a refinement of Σ X . The product fan Σ U × S v can be viewed as a moduli space of pairs ( p, ℓ ) where, as above, ℓ is a parallel translate of ℓ v and p is a point in Σ U . Consider the incidence locus IL := { ( p, [ ℓ ]) : p ∈ ℓ } ⊂ Σ U × S v . This incidence locus has a fan structure by viewing it as the preimage of the diagonal in the pro-jection Σ U × S v → S v × S v where the morphisms are the projection of the universal family and the identity map on the twofactors. Note that since the vertical maps are both combinatorially flat with reduced fibers, thefiber product is canonically a cone complex.Consider any subdivision T of Σ U × S v such that the locus IL is a union of faces. Over T , weobtain two families of subcomplexes – the first is the family of parallel translates of ℓ v while thesecond is simply the universal family U over T . Each of these map to T × Σ U , and we choose aconical structure on the union of their images. The image gives the required conical structure.The Artin fan associated to T gives rise to a moduli space Exp ( X | D ) of expansions of X along D .Note that the fibers of the morphism DT ( X | D ) → Exp ( X | D ) also have a straightforward description. Fixing the target expansion also fixes a -complex in Σ X .For each vertex of Σ X that lies on the line ℓ , we choose a torsor for the one-parameter subgroup v , inthe corresponding component. The closure provides a subscheme Z with the required homologyclass. Therefore, additional moduli is provided by the choice of point in the expansion. In orderto account for the moduli of tangent directions when the point lies on on Z , consider the modulispace of points in the expansion and blowup the incidence locus with Z .6.6. Lines in P . Let ( X | D ) be the toric pair ( P | H ) where H is the union of the coordinate planes.We take the fan of P as formed by the standard basis vectors and the negative of their sum in R . The class of a generic line in P can be described using similar consideration as the one above.First we consider the moduli spaces of lines in P . The generic line in P that meets only thecodimension strata of the toric boundary has tropicalization a single vertex with outgoinginfinite rays, in each of the three coordinate directions and (− − − ) . The relevant embedded -complexes required to construct the moduli space of expansions are balanced tropical curveswhose asymptotic rays are the ones described above. & DHRUV RANGANATHAN
Once again the space of -complexes can naturally be identified with the elements of the mod-uli space of tropical maps, because all tropical maps for this moduli problem are forced to beembeddings. The moduli space coincides with an explicit blowup of M × P , following [55].If we consider a line together with a fat point, thereby raising the holomorphic Euler charac-teristic, the tropical space of expansions is obtained from the one above by adding an additionalpoint, and subdividing the incidence locus where this point lies on the -complex. As before, theDonaldson–Thomas moduli space has a map to the space of targets, whose fibers can be explic-itly described as the blowup of an algebraic incidence locus, stratum-wise on the space of targetexpansions. We leave the details to the reader.7. T RANSVERSE LIMITS AND GENERICALLY EXPANDED TARGETS
We verify the details of the valuative criterion for Theorem 4.5.1 that were postponed i.e. forfamilies with a generic target expansion. An analysis via normalization requires a thorough treat-ment of the rubber geometry. Instead, we handle it with some additional formalism concerninglogarithmic structures and tropicalizations in these circumstances; the overall structure is similarto the special case dealt with previously, so parallel reading may be advisable.7.1.
Preliminaries on valuation rings.
Let K be a complete discretely valued field with alge-braically closed residue field and valuation ring R . Denote the spectra of K and R by S ◦ and S respectively, and the inclusion by j : S ◦ → S . Equip the scheme S ◦ with the standard logarithmicstructure with monoid N . The tropicalization of S ◦ is the real dual and is a ray denoted by ρ .Consider the rank lattice obtained as the direct sum of M gp S ◦ and the group of exponents of auniformizer for R . Let N S be the dual vector space of this group. It comes equipped with a latticeand a canonical quotient map N S → R collapsing ρ and carrying the lattice to the value group.We examine fine and saturated extensions of the logarithmic structure from S ◦ to S . A sufficientclass is obtained as follows. Begin with the direct image logarithmic structure j ⋆ M S ◦ . The char-acteristic monoid of this structure at the closed point is isomorphic to the set of lexicographicallynonnegative elements in Z . The dual of the characteristic group at the closed point is the vectorspace N S defined above and the ray ρ is equipped with an embedding in N S .Collect the set of -dimensional cones σ ⊂ N S containing ρ as a face and whose image in thevalue group is the nonnegative elements. The dual monoids P σ determine fine and saturated sub-monoids of the characteristic monoid of direct image. We obtain, for each σ , a fine and saturatedlogarithmic structure extending the one on S ◦ . Denote it by S σ .For a finite extension K ′ of K equipped with the natural valuation, the corresponding morphism Spec R ′ → Spec R induces a map from the value group of K ′ to the value group of K . The map onvalue groups is an inclusion of a finite index sublattice, and in particular, if we denote by thespectrum by S ′ the associated vector spaces N S and N S ′ above are naturally isomorphic. Since thelogarithmic structure at the generic point is unaffected by the ramified base change, the inducedintegral structure on ρ under this identification is also unaffected by the base change. Terminology.
The logarithmic structures extending the given one on S ◦ above will be called loga-rithmic extensions . The set of such cones σ form a filtered system under inclusion. In the discussionthat follows will say that a statement holds for for sufficiently small σ to indicate that it is true after As this structure is typically not quasi-coherent. It is possible to work with it nonetheless, but as we prefer to usestandard toric machinery, we will avoid the direct use of this logarithmic structure.
OGARITHMIC DT THEORY 45 replacing σ with any element in a lower interval in the filtered system. We will correspondinglyrefer to the extensions as sufficiently small (fine and saturated) extensions .Before proceeding, we remind the reader that a map from S σ to an Artin cone is equivalent tothe data of a map from σ to the corresponding cone [14, Section 6.2].7.2. Expansions and tropicalizations.
Let Y ρ → S ◦ be an expansion of X and let Σ ρ → ρ bethe associated tropical family, noting that Σ ρ comes embedded in Σ X × ρ . Let Z ρ be a stronglytransverse subscheme of Y ρ . As before, we assume that these schemes are defined over a finitelygenerated subfield of K to avoid foundational issues. Lemma 7.2.1.
There exists a rough expansion Y σ of X over a sufficiently small extension S σ of S ◦ , ex-tending the family Y ρ , such that the closure of Z ρ in Y σ is dimensionally transverse. Moreover, the family Y σ → S σ can be chosen to have relative logarithmic rank equal to .Proof. Let σ be any smooth cone giving rise to a fine and saturated extension and consider Σ X × σ .The given family Σ ρ is embedded in Σ X × σ . Since Σ X × σ is smooth it can be embedded in avector space, and by the completion theorem for fans, we obtain a complete fan refining Σ X × σ and extending Σ ρ ; this was referenced in Remark 3.5.2. For any sufficiently small replacement σ of σ , every cone of this subdivision surject onto a cone of σ , guaranteeing flatness. We obtain aproper subdivision Σ of Σ X × σ and therefore a rough expansion of X over S σ .Consider the set of all subdivisions Σ σ of Σ that do not change Σ ρ . We claim any sufficientlyrefined subdivision has the property that the closure Z ρ is dimensionally transverse. To see this,consider the closure of a component Y of the generic fiber of Y ρ . By the tropical compactificationresults for subschemes of logarithmic schemes, there exists a sequence of blowups of strata in theclosure of this component in Y σ , such that the closure of Z ρ ∩ Y is dimensionally transverse [61,Theorems 1.1 & , so any stratum that it intersects has codimension at most and dominatethe base S σ . Therefore the generic fiber is disjoint from the centers of these blowups. Repeat thisfor each component Y of Y ρ to obtain a new total space Y σ for the target with cone complex Σ σ andhe closure of the subscheme Z ρ is dimensionally transverse. After replacing σ with a sufficientlysmall in the filtered system, every cone of Σ σ maps surjectively onto a cone in σ , guaranteeingflatness. Pass to the union of strata that meet the closure of Z ρ nontrivially and obtain a roughexpansion of relative logarithmic rank equal to . (cid:3) Given a subscheme Z ρ ֒ → Y ρ as above, choose an extension Y σ over S σ such that the closure of Z ρ is dimensionally transverse. Let Σ σ be the corresponding embedded complex in Σ X × σ ; wemake the following definition. Definition 7.2.2 (Tropicalization with expanded target) . The tropicalization of Z ρ with respect tothe family Y σ over S σ above is the union of cones in Σ σ such that the closure of Z ρ has nonemptyintersection with the corresponding logarithmic stratum of Y σ . The tropicalization is denoted trop ( Z ρ ֒ → Y σ ) viewed as a subset of Σ X × σ .The tropicalization of a subscheme of the interior of a logarithmically smooth scheme is theimage under a valuation map and is manifestly insensitive to subdivisions. Lemma 7.2.3 (Birational invariance) . Let Y ′ σ ′ and Y σ be two rough expansions over S σ ′ and S σ , bothextending Y ρ . After restricting both families to a sufficient small cone σ ′′ the two tropicalizations trop ( Z ρ ֒ → Y σ ′ ) and trop ( Z ρ ֒ → Y σ ) & DHRUV RANGANATHAN give the same subsets of Σ X × σ ′′ .Proof. Consider two families as in the statement and let Σ ′ σ ′ and Σ σ ′ be the associated cone com-plexes. As we have done previously, we can complete both complexes to proper subdivisionsand guarantee them to be flat after shrinking. By shrinking further we can assume that σ and σ ′ coincide, and examine them as cone complexes embedded in Σ X × σ . Let Σ ′′ σ be the common re-finement. After shrinking again, this common refinement is flat over σ . Since any subdivision of acone complex is dominated by an iterated stellar subdivision, we reduce further to the case where Σ σ is replaced by a single stellar subdivision i.e. a weighted blowup of a stratum, necessarily inthe special fiber. Since the closure Z σ of Z ρ is dimensionally transverse to the strata of Y σ , the set ofstrata that the proper and total transforms intersect nontrivially are the same. The strict transformis equal to the closure of Z ρ in a weighted blowup Y ′ σ so the tropicalizations coincide. (cid:3) Lemma 7.2.4 (Strong transversality) . After replacing S σ with a ramified base change S ′ σ , there existsan expansion Y σ of X over a sufficiently small extension S ′ σ of S ρ , extending the family Y ρ , such that theclosure of Z ρ in Y σ is strongly transverse.Proof. After obtaining a rough expansion whose logarithmic structure has relative rank withthe requisite dimensional transversality property, we perform the ramified base change to obtainan expansion with reduced fibers [7, Section 5]. The passage from dimensional transversality tostrong transversality follows from the arguments of Li and Wu [38] in analogous fashion to thecase dealt with in Section 2.5, and in particular may require a further base change. (cid:3) Verification of the valuative criterion.
We fix a conical structure on the moduli space T oftropical expansions and on its universal family. This determines a moduli space DT ( X | D ) of rela-tive ideal sheaves equipped with a structure map to the stack Exp ( X | D ) of target expansions. Wecheck that for each map Spec K → DT ( X | D ) from the spectrum of a discretely valued field, there exists a canonical base change K ⊂ K ′ and anextension of the map to the spectrum of the corresponding valuation ring.We have shown existence in the case where Spec K maps to the locus in Exp ( X | D ) where thelogarithmic structure is trivial and therefore assume that it maps to the boundary of Exp ( X | D ) . Byreplacing T with a subdivision i.e. by blowing up the stratum to which it maps, we ensure that Spec K maps to a divisor in Exp ( X | D ) . Pull back the logarithmic structure, universal expansion,and subscheme to form a family Z ρ Y ρ S ◦ in keeping with the notation of the preceding discussion. We fix these data for the following. Proposition 7.3.1.
There exists a sufficiently small extension S σ of S ◦ and its logarithmic structure, aramified base change S ′ σ → S σ , and an expansion Y σ ′ over S σ ′ such that, after pullback, the closure of Z ρ is strongly transverse and satisfies the DT stability condition.Proof. Apply Lemma 7.2.4 to obtain a finite extension of K and a sufficiently small extension S ′ σ and an expansion Y σ → S ′ σ such that the closure of Z ρ is strongly transverse. The tropicalizationgives rise to a family of -complexes Σ σ → σ . Choose any point P in the relative interior of Σ and OGARITHMIC DT THEORY 47 let G P be the associated embedded -complex. Each vertex v of the -complex G P corresponds to acomponent of the special fiber of the expansion family. Let Z v ֒ → Y v be the associated subscheme.A -valent vertex that lies on a line in Σ X determines a component of the expansion that is a P -bundle over a surface; the examine whether the associated subscheme Z v is a tube, i.e. the pulledback from a -dimensional subscheme along the projection. Let G P be the -complex obtained byerasing all such tube vertices. Formally, if v is such a vertex that is incident to two edges, deleteboth v and these edges and connect the two vertices that were incident to v by a single edge. Theresult is an embedded -complex. This gives rise to a map from the ray generated by P to thecomplex | T | . After shrinking σ further, for example until P is extremal, this defines a map σ → T. The union of the -complexes G P do not necessarily form a cone complex, though their support iscertainly the same as a that of a cone complex.By pulling back the universal tropical expansion over T to σ we obtain a subdivision Υ σ of Σ X × σ whose support equal to that of Σ σ . Pass to a common refinement ∆ σ , which again, after shrinking σ is flat over σ and therefore gives rise to a rough expansion over S ′ σ . After performing anotherramified base change, we obtain a new expansion and a morphism of expansions associated to themap ∆ σ → Υ σ Y ∆σ → Y Υσ , over S ′′ σ . As strong transversality is stable under additional subdivision, the closure of Z ρ inside Y ∆σ isstrongly transverse. By construction, any component of the special fiber that is contracted by thismap carries a tube subscheme. It follows from arguments of Li–Wu that the resulting family isstrongly transverse [38, 64]. It follows from the definitions that the resulting family satisfies theDT stability condition, so we conclude existence of limits. (cid:3) The order of the base change required for an extension can be controlled. We produced a map σ → T and momentarily view it as a map only at the level of the Q structures. The cone σ can beequipped with an integral structure coming from the value group of R or any of finite extensions.When equipped with the integral structure coming from R the map to T above may not be given byan integer linear map. The restriction to ρ is an inclusion. There is a minimal base change in thesecircumstances, obtained passage to a finite index sublattice of the standard lattice in N S , such thatthe map can be extended, giving rise to a family of stable ideal sheaves. Strong transversality andDT stability are unaffected by further base change, we can conclude a posteriori that the familyproduced by this minimal extension is transverse and stable.Call the minimal extension constructed above as the distinguished extension. We conclude theproof of properness by verifying the distinguished extension is the only one. Proposition 7.3.2.
Given a morphism
Spec K → DT ( X | D ) and an extension Spec R ′′ → DT ( X | D ) froma ramified base change, it is obtained by pulling back the distinguished extension above.Proof. Given the morphism from S ◦ → DT ( X | D ) we have obtained the distinguished extension toa logarithmic morphism S ′ σ → DT ( X | D ) as in the previous proposition. Let R ⊂ R ′′ be a finite basechange and consider an extension Spec R ′′ → DT ( X | D ) . By the universal property of the direct image, this lifts to a logarithmic map from the direct imagelogarithmic structure on
Spec R ′′ to DT ( X | D ) . The target is fine and saturated so for any sufficiently & DHRUV RANGANATHAN small fine and saturated extension, the schematic map is enhanced to a logarithmic map S ′′ τ → DT ( X | D ) . This can be seen by direct analysis of the monoids, see also [48, Section 2.2.5].We view τ and σ as two cones in the same vector space N S , but that span possibly differentrank lattices. By shrinking, we may and do assume that τ and σ coincide set theoretically on N S i.e. their real points coincide, and we obtain two families of embedded -complexes in Σ X × σ .Denote these by Υ and Υ ′ respectively, and note that they are different polyhedral structures onthe same set in Σ X × σ . We may therefore pass to the common refinement of these two complexesand choose a sublattice of σ such that the associated map has reduced fibers.Let P be a point in the relative interior of σ and let G P be the fiber of this common refinementover P . As in the previous proposition, we may examine the closure of Z ρ for vertices in G P that carry tube subschemes, and erasing the corresponding tube vertices. This gives rise to a newgraph and map from σ to | T | as discussed in the previous proposition. Now note that the outcomeof erasing the vertices with tube subschemes for G P is the same as erasing such vertices for thefibers over p in Υ and Υ ′ . The map from the base cone σ to T that is induced by the map abovetherefore coincides in both cases, at the level of the real points.For the base change, observe that as discussed immediately before the proposition statement,there is a minimum order of base change required to extend the given map from Spec K . It followsthat up to a further ramified base change, the two extensions coincide, establishing separatedness. (cid:3) R EFERENCES [1] D. A
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