LOGARITHMIC GROMOV–WITTEN THEORY AND DOUBLE RAMIFICATION CYCLES

A BSTRACT . We examine the logarithmic Gromov–Witten cycles of a toric variety relative to its full toric boundary. The cycles are expressed as products of double ramiﬁcation cycles and natural tau-tological classes in the logarithmic Chow ring of the moduli space of curves. We introduce a simple new technique that relates the Gromov–Witten cycles of rigid and rubber geometries; the technique is based on a study of maps to the logarithmic algebraic torus. By combining this with recent work on logarithmic double ramiﬁcation cycles, we deduce that all logarithmic Gromov–Witten pushfor-wards, for maps to a toric variety relative to its full toric boundary, lie in the tautological ring of the moduli space of curves. A feature of the approach is that it avoids the as yet undeveloped log-arithmic virtual localization formula, instead relying directly on piecewise polynomial functions to capture the structure that would be provided by such a formula. The results give a common generalization of work of Faber–Pandharipande, and more recent work of Holmes–Schwarz and Molcho– Ranganathan. The proof passes through general structure results on the space of stable maps to the logarithmic algebraic torus, which may be of independent interest.


Introduction
The logarithmic Chow theory of the moduli space of curves M g;n concerns the cycle theory of the system of birational models of M g;n obtained by blowups along boundary strata, considered simultaneously.It has attracted significant recent interest [33-35, 46, 47].Logarithmic Gromov-Witten theory is a basic source of classes in this ring, parallel to the manner in which Gromov-Witten theory is a source of ordinary Chow classes in M g;n .The simplest of these classes arise when the target is a pair .X; D/ where X is a smooth projective toric variety and D is the full toric boundary.The purpose of this article is to explain how to relate these logarithmic Gromov-Witten cycles to the logarithmic double ramification cycle, and deduce that the cycles lie in the tautological part of the logarithmic Chow ring of M g;n .
0.1.Logarithmic Chow rings.We recall the logarithmic Chow ring of M g;n .Given a stack B with a normal crossings divisor E, a simple blowup is the blowup of B at a smooth stratum.The blowup has a normal crossings divisor given by the reduced preimage of E. A simple blowup sequence is obtained by successive simple blowups.If B 0 !B is a simple blowup sequence, pullback gives rise to an injective homomorphism on Chow rings.The logarithmic Chow ring is the colimit of Chow rings over all such blowups.The logarithmic Chow ring of M g;n with respect to the divisor of singular curves is denoted logCH ?.M g;n /.See [46,Section 3] for an introduction.There are two simple ways to produce classes in the logarithmic Chow ring of B.
Homological.Let Z be another simple normal crossings pair and Z !B a proper morphism of pairs.We produce a class in the Chow ring of X by pushing forward the class OEZ.If B 0 !B is a simple blowup sequence, then the strict transform of Z gives a class in the Chow ring of B 0 by the same construction.The class on B 0 and B are related by pushforward along B 0 !B, but are not necessarily related by pullback.However, for sufficiently fine blowups, the classes stabilize under pullback and so define a class in the logarithmic Chow ring of B. See [47,Sections 2,3].
Cohomological.Associated to B is a generalized cone complex †.B/.To each subdivision of †.B/ we associate the ring of strict piecewise polynomials.This is a very concrete ring: if †.B/ is a simplicial fan, it is the face algebra of the fan.The ring of piecewise polynomials PP. †.B// is the colimit of the rings obtained from all subdivisions.It has a ring homomorphism to the logarithmic Chow ring of B. The classes can roughly be thought of as polynomials in the Chern classes of bundles built from the strata of B and its blowups.See [35,46,47] for details.0.2.Logarithmic Gromov-Witten cycles.Let M .X / be the moduli space of logarithmic stable maps to the pair .X; D/ where X is a projective toric variety and D is the full toric boundary.The discrete data records the genus g, number of markings n, and their contact orders with the toric boundary [3,18,29].The moduli space is equipped with a basic diagram: M .X/ Ev .X/ M g;n . ev The evaluation space Ev is constructed as follows.For a marked point p i and a divisor D j there is an associated nonnegative contact order c ij .For a fixed point p i , the intersection of divisors with which p i has positive contact order is a stratum of X.If the contact order is 0 with all divisors, the stratum is defined to be X .The evaluation space is the product of these strata associated taken over all marked points.A primary logarithmic Gromov-Witten cycle is a class of the form back from CH ?.M g;n /, and (ii) classes from piecewise polynomials.A class is tautological in logarithmic Chow if it is a polynomial combination of these.
Theorem A. Primary logarithmic Gromov-Witten cycles of toric pairs .X; D/ lie in the tautological subring of the logarithmic Chow ring logCH ?.M g;n /.In particular, after pushing forward to the standard Chow ring CH ?.M g;n /, the cycles lie in the tautological subring.
Note that descendants at marked points with trivial contact order are pulled back from M g;n , see [41,Section 3].By the projection formula, these can be included in the theorem.Other descendants are not pulled back, but the correction terms are boundary strata on the moduli space of maps, so we expect they can be treated inductively.We do not attempt this as we feel it would obfuscate the geometry in our arguments.
The above result is a parameterized version of the results of Holmes and Schwarz [35] and Molcho and Ranganathan [47].When X is P 1 , Faber and Pandharipande proved this statement in their study of relative maps and tautological classes [21].0.3.Enumerative invariants via piecewise polynomials.Theorem A above is a corollary of the following stronger result, which relates logarithmic Gromov-Witten cycles for toric pairs to logarithmic double ramification cycles on the moduli space of curves, as studied in [33,35,47].
The theorem consists of two pieces.First, in the main text, we associate to every cohomology class on Ev .X / a logarithmic cohomology class rub on M g;n .It is given by a piecewise polynomial on the tropical moduli space of curves [46] and can be calculated by a combinatorial procedure.Second, the discrete data determines a class in the logarithmic Chow ring of M g;n called the toric contact cycle or higher double ramification cycle.It is tautological in the sense above, and on restriction to the moduli of smooth curves, it is the locus of curves that admit a map to a toric variety with discrete data , see [35,47].We denote it TC g ./.
Theorem B. There is an equality of classes ?.ev? ./ \ OEM .X; D/ vir / D rub \ TC g ./ in logCH ?.M g;n /; where TC g ./ is the logarithmic toric contact cycle associated to the data .
The result is a replacement, in the setting of logarithmic maps, for virtual localization on maps to projective space, proved by Graber-Pandharipande [25,Section 4].Parallel to that result, this one gives a complete in-principle solution to the logarithmic Gromov-Witten theory of toric varieties, relative to their full toric boundary.
The class TC g ./ is a product of logarithmic double ramification cycles in the ring logCH ?.M g;n /.A formula for the cycles has recently been established [33] and together with the admCycles package [20] this gives a practical route to calculations and reduces the computation of logarithmic Gromov-Witten invariants for toric varieties to tautological integrals.Although the complexity of the procedure is significant, this is the first known procedure that completely determines the logarithmic Gromov-Witten theory of a toric variety from known calculations.
One can approach the logarithmic Gromov-Witten theory of toric varieties via many techniques, several based on tropical curves.See [11,15,27,28,41,49,51,63] and the references therein.Via tropical geometry, the result above can be seen to have concrete consequences.For example, it can be used to calculate the Severi degrees of P 2 and the Hurwitz numbers of P 1 , see Section 3.5.
The calculation of the class rub is explained in the proof of the theorem, but we summarize it here.In Section 2 we construct a toric variety Ev rub .X / together with an equivariant, flat, and proper morphism Ev .X / !Ev rub .X /.Given a class on the evaluation space, we can push it forward and then lift it to an equivariant class on Ev rub .X /.Interpreting this as a class 0 on the Artin fan, we argue that the morphism from TC g ./ to this Artin fan factors through a blowup of the moduli space of curves.The pullback of 0 to this blowup is a piecewise polynomial rub .The equality in the theorem is proved at the end of the final section.0.4.Conjectures.The ordinary Gromov-Witten cycles of any toric variety lie in the tautological ring of the moduli space of curves by the virtual localization formula.This is perhaps the first important consequence of the formula [25].A parallel formula in logarithmic Gromov-Witten theory has not yet materialized, although progress has been made by Graber [24].Even if such a formula appears, it seems likely that its complexity will be similar to that of the degeneration formula [56,Theorem B].The result presented here, by comparison, is rather more direct, though it only handles the "full toric boundary case".We record the following statements as conjectures, so they might attract some interest.We begin with a weak version.
Conjecture C. Let X be a smooth projective toric variety and let D be a subset of the toric boundary.Logarithmic Gromov-Witten cycles lie in the tautological subring of the logarithmic Chow ring logCH ?.M g;n /.
The conjecture above would likely be a consequence of a logarithmic virtual localization formalism.The next, stronger conjecture, is unlikely to fall to localization alone.
Conjecture D. Let X be a smooth projective toric variety and D a simple normal crossings divisor.Logarithmic Gromov-Witten cycles lie in the tautological subring of the logarithmic Chow ring logCH ?.M g;n /.
The case where X is P r and D is a normal crossings union of hyperplanes is already very interesting, due to the interplay with the theory of matroids [58].If X is P 1 , the conjecture holds by work of Faber and Pandharipande [21].In addition to being stronger, it is slightly less clear that the result should hold.One might believe it because of descendant reconstruction, i.e. the expectation that numerical logarithmic Gromov-Witten theory is determined to leading order by the numerical ordinary Gromov-Witten theory with descendants.This is a theorem in the smooth pair case [43].If the transfer to the ordinary theory can be made and the subleading terms controlled, virtual localization would yield the result.Orbifold methods may provide another route to the use of torus actions, see for instance [10].
The point of our approach here is to avoid localization, but torus actions do lurk in the background.Torus localization is still a crucial part of the study of the double ramification cycle [33,37].Also, in the proof of main results, classes from the evaluation space are replaced by torus equivariant lifts, given by piecewise polynomials on the cone complex of the evaluation space.0.5.Overview.The high level strategy in the paper is simple: apply the Abramovich-Wise birational invariance theorem to replace the toric varieties by products of copies of P 1 , then apply the product formula in logarithmic Gromov-Witten theory, and finally reduce to the double ramification cycle via the rubber geometry [8,30,33,37,47,55].However, as stated, the strategy is nonsensical.0.5.1.Gaps in the naive strategy.The first issue is that birational invariance only relates the virtual classes of curves in X and in .P 1 / r by pushforward, but an arbitrary insertion is unlikely to be pulled back along X !.P 1 / r so this does not help; this was raised by Abramovich and Wise [8,Section 1.4] as a missing piece in their result.We realize that if one works in the logarithmic Chow ring, the missing insertions become accessible.The relevant strengthening of the Abramovich-Wise result was accidentally recorded in [56,Section 3].
The second issue, which is closely related, is that the simplest form of the product formula does not hold in logarithmic Gromov-Witten theory.However, it does hold if one works over a blowup of the moduli space of curves, or equivalently in the logarithmic Chow ring [30,55].
The third issue, which is of a different nature, is that the double ramification cycle deals with the rubber geometry rather than the rigid geometry, so we need to relate the virtual classes of the rubber and rigid mapping spaces.And even if this is done, the insertions need to be accessed on the rubber geometry, and evaluation spaces for rubber have not previously been considered.0.5.2.Torus quotient problems.Another key geometric idea in the paper is the rubber evaluation space and associated evaluation maps from the space of rubber maps.These have a simple relationship with the corresponding rigid evaluation spaces and maps.
The basic idea is as follows: just as the rubber moduli space parameterizes maps from curves up to the action of the dense torus T of X, the rubber evaluation space should parameterize configurations of points on strata up to the action of T .The elementary but crucial observation is that Ev .X / should be considered as a single object with the diagonal action of the torus T , via the natural action of T on its strata.We are guided here by observations of Carocci and Nabijou [13].
A geometric complication in this idea is that while we have spoken about the rubber moduli space of stable maps above, the only sensible meaning for such a space is via an intersection problem in the Picard stack, using [47] and building on [32,42].This needs to be related to logarithmic mapping spaces.The mapping space with a toric target appears, on the interior, to be a torus bundle over the rubber.However, the extension of the description to the boundary is not clear, even in rank 1.The construction of our proposed rubber evaluation spaces presents the same issues: it is straightforward on the interior, where the torus action is free, but complicated in the boundary.
In principle, the resulting quotient problems are as complicated as the problem of constructing quotients of toric variety by subtori.Even if one assumes a solution to this, the global structure of the map from the space of rigid maps to a toric variety to the space of rubber maps is complicated.0.5.3.Logarithmic algebraic tori.While these quotient problems are subtle, one might expect that by birational invariance results in logarithmic geometry, the combinatorial choices made in the toric quotient problem above are immaterial [8,56].In order to use this flexibility, we work with the space of stable maps to the logarithmic algebraic torus.This is a non-representable group-valued functor on logarithmic schemes that is, in some sense, birational to the algebraic torus, see [59].Although not representable, it is compact, and therefore quotient constructions by this group are straightforward.Stable maps to the logarithmic torus bundles arise implicitly throughout logarithmic Gromov-Witten theory, via the expansions in [56].We believe they will find use wherever immaterial non-canonical polyhedral choices need to be made.
Allowing logarithmic tori, there are essentially no difficulties in setting up the quotient problems.We easily construct the space of rubber maps, the rubber evaluation space, and compare with the rigid geometry.Once everything is constructed, birational modifications can be made to make things representable.We then move the problem into the logarithmic Chow ring of M g;n .0.6.Broader context.A question motivating this work is whether Gromov-Witten cycles always lie in the tautological singular cohomology of the moduli space of curves; this is a central conjecture in the subject going back to work of Levine and Pandharipande [40, Section 0.8].The reader may refer to [9,36,43] for a sampling of results.The statement is interesting for several reasons.For one, the tautological ring is a very small part of the cohomology ring, so the constraint is strong.Second, the tautological ring is highly structured: for example, it has an additive set of generators with a conjecturally complete set of relations [50].And finally, it is sufficiently well-understood that it can be explored using computer algebra methods [20].
A promising approach to this conjecture is via the logarithmic degeneration methods in [56] and the study of vanishing cycles in [9].The crucial inputs into this are the study of toric varieties and toric fibrations over logarithmic targets.We initiate such a study for toric varieties with respect to the full toric boundary, and will further develop these methods to study toric fibrations in future work.The conjectures in Section 0.4 are central to this question.
The results here fit into a broader study of the logarithmic tautological ring, which has seen a flurry of recent activity [33-35, 46, 47].The present paper illustrates the utility of understanding the cycle theory of this ring, showing for example that the multiplicative structure of logarithmic tautological classes recovers the enumerative geometry of all toric pairs.It is parallel to the recovery of absolute Gromov-Witten invariants by Hodge integrals.
While the present paper gives a complete in-principle solution to the logarithmic Gromov-Witten theory of toric varieties, it an interesting problem to make the result explicit for various classes of invariants.For example, double Hurwitz numbers are explored in [17].More recently, Kennedy-Hunt, Shafi, and the second author use the perspective in the present paper to give explicit formulas for the stationary descendant Gromov-Witten theory of S A 1 , where S is a toric surface relative to its toric boundary [39], recovering the formulas of Bousseau [11].
A more speculative question is the link to traditional virtual localization.When the logarithmic boundary is empty, localization describes the invariants via Hodge integrals on the moduli space of curves.When the boundary is toric, our methods here describe the invariants via double ramification integrals.It is likely that future applications will demand that this gap is bridged.0.7.User's guide.In Section 1 we recall the logarithmic torus and discuss stable maps into it.The section includes a discussion of stable maps to toric varieties and Artin fans.The main content of the section beyond the construction is the relationship between the rigid and rubber mapping spaces.In Section 2 we construct evaluation spaces for stable maps to logarithmic tori, and for its rubber variant.In Section 3 we prove the main results.After recalling the construction of logarithmic Gromov-Witten cycles, we compare the rigid/rubber virtual structures, and use this, with the logarithmic intersection yoga of [47], to deduce the results.
Throughout the paper, we work with fine and saturated logarithmic schemes and stacks, and with stacks over this category.The logarithmic structures will typically be defined in the étale topology, with the exception of several Artin stacks that appear, whose logarithmic structures are defined in the lisse étale or fppf sites.Fiber squares will be understood to be in the category of fine and saturated logarithmic schemes (or stacks, as appropriate).

Stable maps to the logarithmic torus
The main constructions in this paper are based on torus quotient problems.Quotient problems for algebraic tori can be subtle, even for quotients of a toric variety by a subtorus of the dense torus [38].The source of the complexity is the non-compactness of the torus.In logarithmic geometry, there is a canonical group compactifying the G m , and quotients by it are better behaved.We use this and birational invariance for logarithmic maps [8] to prove our main results.
1.1.Subdivisions and Artin fans.A logarithmic morphism Y !X with X logarithmically smooth is a subdivision if it is proper, birational, and logarithmically étale.More generally, if X is any logarithmic scheme, Y !X is a subdivision if, locally on X , it is the strict base change of a subdivision of a logarithmically smooth scheme.We do not require the map to be representable by schemes, but only by Deligne-Mumford stacks, so generalized root constructions are permitted; this is what is termed as a logarithmic modification by Abramovich and Wise [8], but is called a logarithmic alteration by some authors.The relevant subdivisions for us will be globally expressed as pullbacks of subdivisions of logarithmically smooth stacks.
Let be a cone in the sense of toric geometry and let X. / be the associated affine toric variety with torus T .The Artin cone associated to is the algebraic stack A D OEX. /=T .It carries a logarithmic structure in the smooth topology, descended from X. / and has the property that A ! Spec C is logarithmically étale.
More generally, one can consider stacks A with logarithmic structure such that the map A ! Spec C is logarithmically étale.A stack A of this form is an Artin fan if it has a strict étale cover by Artin cones.Typical examples are stacks A X D OEX=T , for X a toric variety.These stacks will arise as targets in logarithmic mapping stacks.We will only need a few basic facts about Artin fans beyond the definition, and refer the reader to the references [5,14] which are both well-adapted to our point of view here.
1.2.Setup and numerical data.Let X be a smooth and projective toric variety of dimension r.Equip it with its toric logarithmic structure.Let N denote the cocharacter lattice of the torus.
We will consider maps from logarithmic curves to the target X.The symbol will denote the numerical data of the moduli problem for such maps.If we fix an isomorphism of N with Z r this is: a genus g, a number n of marked points, and a contact order matrix A with r rows and n columns. 1) We assume the row sums are equal to 0; the relevant moduli spaces are empty otherwise.The j th column of A should be thought of as attached to j th marked point.
We pause to explain the contact order.Let † be the fan of X. Fix an index j , with 1 Ä j Ä n.The vector v j is contained in the relative interior of a unique cone, say .Let u i 1 ; : : : ; u i p be the primitive generators of .Write Given this, we can restrict attention to maps from pointed curves .C; p 1 ; : : : ; p n / !.X; D/; where the divisor D i k corresponding to the generator u i k has contact order a j i k with the marked point p j .In this way, the matrix A specifies the contact orders.More generally, we can consider logarithmic maps with these contact data.
1.3.Maps to toric varieties and Artin fans.We consider moduli M .X / of logarithmic stable maps to the toric variety X with its toric logarithmic structure.The moduli functor is representable by a proper Deligne-Mumford stack with a logarithmic structure and a virtual class [3,18,29].
Let A X be the stack OEX=T equipped with the logarithmic structure descended from X .It is called the Artin fan of X.One can consider moduli of logarithmic maps from prestable curves to A X with numerical data .The spaces are denoted M .A X / and they are Artin stacks with logarithmic structure, see [8].The natural morphism M .X / !M .A X / is strict.
We impose a "stability condition" on M .A X /.The idea is that the condition that a prestable map C ! X is stable is visible after projecting to A X -this is a toric phenomenon.Indeed, stability needs to be examined on rational components with one or two special points.A 1-pointed component is automatically contracted, for example by the balancing condition.A 2-pointed component is contracted if and only if the two special points both map to the same locally closed toric stratum. 2)Thus, the conditions depend only on the dual graph of C and the strata to which the special points map.We can therefore impose the conditions on logarithmic maps C !A X . 1) If we do not choose a basis, the contact order matrix should be thought of as a n-tuple of vectors in N .
2) The discussion and pictures in [53, Section 3] may be helpful for the reader.
We now do this formally.A prestable logarithmic map C !A X over a logarithmic point has a combinatorial type [29,Section 1].It includes (i) a dual graph , (ii) a cone of † X for each vertex and edge of , (iii) a primitive direction for each vertexedge flag, and (iv) a nonnegative weight associated to each such direction.The directions are primitive elements in the lattice N .Definition 1.3.1 (Linear, balanced, and stable).Let C !A X be a prestable map.A vertex of the dual graph is linear bivalent if it has genus 0, is bivalent, and the slopes of the two flags based at the vertex are opposite, i.e. the edge lies on a line.A linear bivalent vertex is stable if the cones assigned to the vertex and its two flags do not coincide.
A vertex of is contracted if the slopes of all flags based at this vertex are 0. A contracted vertex is stable if it either has positive genus, or it has genus 0 and at least three incident flags.
A prestable map C !A X is balanced if at every vertex of , the weighted sum of the outgoing edge directions at that vertex is 0.
A prestable map C !A X is stable if it is balanced and all linear bivalent vertices and contracted vertices are stable.
The balancing condition and the stability condition are both open in moduli.Both conditions can be checked at the level of dual graphs, so it suffices to check that the balancing and stability conditions are preserved when specializing the combinatorial type.These can both be checked explicitly.The reader can find the verifications, although unfortunately with slightly different terminology, in [54, Section 2.2].Definition 1.3.2.The moduli stack of stable maps to the Artin fan A X is the open substack of M .A X / parameterizing maps that are stable in the sense above.It is denoted M .A X /.
In short, the space M .A X / are those maps to the Artin fan whose combinatorial type could come from a logarithmic stable map to a toric variety.The balancing condition on C !A X is a necessary condition for the existence of a lift to X , though it is typically far from sufficient.
Warning 1.3.3.The space of stable maps to the Artin fan is of finite type, but it fails to be universally closed and is typically an Artin stack.A clearer picture of its properties can be deduced from its description as a subdivision of the functor of maps to G r trop , which is developed in later sections.However, we do not use its geometric properties in a serious way here, so we leave the details to an interested reader.
By the definition of stability, we have a natural morphism The latter two are logarithmically smooth of dimension 3g 3 C n; the space on the left is typically singular.The morphism is strict and equipped with a relative perfect obstruction theory [8].A logarithmic scheme has an associated generalized cone complex or cone stack, and these are useful when considering maps to G trop and G log .The generalized cone complex is a colimit of a diagram of cones with face maps between them, see [29, Appendix B] or [5] for the association, and [14] for a detailed discussion of cone stacks.The cone stack is similar but retains some higher categorical data about the diagram.Whenever we use this, either the distinction does not arise or either can be used.
In more detail, a section of M gp S is equivalent to combinatorial data.For families of logarithmic curves C =S , we record the relevant notions in the next two remarks.
Remark 1.4.2(Moduli of tropical curves).In a few places in this section, we will need a part of the theory of the tropicalization for the moduli stack of curves [14].The authors construct an object M trop g;n , a certain stack on the category of cone complexes.Let be a strictly convex cone with dual monoid S .A tropical curve over , denoted C trop , is a stable dual graph of type .g;n/ together with a decoration of each edge by a nonzero element of S .This decoration is the length of the edge.Dualizing, a tropical curve over produces a map of cone complexes C trop !, such that the fibers of this map over the relative interior are metric graphs enhancing .We are mildly abusing notation here, using C trop both for the decorated graph and the cone complex.The first main result concerns the fibered category M trop g;n over cone complexes whose value on a cone is the groupoid of tropical curves over .It is proved in [14] that M trop g;n is representable by a "cone stack".The second main result is that there is a canonically associated Artin fan, deoted a ?M trop g;n and a strict, smooth, and surjective map .Suppose C is a logarithmic curve over a closed logarithmic point S and let be its stable dual graph, with genus and marking decorations.Building on the previous remark, the dual graph can be enhanced with an element of M S at each edge; this monoid takes the place of S in the previous remark.An edge E of is dual to a node q E of C .The local equation at q E is xy D t , where t is an element of the monoid sheaf M S that maps to 0 in O S .The image of t in the characteristic monoid M S is denoted t .The decorated graph is denoted C trop , consistent with the discussion above.The element t is the length of the edge E. As above, the decorated graph C trop is equivalent to a cone complex over the dual cone of M S .
Suppose that ˛2 M We may view ˛as a piecewise linear function on C trop with slope m on the edge E, directed from V to W . Globally, we obtain a piecewise linear function on the cone complex C trop .
The functors are not representable by schemes, or even stacks, with logarithmic structure.We say that they are "schematically non-representable".The following is a replacement: Proposition 1.4.4.The scheme P 1 equipped with its toric logarithmic structure is a subdivision of G log .

Proof. See [59, Section 1].
Any complete toric variety X completing G m ˝N arises uniquely as a subdivision X !G log ˝N .Analogously, there is a subdivision A X !G r trop .
Notation.The symbol N will denote the cocharacter lattice of the dense torus of X.However, to compactify notation, we fix an identification of N with Z r and denote the logarithmic torus N ˝Glog by G r log .
Remark 1.4.5.One can visualize G r log as the "toric variety" defined by a fan with exactly one non-strictly convex cone D R r .It is not representable by a scheme (or algebraic stack) with logarithmic structure.Generally, if "strict convexity" in the definition of a cone from toric geometry is dropped, we obtain non-representable functors "in between" toric varieties and G r log .
1.5.Maps to logarithmic tori.We define moduli spaces of logarithmic maps with target G r log .Reflecting the schematic non-representability of the target, the moduli space is also non-representable.
We noted that a complete toric variety gives rise to a canonical subdivision X !G r log .In Proposition 1.5.5, we show that this induces subdivision of the corresponding moduli spaces; the analogous statement for schematic targets is proved in [8].
1.5.1.Setup and discrete data.Fix discrete data D .g;A D .A 1 ; : : : ; A n //, where A is a tuple of integer vectors each of length r.In other words, the contact data r n matrix.Each row of this matrix records a contact order for the moduli problem of logarithmic maps.
In geometric contexts, contact orders keep track of orders of vanishing of boundary divisors of the target along marked points on the curve.Since we work in this non-representable context, we explain how to think about the contact order.A morphism Projecting along the natural map M gp s !M gp s , we obtain r integers.Given a morphism from a logarithmic scheme S, one can collect this tuple of integers for all geometric points.These data are locally constant.
Given a logarithmic morphism to G r log from a logarithmic curve C over S , the marked points are a family of Spec.N !C/-points over S .We can analogously fix the contact order at each marked point.We can similarly make sense of the contact order for a flag consisting of a component and a node on it.The contact order of C ! G r log only depends on the map C ! G r trop .
1.5.2.Geometry of the mapping space.We describe moduli of maps to logarithmic and tropical tori.A morphism between two such pairs is a cartesian diagram such that the two r-tuples of sections are related by pullback.We have a natural map where M log g;n is the moduli space of genus g log curves with n markings.The moduli space of stable logarithmic maps to G r log , denoted M .G r log /, is the subcategory of M .G r log / where the source curves are stable.
In parallel, we define logarithmic maps to G r trop as r sections of trop is called stable if the underlying curve is stable and the map is balanced at all vertices.The moduli of stable maps to G r trop is denoted M .G r trop /.
We come to the relationship between the spaces of maps to logarithmic tori and to tropical tori.If C ! G trop factors through G log , then it is automatically balanced, so there is a morphism Although neither M .G r log / nor M .G r trop / is schematically representable, we have the following proposition: Proposition 1.5.2.The morphism is representable and strict.
Proof.Let S be a logarithmic scheme and fix S !M .G r trop /.We show that the base change map trop / is equivalent to the data of a family of logarithmic curves C =S and a map ˛W C ! G trop .The torsor are equivalent to the data of sections of G.˛/ !C .The space S 0 is therefore precisely the moduli space of sections of G.˛/ !C .As the map G log !G trop is strict, the map G.˛/ !C is too.The space S 0 of sections of the torsor is therefore representable and strict over S .
Warning 1.5.3.Although we call it a stability condition, the space M .G r trop / should not be thought of as being proper.Since the space is schematically non-representable though, only maps to it from logarithmic schemes are defined, rather than maps from ordinary schemes.Therefore it is slightly awkward to precisely define properness.One manifestation of the nonproperness is that M .G r trop / fails the logarithmic valuative criterion for properness [42, Theorem 3.5.2].In this criterion, we start with the spectrum S of a discrete valuation ring, and let Á be its generic point.We then equip Á with a logarithmic structure consider maps from it to M .G r trop / and ask when such maps can be extended.In this setup, there exist maps from Á that do not extend to maps S !M .G r trop / for any logarithmic structure on S that restricts to the chosen one on Á.We leave the details of this to the reader, as the it is orthogonal to the main results of the paper.
Another manifestation of the non-properness will come later in the paper, when we argue that after a subdivision, the space M .G r trop / is representable by a non-proper Deligne-Mumford stack.
Remark 1.5.4.Consider the moduli of logarithmic maps to G r log from prestable curves of type .g;n/, without a specified contact profile, denoted M g;n .G r log /.We can view this as a sheaf of abelian groups on the étale site of M g;n .We have a sheaf homomorphism which sends a log map f to the contact profile of f .
The following proposition can essentially be found in [57, Section 2].We take the opportunity to present a detailed discussion.Proposition 1.5.5.Let X be a smooth projective toric variety with torus G r m , equipped with its toric logarithmic structure.The subdivision X !G r log induces a subdivision M .X / !M .G r log /: We prove this result via a related result at the Artin fan level.To state it, consider a family We stabilize the source curve to obtain a curve C ! S, for example by using the moduli map M g;n !M g;n .We observe that the only unstable components that could be contracted by C ! C are 2-pointed P 1 -components.By [4, Proposition 2.10], the map C !A X has a concrete description.Precisely, recall that C admits a generalized cone complex, given by a colimit of cones as in [29, Appendix B]. 3)  It is denoted C trop .The cited proposition tells us that the map C !A X is equivalent to the data of a piecewise linear map where † X is the fan of X.Similarly, the curve C has an associated cone complex, and there is a map Now, the curves C and C differ only by 2-pointed P 1 -components.But by the stability in Definition 1.3.1, the composition factors through C trop .Indeed, linearity is exactly the condition that the map factors through contracting the component, and stability is the condition that the map to † X does not factor through the contraction.
As previously noted, an R r -valued piecewise linear function on C trop is precisely r sections of H 0 .C ; M gp C /, we have produced a map M .A X / !M .G r trop /: We now prove Proposition 1.5.5, via its "tropical" incarnation.Proposition 1.5.6.Let X be a smooth projective toric variety with torus G r m , equipped with its toric logarithmic structure.The subdivision In the proof that follows, we use the tropical curve associated to a family of logarithmic curves over an atomic base.We will also use the description of sections of the characteristic abelian sheaf via piecewise linear functions.The reader may consult Remarks 1.4.2 and 1.4.3.
3) See also [2] for details on the combinatorics and [5] for a discussion in the appropriate conceptual context.
Proof.Consider a logarithmic scheme S with a map S !M .G trop /.Denote the pullback of M .A X / !M .G r trop / by S 0 .It suffices to prove the following: Claim.The map S 0 !S is a subdivision.
Step I. Reduction to a toroidal base.It suffices to treat the case where S is atomic [8, Section 2].The relevant part of atomicity is that S has an Artin fan, and this Artin fan is in fact the one associated to a cone .We claim there is a factorization of S !M .G r trop / as S !U ! M .G r trop /; where S !U is strict and U ! M .G r trop / is logarithmically étale.If M .G r trop / were representable, one can show this easily using relative Artin fans.In this non-representable context, we argue directly.
The map S !M .G r trop / consists of a logarithmic curve C =S together with r sections of H 0 .C; M gp C /.The latter datum can be packaged combinatorially.The curve C =S determines a cone complex C trop with a map C trop ! and the r sections can be understood as a R r -valued piecewise linear function on C trop .
By the results of [14] we have a moduli map We can pass to the associated Artin fans A ! a ?M trop g;n , and note that this is logarithmically étale.By using the strict and smooth map M g;n ! a ?M trop g;n we define By the mapping property of this fiber product, we have a map S !U ; and since S !A is strict by hypothesis so is this map.Furthermore, if C ! U is the pullback of the universal curve, then the associated cone complex of this family, by construction, is C trop ! .Recall it is exactly this family that possess the map to R r .By reversing the combinatorial dictionary describing sections of the characteristic abelian sheaf, there is Abstract nonsense reduces the claim to the case when S !M .G r trop / is logarithmically étale, i.e. S D U .
We make a further reduction.The map M .G r trop / !Spec.C/ is logarithmically smooth.To see this, we consider the infinitesimal lifting criterion for logarithmically étale maps, applied to M .G r trop / !M g;n .Fix T a logarithmic scheme and T !M .G r trop / is a morphism and let T T 0 be a strict square zero extension, together with a compatible map The lifting problem for T 0 Ü M .G r trop / depends only on the characteristic sheaf of the logarithmic structure on T 0 and on the associated family of curves.Since the extension is strict, these data are unchanged, so there is a unique lift.Thus M .G r trop / !M g;n is logarithmically étale.
We therefore conclude that the composite map S !M .G r trop / !M g;n !Spec C is logarithmically smooth.In particular, it suffices to prove our claim above when the test scheme S is logarithmically smooth over a point, i.e. "toroidal".
Step II.The case of a toroidal base.We take a logarithmically smooth and atomic scheme S with S !M .G r trop /.One can now directly check that the map S 0 !S is birational, logarithmically étale, and proper and is therefore a subdivision.We say a word about the verifications.For birationality, observe that because S is logarithmically smooth over a point, the locus S ı S where the logarithmic structure is trivial is dense.The stacks M .A X / and M .G r trop / are identical on schemes with trivial logarithmic structure, so S 0 !S carries S ı isomorphically onto its image.To see the map is logarithmically étale, one can again apply the infinitesimal lifting criterion; the logic is very similar to the application in the previous step -the lifting criterion concerns strict square-zero extensions, and the data to lift depend only on the characteristic sheaf.Finally, the map is proper because it satisfies the valuative criterion for properness.For this, one can directly apply the proof of [6, Proposition 4.7.2] with cosmetic changes; this proof treats the case where A X !G r trop is replaced by a subdivision of Artin fans.But the representability is not used in the verification of the valuative criterion, and our stability condition plays the role of the stability condition on the spaces denoted there as M.Y !X/.
We prove the "geometric" version.The discussion in [8, Section 4] proves a very similar result.
In Proposition 1.5.6 we proved the bottom horizontal was a subdivision, so it will suffice to show that the square is fibered.Let G denote the fiber product M .A X / M .G r trop / M .G r log /.The universal property of the fiber product immediately gives us a map M .X / !G.
Conversely, the fiber product fitting into the square parameterizes unfilled diagrams: The right square is fibered so we get a filling of the dashed arrow canonically.We claim the map C ! X is stable.In fact, this has been reverse engineered -the curves C and C differ only by semistable, i.e. 2-pointed P 1 -components.By the stability condition, the two special points on such a component are mapped to different locally closed strata in A X , and therefore in X.The new semistable components therefore cannot be contracted in the map to X, and so are stable.We therefore have a moduli map G ! M .X /.
The maps in the two paragraphs above are clearly inverse, and the assertion of the proposition follows.
Remark 1.5.7 (Previous sightings).Maps to logarithmic tori have appeared in the literature in a few places already.In the genus 0 case, the spaces were studied in [59] and used to give a unified perspective on earlier work on the genus 0 logarithmic Gromov-Witten theory of toric varieties [16,19,53].It appeared in earlier work [57] on stable maps to genus 1 and is the heart of interactions between logarithmic geometry and the Picard variety [42,48].
1.6.Maps to rubber G r log and G r trop , and the work of Marcus-Wise.We define "rubber" moduli spaces of maps with target G r log .In these spaces, maps to logarithmic tori are identified if they differ by the translation action of G r log .The rank 1 case is the subject of [42].Perhaps interestingly, these spaces are easier to define than rubber spaces of relative stable maps [26,43], since stability for ordinary and rubber maps amount to stability of the domain.1.6.1.Summarizing the work of Marcus and Wise.Unlike M .G r log / the space M rub .G r log / is representable by an algebraic stack with logarithmic structure.This can be understood using observations due to Marcus and Wise [42].Let us collect the basic facts from this paper.
First, if A is a contact order vector, the authors construct a mapping stack M rub g;A .G trop /, consisting of prestable curves equipped with a section of the characteristic sheaf.The contact order is given by A. This is an Artin stack equipped with logarithmic structure denoted Div g;A .
Second, they show that there is an Abel-Jacobi map to the universal Picard stack sending a section of the characteristic sheaf on a logarithmic curve C to its associated O ?C -torsor.The locus where the torsor is trivial was denoted there as Div g;A .O/.It is exactly the space we call M rub g;A .G log /.Third, stable versions of these spaces may be defined by restriction along M g;n !M g;n , i.e. imposing stability of the underlying curve, ignoring the logarithmic structure.The two spaces are respectively denoted R g;A .G trop / and R g;A .G log /.The stable versions are Deligne-Mumford stacks, rather than merely Artin stacks.
Fourth, they show that the natural maps M rub g;A .G log / !M g;n and R .G log / !M g;n are logarithmic monomorphisms.For example, in the case of the stable map space, there is a factorization where the first map is a strict closed immersion and the second map is a logarithmically étale monomorphism.In particular, R g;A .G trop / !M g;n can be obtained composing a representable logarithmic modification, a generalized root construction, and an open immersion.An analogous statement holds for the prestable mapping spaces, though we will not need this.The higher rank case behaves similarly on all fronts.We have the observation that where the fiber product is taken in the category of fine and saturated logarithmic schemes.We also have the identification log is a group sheaf on the étale site of LogSch, one can make sense of G r log -torsors, and one expects that Á is such a torsor.We give the basic definitions of these objects.The discussion is essentially standard, but we provide some details, as the notion appears prominently in the text.Definition 1.7.1.Let P be a sheaf valued in sets on the strict étale site of a logarithmic scheme S .We assume we also have a fixed group action of a group sheaf G on P .We call P a G -torsor over S if the following conditions are satisfied: (i) If P .U / is nonempty, then the action of G .U / on P .U / is simply transitive.
(ii) There exists an open cover ¹U i º i 2I of X such that P .U i / is nonempty for all i 2 I .Remark 1.7.2.Note that condition (i) implies P .U / can be G -equivariantly identified with G .U /.For an algebraic group G if we set G D Mor. ; G/ we recover the more familiar notion of a G-torsor.Definition 1.7.3.Let F be a functor valued in Sets on LogSch.Let G be a group sheaf on the strict étale site of LogSch with an action on F .Fix a logarithmic scheme S .We call a G -invariant morphism W F ! S a G -torsor if the following functor on the strict étale site of S , denoted F , is a G -torsor: an étale open U ! S we assign the set The following proposition gives a transparent relationship between rigid and rubber, and illustrates the utility of working with the logarithmic torus.Proposition 1.7.4.The morphism Á is a G r log -torsor.
Proof.We first assume r D 1.For a logarithmic scheme S with a map to Div g;A .O/ we consider the following fiber product: which is also a G r log -torsor.
Proof.Stability for maps to G r log and for rubber maps to G r log is the condition that the underlying curve is stable.So if we restrict the domain of Á from M ƒ .G r log / to M .G r log /, the resulting map to M rub ƒ .G r log / factors through R .G r log /.Since Á is a G r log -torsor, the corollary follows.
Proposition 1.7.6.There is an exact sequence of sheaves of abelian groups on the étale site of M g;n : Remark 1.7.7.The exact sequence of [59, Theorem 4] identifies as sheaves on the étale site of M 0;n ; the right-hand side is the constant sheaf.This is a genus 0 phenomenon, essentially coming from the fact that degree 0 divisors and principal divisors on genus 0 curves coincide.The Z 0 above can be interpreted as the group of degree 0 divisors supported on the marked points.In higher genus, this is certainly not true, and there is a virtual codimension g condition cutting out the space of principal divisors inside the degree 0 divisors.

Evaluation spaces
We construct evaluation maps and evaluation spaces for moduli of maps to G r log .The torsor presentation of the rubber mapping space gives rise to a natural rubber evaluation space.We introduce rubber evaluation spaces, which play the key role in transferring integrals from rigid to rubber spaces.On the locus of non-degenerate maps the rubber evaluation has an elementary description, and the logarithmic torus allows us to extend this description to the boundary.

A review of evaluation maps.
In this subsection, we review evaluation maps in logarithmic Gromov-Witten theory, focusing on certain subtleties that have not received an abundance of exposition.Much of the discussion applies to logarithmically smooth pairs, but we restrict to toric varieties.
Let M .X / be the moduli of logarithmic stable maps to X with numerical data .An S -point of the space is a diagram C X S , equipped with sections p i W S !C corresponding to the marked points.In ordinary Gromov-Witten theory, composing p i with the universal map gives evaluations ev i from M .X / to X. Logarithmic structures bring subtleties.Typically, the curve C is given logarithmic structure by pulling back the natural logarithmic structure on the universal curve on the moduli space of (not necessarily minimal) logarithmic curves.Given a marked point, the section is not logarithmic -the logarithmic structure along the section is nontrivial relative to the base.
There are two ways out of the issue.One can add logarithmic structure on S (i.e. to the M .X /) or remove logarithmic structure along the image of p i .We opt for the latter.
2.1.1.Removing logarithmic structure.The universal curve C over M log g;n has a divisor given by the image of the section p i .The logarithmic structure on the universal curve C comes from a normal crossings divisor, and removing this divisor produces a new logarithmic structure on C. The map C ! M log g;n is still logarithmic, since the divisor is horizontal.We pull this back to any given family of curves, and refer to it as removing the logarithmic structure along p i .Removing the logarithmic structure along all p i is known in the literature as the vertical part of the logarithmic structure.
2.1.2.Trivial contact order points.We construct evaluation maps.First, assume the contact order of p i is trivial, or geometrically, that the marking generically maps to the inte-rior of X.Since the contact order is trivial, the morphism C ! X remains logarithmic after removing logarithmic structure along p i .We obtain morphisms ev i W M .X / !X: 2.1.3.Nontrivial contact order points.When p i has nontrivial contact order, we use the following trick.The contact order at p i determines a vector v i in the cocharacter lattice N .The vector v i lies in the relative interior of a cone i in the fan of X .Let W i be the closed toric stratum dual to i .Now, there is a rational map X Ü W i , since the torus in W i is naturally a quotient of the torus in X.We can blow up X at strata that do not intersect W i such that the rational map extends to a morphism X 0 !W i .Note that this map is torus equivariant, so automatically logarithmic.We have a morphism on moduli spaces The induced discrete data on the right is obvious for everything except the contact order.For that, note there is a projection from the cocharacter lattice of X to that of W i .The image of the contact order for each point determines the contact order for this point on the right-hand side.
Lemma 2.1.1.Let OEC !X 0 be a logarithmic stable map with discrete data .In the composite map OEC !X 0 !W i the point p i has contact order 0.
Proof.The cocharacter space of W i is the quotient of the cocharacter space of X by the linear span of the cone i .The contact order for p i lies in i , so it is 0 in the quotient.
By the lemma, the previous discussion about evaluations with trivial contact order, and the morphism M .X 0 / !M .W i / we have a logarithmic evaluation map This will actually suffice for our purposes, as we are always prepared to blowup up the target further.The following is recorded for future reference: Proposition 2.1.2.The logarithmic evaluation map ev 0 i descends to Proof.Consider the composition M .X 0 / !X 0 !X of ev 0 i , viewed as a map to X 0 , with the blowup.We argued that the image of ev 0 i lies in W i .The composed map descends to the evaluation from M .X / to X.However, the map M .X 0 / !M .X / is proper and surjective, so the scheme theoretic image of M .X / in X is contained inside the image of W i inside X, as required.
Remark 2.1.3(Dimensionally stable evaluation spaces).By blowing up the target X, one can assume that the strata W i associated to the markings are either divisors or X .It is convenient to do this, as the evaluation space does not change dimension upon further blowup.There is no loss of generality, since evaluation constraints can always be pulled back to such a blowup.

2.2.
Evaluating to logarithmic tori.We construct evaluations for logarithmic maps to G r log .Each marking p i with nontrivial contact order determines a non-constant 1-parameter subgroup by taking the span of the contact vector and tensoring the resulting map of lattices with G log : G log ,! G r log : We can therefore define: Definition 2.2.1 (The evaluation space).The rigid evaluation space of M .G r log / for p i is the quotient G r log =G log by the subtorus above.If p i has trivial contact, the rigid evaluation space is G r log itself.The consolidated rigid evaluation space Ev .G r log / is the product of these spaces over all markings.
We can define evaluation maps similarly.Let us first assume that we have marked point p i where the contact order is equal to 0. Given a family of logarithmic curves C over S , with a map to G r log , we may remove the logarithmic structure along the section corresponding to p i .By composing the natural section S !C with the map to G r log , we obtain an S-valued point of G r log .Globally, we have In order to record some consistency between the evaluations for toric targets and for the logarithmic torus, and to provide some geometric intuition, let us give another view on these maps.We can use the fact that evaluation maps already exist for M .X /, where X is a subdivision of G r log and ƒ is the induced data.Let us again first assume the contact order is 0. Consider the following diagram: The contact order is 0, so we remove the logarithmic structure along the marked point p i .The dashed arrow is filled in, as above, by composing the natural section from M .G r log / to the universal curve, with the morphism to G r log .The resulting square commutes and gives the evaluation morphism ev i W M .G r log / !G r log : Intuitively, the morphism is obtained simply by "blowing down" both sides of evaluation map in the schematically representable case.Now if the i th contact order is nonzero we can use the projection trick from the previous section.Precisely, we have seen that the contact order of p i determines a 1-parameter subgroup G log ,! G r log .By projecting a curve onto the quotient by this subgroup, the contact order at p i becomes 0. By applying the previous construction we get an evaluation morphism We will combine these to get a consolidated evaluation map 2.3.Effectivity of the torus action.We introduce an assumption for the remainder of the paper; its helps to condense some diagrams that appear later.Consider the action of G r log on Ev .G r log /.It is induced by a group action of the lattice N on the lattice N 0 associated to Ev .G r log /.Explicitly, N 0 D N=Zw 1 ˚ ˚N=Zw n ; where w i is the primitive integral vector in the direction of the contact order v i for i D 1; : : : ; n.And the action of N on N 0 is induced by the quotient group homomorphism W N !N 0 .The orbit of the origin (i.e. the image of ) is a sublattice L of N 0 .
Effectivity assumption.The lattice L above is a torsor for N , i.e. the homomorphism has trivial kernel.Furthermore, by replacing N 0 with a finite index sublattice, assume that N 0 = .N / is free.
Passing to a finite index sublattice amounts to a root construction, and does not affect the Gromov-Witten theory [8].A typical case where the effectivity fails is when the target is P 1 , and all marked points have nonzero contact order.The evaluation space is a point, so effectivity fails.

Justifying the effectivity. Effectivity implies we have the following split short exact sequence:
0 !N !N 0 !N 0 = .N / !0: If we tensor this with G m , we get that the dense torus of X injects into the dense torus of Ev .X/.Therefore, the action of the torus of X on the torus of Ev .X / is effective.We only ever care about the effective case, as if we had a toric variety where this action was not effective, its primary logarithmic Gromov-Witten cycles would vanish, as we now argue.Fix a cohomology class on some subdivision Ev .X /.The torus N ˝Gm maps to Ev .X / via the homomorphism above.In the absence of effectivity, this map has positive-dimensional fibers, and contains a copy of G m that is contracted in Ev .X /.Choose such a G m .Up to birational modifications, we can write X as Y P 1 , where P 1 is the compactification of this G m .The evaluation factors as Identify with a cohomology class on a subdivision of Ev .Y /.The vanishing follows from the product formula [30,55], together with the vanishing of the pushforward of OEM .P 1 / vir to M g;n .
2.4.The rubber evaluation space.The group G r log acts on itself and on any group quotient of it.Therefore G r log acts on the consolidated (rigid) evaluation space by the diagonal action on the factors of its presentation as a product.where the quotient is taken in the category of AbGrp-valued sheaves on the étale site of LogSch.
By Proposition 1.7.4,the rigid-to-rubber map on the mapping spaces is a G r log -torsor.The consolidated rigid evaluation map is equivariant for this group, so it descends to a rubber evaluation ev rub W R .G r log / !Ev rub .G r log /: We now define an analogous rubber evaluation space Ev rub .X / for R .X / for representable targets X !G r log .Fix such a toric variety X.
Definition 2.4.2.We define a rubber evaluation space to be a representable smooth subdivision Q of Ev rub .G r log / such that there exists a smooth subdivision B of Ev .X / with a combinatorially flat morphism b W B ! Q such that the following diagram commutes: b Remark 2.4.3.Rubber evaluation spaces always exist due to the following construction.Start with any representable subdivision Q of Ev rub .G r log /.Then we consider Then by applying semistable reduction we can assume b is combinatorially flat with smooth source and target.
One should think of a rubber evaluation space as a quotient of the rigid evaluation space by a subtorus of its dense torus.The subtorus is given by the inclusion The exact choice of rubber evaluation space is not relevant to our paper but an explicit construction can be obtained from the construction of [38].Further combinatorial details about this construction can be found in work of Molcho [45].
Proposition 2.4.4.The following diagram is cartesian: We show that for any base change of " via a map S !Ev rub .G r log / from a logarithmic scheme, the resulting outer square is cartesian: Since is G n log -torsor, we have a cover of S by étale open sets U ! S such that and the restriction of to U G r log is the projection on to the first factor.The resulting diagram is which is clearly cartesian.By gluing we get that the diagram in equation (2.1) is cartesian.
We now lift this statement from G r log to the toric variety X.
Proposition 2.4.5.There exists a cartesian square of fine and saturated logarithmic schemes: where X 1 ; X 2 ; X 3 ; X 4 are representable subdivisions of the 4 spaces appearing in Proposition 2.4.4 such that the following diagram commutes: Furthermore, we require X 3 to be a subdivision of Ev .X / with the morphism X 3 !X 4 being flat.The properties can all be ensured while also demanding X 3 and X 4 are smooth.
Proof.We define X 4 to be a rubber evaluation space as in Definition 2.4.2.Therefore we have X 3 a subdivision of Ev .X / with a flat morphism to X 4 .Both X 3 and X 4 are smooth.Now define X 2 as We get a map from X 1 !X 2 which gives a commutative cube diagram as in (2.4.5).Since both the square in Proposition 2.4.4 and the square involving M .G r log /, Ev .G r log /, X 1 and X 3 are cartesian, their composition is cartesian.This implies the outer rectangle in is cartesian, so the left square is too.The representability of X 2 comes from being a subdivision of a space that is representable by a stack (cf.Section 1.6.1).As a fiber product of representable spaces we get that X 1 is representable.

Logarithmic GW & DR cycles
We continue to let X be a toric variety and let A X be its Artin fan.Logarithmic Gromov-Witten theory produces logarithmic mapping stacks to X and to A X respectively [3,18,29].The stack of maps to X is always strict over the stack of maps to A X .We consider subdivisions of these mapping spaces.But we will only ever consider subdivisions that arise via strict maps where M .A X / is a subdivision of the usual logarithmic mapping space, with the stability condition of Section 1.3.The space M .X / is the pullback of M .X / along this subdivision.Each M .X / has a virtual class in Chow homology, compatible under pushforward along subdivisions [56, Section 3].We consider the analogous class of subdivisions of rubber mapping spaces.Recall the space of rubber stable maps to the logarithmic torus G r log from Section 1.3.As in that section, this space has a map to the space of rubber maps to G r trop .The latter is equipped with a natural stability condition.As above we will consider subdivisions of both spaces, but will always demand a strict map .G r log / carry virtual classes, given by the Fulton-Macpherson intersection class from its presentation as an intersection of regular embeddings in a product of Picard varieties over an open in a subdivision of M g;n , see [47,Section 4].
And finally, we will also consider the analogous class of subdivisions for maps to G r log and G r trop .
3.1.Conventions on notation and the minimal models.In order to avoid an overabundance of decorations, we adopt the following notation.We use subdivision-insensitive notation from here forward, and explicitly note when spaces have to be replaced by subdivisions, rather than decorate the spaces themselves; decorations are used only when two subdivisions need to be compared.
So from here forward, the notation M .X / stands for a subdivision of the usual moduli space of logarithmic stable maps.Note that subdivisions of the rubber variant are denoted R .G r log / rather than R .X /.There is no meaningful space of rubber maps to a general X, so it is not really appropriate to use the symbol "X" here.The symbol M will be replaced by M for prestable variants, where the stability condition is dropped.The moduli space of prestable logarithmic curves is denoted M log g;n .Given a toric variety X , among the subdivisions of the mapping space we consider, there is a trivial subdivision, namely, the usual space of maps to X from [3,18,29].We denote this M min .X/.The superscript stands for minimal model.Similarly, there is a distinguished functor M min .G r log / on logarithmic schemes.It is typically schematically non-representable.We use the superscript similarly for maps to Artin fans A X , to the tropical torus G r trop , and for the spaces of rubber maps to G r log and G r trop .
3.2.Logarithmic Gromov-Witten cycles.We have a forgetful morphism Given a class in the Chow ring of Ev .X /, we produce a class in the ordinary Chow ring M g;n by the pull/push formula ' ?.ev? ./ \ OEM .X / vir /: We explain how to lift this to logarithmic Chow.Recall the ring is the colimit taken over smooth subdivisions, with transitions given by pullback.Given M g;n !M g;n a subdivision with smooth domain, we can perform a fine and saturated logarithmic base change along ' to produce a map from a subdivision of M .X / to M g;n .The virtual class produces an element in the Chow ring of this blowup.
In Chow cohomology, pullback along a blowup sequence with smooth centers is split injective, so we have an inclusion logCH ?.M g;n / WD lim !CH ?.M g;n / lim CH ?.M g;n / WD logCH ?.M g;n /: The colimit is recognized inside the limit as those systems of compatible classes that are eventually related by pullback.We show the logarithmic Gromov-Witten class above is such a class.Any subdivision of M .X / that maps to the fine and saturated pullback also maps to M g;n .Note that this pullback is strict over the analogous fiber product with X replaced by A X , so it falls under the class of subdivisions we are considering.As noted earlier, these subdivisions carry virtual classes [56,Section 3].It is proved in [56] that the classes are related by proper pushforward.The evaluation class ev ? ./ can be pulled back along further subdivisions.If we cap with the virtual class of each one of these subdivisions and push forward, we get a class on every smooth subdivision of M g;n .This defines an element of the inverse limit above by pushforward compatibility.It is called the the logarithmic Gromov-Witten cycle.
Assertion.The logarithmic Gromov-Witten cycle in lim CH ?.M g;n / from the construction above is contained in the colimit of Chow rings, i.e. in logCH ?.M g;n /.
In their forthcoming survey, Herr, Molcho, Pandharipande and Wise explain why the system of classes on the M g;n spaces are eventually compatible under cohomological pullback and therefore defines an element in the logarithmic Chow ring [31].Since the survey has not appeared, we say a word about why the result is true; another version appears in [47,Section 4].
By toroidal semistable reduction, we can ensure that the forgetful map factorizes as such that the second map is flat; the first map is strict and virtually smooth [56, Section 3].For a further smooth subdivision M g;n !M g;n , this flatness ensures the fine and saturated fiber product is a fiber product in ordinary stacks.The refined pullback for the horizontal arrows therefore coincide.By a diagram chase, compatibility of refined pullback and virtual pullback ensures the virtual classes are related by Gysin pullback along The assertion is a consequence.

Comparing virtual structures.
We compare the virtual classes of M .X / and R .G r log /.Morally, the morphism from rigid to rubber is logarithmically smooth, so classes should be related by pullback, at least once subdivisions have been made to make the morphism flat.This is the content of Theorem 3.3.2,whose statement and proof are our next tasks.We begin by recalling the obstruction theory in each case.

The rigid geometry.
We have a strict morphism of subdivisions of logarithmic mapping stacks M .X / !M .A X /; where in the latter mapping stack we have imposed the stability condition of Section 1.5.Abramovich and Wise equip the map above with a relative perfect obstruction theory.If W C ! M .X / is the universal curve and f W C ! X is the universal map, the domain of the obstruction theory is given by .R ?f ?T log X / _ , see [8,Section 6].The vector bundle f ?T log X is isomorphic to O ˚r C , since the logarithmic tangent bundle of X is trivial.Note that the substack M .A X / M .A X / from Section 1.3 is open, so either can be used as the base of this obstruction theory.

3.3.2.
A variant of the rigid geometry.We have noted that M .G r log / is typically not schematically representable.One way to produce representable subdivisions is to consider the M .X / for a toric variety X .We will come across a more general source of representable subdivisions.If B ! M .G r trop / is any subdivision by an algebraic stack with logarithmic structure, the base change along M .G r log / !M .G r trop / yields d W M !B: By Proposition 1.5.2 this is schematically representable and strict.The stack B need not be a subdivision of M min .A X /, so the spaces M are more general than the mapping stacks M .X / above.Nevertheless, the morphism d is equipped with a canonical perfect obstruction theory, with obstruction bundle .R ?O ˚r C / _ .Roughly, this is because the map from is representable with obstruction theory given by the standard pull/push construction associated to the logarithmic tangent bundle of G r log , which is trivial.More formally, one can copy the analysis of [8,Section 6].We provide a sketch.Consider the lifting problem associated to d .Let S !M be a map of logarithmic schemes.Let S S 0 be a strict square zero extension with ideal J , and S 0 !B a compatible map Lifts in this diagram are controlled by the relative tangent bundle of G r log !G r trop ; note that although source and target are not schematically representable, the morphism is the stack quotient by G r m , so it makes sense to speak of its relative tangent bundle (or at least the pullback to a scheme) and it is a rank r trivial vector bundle.
It follows that lifts of the diagram form a torsor on C under the sheaf of abelian groups O ˚r C ˝J , and we obtain an obstruction theory in the sense of Wise [62].The arguments of [56,Section 3] and of [8] carry over verbatim to show the following compatibility.Lemma 3.3.1.Let M .X / be (a subdivision of) the moduli space of logarithmic stable maps to X , and assume the morphism to M .G r log / factors through M to give The virtual classes of M .X / and M are identified by pushforward along h.
3.3.3.The rubber geometry.We move on to the rubber theory.Recall that R .G r log / can be understood as an intersection problem involving the Picard stack, after Marcus and Wise [42].We describe the rank 1 case for notational brevity, and indicate the required changes.In our notation, rubber moduli space is defined as the fiber product: Note that R .G trop / is (any subdivision of) the space Div in the work of Marcus and Wise [42] and is denoted tDR in [47]. 4)The morphism aj is the Abel-Jacobi section to the universal Picard variety, determined by the contact order .The bottom map is the 0 section.It is a section of a smooth fibration, and therefore a regular embedding; this equips the top map also with an obstruction theory: explicitly, it gives rise to an embedding of the normal cone of the top arrow into the pullback of the normal bundle to 0.
The normal bundle to the map 0 at a curve C is H 1 .C; O C /: the obstruction to deforming OEC as an object in the image of the 0 section is precisely a deformation of the trivial bundle on C , i.e. an element of the tangent space to the identity in the Picard variety of C .Globally, the obstruction bundle is .R 1 ?O C / _ .In the higher rank case, the bottom right in the square is replaced by the r th fibered power of Picard varieties over M log g;n , the 0 map is replaced by the factorwise 0 map, and the aj map is replaced by the factorwise aj map given by A. The obstruction bundle is .R 1 ?O ˚r C / _ .

Global charts and Siebert's formula.
In what follows, we compare virtual classes obtained by two constructions.As our setup is rather concrete, we can do so by means of traditional intersection theory, rather than chasing obstruction theories.Specifically, we use an elegant formula for the virtual class in terms of the Chern-Fulton class and the Segre class of the obstruction bundle, due to Siebert [60].We were led to the argument while puzzling over the related papers [1,61].
We recall Siebert's idea.Let K !V be a morphism with a perfect obstruction theory E ; assume V is pure dimensional.Suppose K can be embedded in a space G which is smooth over V .In this case, we make sense of the Chern-Fulton class c F .K=V / of K relative to V , by using the Chern class of the relative tangent bundle of G over V and the Segre class of K in G.This is a Chow homology class that agrees with the total Chern class of the relative tangent bundle when the latter is defined.For further details, see [22,Example 4.2.6] and [60, Section 1].
Siebert proves that the virtual fundamental class of K, which is by definition the virtual pullback of the fundamental class OEV , is given by where the first term is the Segre class of the virtual vector bundle of deformations and obstructions.The final subscript takes the expected dimensional component.
To apply this, in addition to the obstruction bundles above, we need to know that our moduli spaces are embeddable in something smooth over the base of their obstruction theory.This is required for the existence of the Chern-Fulton class; see [60,Introduction].
In the rubber geometry, we have R .G r log / !R .G r trop / playing the role of the map K !V above.We are prepared to replace spaces by subdivisions, so the target of this morphism can be chosen to be smooth.Siebert's formula gives us control of the virtual class.Note that in this case, the definition of the Chern-Fulton class of K over V simplifies since the map in question is an embedding.If K !V is an embedding, then where c F .K/ is the usual Chern-Fulton class of the scheme K, see [22,Example 4.2.6].
In the case of the rigid geometry, we look at M .X / !M .A X /: The target can be chosen to be smooth.The map is not an embedding, but as we will see in the course of the proof of Theorem 3.3.2, it is a torus torsor over a closed substack of M .A X /.
The torsor is pulled back from M .A X / so again, we can use Siebert's formula.
3.3.5.Comparison.We state and prove the rigid/rubber comparison.Recall there is a natural rigid-to-rubber morphism between minimal models The morphism is not schematically representable, as it is a G r log -torsor.Let be a morphism of subdivisions of these spaces, with the map induced by .The term "induced" means that the morphism is obtained by first subdividing the tropical space R min .G r trop /, pulling back this subdivision to M min .G r log /, and then subdividing further.In order to state the comparison, we impose that M is schematically representable and that is flat; since is a subdivision of a logarithmic torus torsor, it is logarithmically smooth, and flatness can always be guaranteed by appropriate subdivision [7,45].We show the following: Theorem 3.3.2.For a subdivision of M !R .G r log / as above, we have an equality OEM vir D ?.OER .G r log / vir / in the Chow homology of M in homological degree 3g 3 C n r.g 1/, where n is the number of marked points in .
Proof.We have the following diagram, which is commutative but not cartesian: Here, B is a subdivision of the space M min .G r trop /, and M is defined by B R min .G r trop / M min .G r trop /.Therefore M !B is strict.The symbols M and B reprise their roles from Section 3.3.2.
Step I. Observations about arrows.The right vertical is a subdivision of the following G r trop -torsor: trop / is logarithmically étale.The subdivisions in the setup are chosen such that the map B ! R .G r trop / is flat.The left vertical arrow is a subdivision of a G r log -torsor, induced by the corresponding subdivision of the G r trop -torsor B ! R .G r trop /.Thus, it is logarithmically smooth and flat of relative dimension r.The difference in dimensions illustrates the failure of the square to be cartesian.
The lower horizontal is a closed immersion: it is the inclusion of the intersections of the double ramification spaces.The upper horizontal in the commutative square is not a closed immersion.While the square is not cartesian, we can consider the fiber product F instead, and there is a map M !F: The morphism from G log !G trop is a quotient by G m , so it is smooth.Therefore M !F is also a torus torsor, and therefore is also smooth.It will be useful to describe the morphism M !F more explicitly; we set r D 1 for simplicity.The map M !F strict, and it can be understood via the fiber product description of F :  [57,Section 2.6]; the sign convention is immaterial but we choose it for consistency with the formula given for it in [57].The fiber product imposes the condition that this torsor is isomorphic to the trivial torsor.The space M has, in addition to these data, a distinguished trivialization of the torsor O C .˛/.We can therefore view the G m -torsor M !F , as the torsor associated to the line bundle R 0 ?O C , where as before W C ! M is the universal curve.More canonically, this torsor can be viewed as the torsor of isomorphisms Iso.O C .˛/; O C /.For higher values of r, one obtains r copies of this torsor.
We make a final note before proceeding.The space M is proper, and yet it is a torus torsor over the space F .While this might be slightly disorienting at first, the reader may note that F ! R .G r log / is relatively 0-dimensional, but of Artin type.A proper space can certainly be exhibited as a torus torsor over an Artin stack, e.g., consider P 1 !OEP 1 =G m .
Step II.Obstruction theories.We compare the virtual classes on these spaces.Consider the closed immersion F ! B. We can view it as a fiber product in two different ways: where Q r i D1 Pic is the product, over M log g;n of r copies of the Picard scheme.The 0 denotes the coordinatewise trivial bundle.Now both squares are cartesian.The embedding of the 0 section has normal bundle H 1 .C; O C / ˚r at the curve C and forms a rank g vector bundle.Pullback equips the middle and top arrow with a perfect obstruction theory, i.e. an embedding of the cones of these horizontal morphisms into this vector bundle.Since the right vertical map on the top square is flat, the two horizontals in the top square have compatible perfect obstruction theories.If W C ! F is the universal curve, the obstruction bundle is R 1 ?.O ˚r C / _ , see [42,Proposition 5.3.6.2].
The morphism M !F is smooth, and we have equipped F with a perfect obstruction theory over B, we have two ways of endowing M with a virtual class.First, we can perform the standard virtual pullback of the fundamental class along the morphism using the obstruction theory defined earlier in this section.Second, we can use the obstruction theory on F ! B to obtain a virtual class OEF vir and then perform flat pullback for the class OEF vir along M !F .We claim that these homology classes on M coincide.
The claim follows from Siebert's formula, discussed in Section 3.3.4.Indeed, M is a torus bundle over a closed substack F inside B, and as we have described above, this torus torsor is isomorphic to the one associated to the bundle R 0 ?.O C ˝N /, which is certainly pulled back from B. We are free to use Siebert's formula, and writing out the classes yields the result.To spell out the details, recall that the formula requires the Chern-Fulton class and the obstruction bundle, as a virtual vector bundle.The obstruction bundle on F is R 1 ?.O C ˝N /, while the obstruction bundle on M relative to B is R 1 ?.O C ˝N / R 0 ?.O C ˝N /.Observe the R 0 term is a trivial vector bundle, so the total Segre classes of these two virtual vector bundles coincide, and their ranks differ by r.
The relative tangent bundle of M !F is trivial, since it is a torus torsor.Therefore, the Fulton-Chern class of M is equal to the pullback of that of F .The result follows from Siebert's formula.
We now prove Theorem B. The construction of Proposition 2.4.5 will be used below.

Proof of Theorem B.
There are three steps in the proof, starting from an evaluation class operating on the class OEM .X / vir .First, we express the pushforward of this to the moduli space of curves as an analogous pairing on the rubber space.Next, we replace the rubber cohomology class with a piecewise polynomial.Finally, we move this to a piecewise polynomial class over the higher double ramification cycle on the moduli space of curves.
Step I. Rigid-to-rubber.We start with the diagram of rubber, rigid, and evaluation spaces The morphism from R .G r trop / to A E is specified by a set of piecewise linear functions on some subset of rays in †, i.e. those rays that lie in the subcomplex that defines R .G r trop / as an open.By choosing the values on the remaining rays (arbitrarily) and then reversing the dictionary, we obtain a morphism from a subdivision of M g;n to A E .Summarizing, this blowup which we denote M , contains R .G r trop / and therefore R .G r log / as a substack, and upon restriction to these substacks, we recover the evaluation maps above.
Apply the projection formula for the pushforward from R .G r log / into the blowup M with the class 0 .We have expressed the required Gromov-Witten cycle as a class in the logarithmic Chow ring of M g;n .By applying [33,35,47], the pushforward of R .G r log / is tautological in the logarithmic Chow ring.The result follows.
3.5.Classical connections.We conclude with a discussion of two practical examples: Hurwitz numbers of P 1 and Severi degrees of P 2 .We begin with the latter, since the former are treated in some detail in [17,Section 3].Further links with tropical curve counting may be found in [39].
3.5.1.Severi degrees.The Severi degrees are invariants from classical enumerative geometry, and are recalled below.They are also the ordinary Gromov-Witten invariants of P 2 with point insertions.We connect the Severi degrees of P 2 to rank 2 higher double ramification cycles.
The Severi degrees concern the enumeration of genus g curves of degree d in P 2 , through 3d 1 C g general points.The term Severi degree comes from the equality with the degree of the variety parameterizing genus g curves in the family of degree d plane curves, namely the Severi variety.
The Severi degrees can be computed by tropical geometry, via Mikhalkin's correspondence theorem [44], though we think of this via its logarithmic geometry incarnation [41,49].The number N d;g also equals .dŠ/ 3 times a logarithmic Gromov-Witten invariant of P 2 with its toric log structure. 5)The discrete data is as follows: the genus is g and the contact order matrix is The matrix has 3d nonzero columns, written in three groups of d each.The number of zero columns is equal to 3d 1 C g.We obtain a logarithmic Gromov-Witten invariant by pulling back a point constraint corresponding to each of these 3d 1 C g points.
Our goal is to use the method in the proof of Theorem B to provide a piecewise polynomial class in the logarithmic Chow ring of the moduli space of curves, which when intersected with the logarithmic toric contact cycle computes the Severi degree N d;g .We recall the relevant diagram M .P  5) The implicit equality of logarithmic and ordinary Gromov-Witten invariants of P 2 here is a happy accident; it holds only for point insertions, and already fails for non-Fano toric surfaces.The logarithmic invariants of a toric surface are always equal to toric Severi degrees but this typically fails for the ordinary Gromov-Witten invariants of Hirzebruch surfaces [52].
By convention, we take the evaluation spaces to be a restricted one, corresponding only to the markings with contact order 0. We set n D 3d g C 1 and take Ev .P 2 / to be a subdivision of .P 2 / n .Similarly Ev rub .P 2 / will be a subdivision of a toric quotient of .P 2 / n .
The cocharacter space of the torus in Ev rub .P 2 / can be viewed as a moduli space of configurations of n points in R 2 up to translation.This furnishes a tropical interpretation of ev rub .Recall that ev rub factors through R .G r trop / !A E .We denote the induced map from the generalized cone complex of R .G r trop / to the fan of Ev rub .P 2 / by ev trop rub .We can view this generalized cone complex as a moduli space of tropical maps from tropical curves to R 2 up to translation.The contact order 0 markings correspond to infinite legs of the tropical curve which are contracted to points in R 2 .Now ev trop rub sends a tropical curve to this configuration of points in R 2 up to translation.We have a class D .pt/˝n in CH ?.Ev .P 2 // which is Poincaré dual to a point class.This implies that ı ? ./ is Poincaré dual to a point class.The choice of a maximal-dimensional cone in the fan of E gives us a piecewise polynomial which is a lift of ı ? ./ to CH ?.A E /.Since ev rub is logarithmic, the pullback of ı ? ./ to R .G r trop / is given by the piecewise polynomial ı ev trop rub .As in the proof of Theorem B, choose a smooth subdivision M of M g;n from which ı ev trop rub is pulled back.By the projection formula, we can intersect the virtual class of R .P 2 / with ev ?rub .ı? .// in the Chow ring of M .The intersection number is the Severi degree.

The Severi degrees in practice.
Let us sketch what the piecewise polynomial looks like in one of the simplest cases -degree 2 curves in P 2 of genus 0 through five points.Of course, the number is 1.In keeping with the discussion above, the integral can be expressed on M 0;11 .Among the eleven points, six of them correspond to the boundary contact points in P 2 and the remaining five are contact order 0.
The insertion is .pt/˝5 in .P 2 / 5 .The appropriate logarithmic cohomology class rub is the image of a piecewise polynomial in logarithmic Chow, but we abuse notation slightly and let rub denote the piecewise polynomial itself: rub W M trop 0;11 !R: It associates a real number to each 11-pointed, genus 0 tropical curve .Such a tropical curve admits a unique balanced map to R 2 subject to the following conditions: (i) the marked point p 7 is sent to the origin, and (ii) the slopes along the first six marked points are as given by the contact order.Note this uniqueness follows from, e.g., [23,59].A typical 11-pointed genus 0 tropical curve can be found in Figure 1.
For each of the marked points p 8 ; : : : ; p 11 we attach a real number; the product of these will be the value of rub on .Each marked point is supported at a vertex of the tropical curve .For each j in ¹8 : : : ; 11º, there is a unique path in connecting the vertex supporting p 7 with p j .This path maps piecewise linearly to R 2 , and we can project onto the x-axis.Define the value x ;j to be the length of this image if it is contained in the positive axis, and 0 if this path intersects the negative x-axis.Define y ;j similarly.The function rub takes the value on to be Q 11 j D8 x ;j y ;j .In this case, we have taken the rubber evaluation space to be .P 2 / 4 .We have chosen the piecewise polynomial that represents the equivariant class corresponding to the point .1 W 0 W 0/ in each factor and the class rub is its pullback to a blowup of M 0;11 via the procedure above.It is combinatorially somewhat nontrivial to evaluate this piecewise polynomial and get the Figure 1.An 11-pointed genus 0 tropical curve, drawn in R 2 using the unique balanced map with the given contact orders.Note that the dotted lines correspond to the sive marked points of contact order 0 while the six "ends" with arrow heads correspond to the boundary markings.
number 1, but this is the subject of tropical intersection theory [23].We do not explain this here, but in this case, it is relatively easy to check that the piecewise polynomial is 0 on most cones of M trop 0;11 , and is nonzero on exactly one 8-dimensional cone of a subdivision.The polynomial on this cone is equal to the product of the canonical "slope 1" functions associated to the rays so the intersection number is 1.We leave further details of the calculation to the reader.
3.5.3.Hurwitz theory.Intersections against the logarithmic double ramification cycle in rank 1 also contain information related to classical algebraic geometry.It was recently shown that double Hurwitz numbers of P 1 are intersections of the logarithmic double ramification cycle with piecewise polynomials [17].Note this includes the classical Hurwitz numbers counting covers of the Riemann sphere, with simple branching.We treat only this case for simplicity.The result is not, strictly speaking, a special case of our main result.However, it is close in spirit so we record it.
We take the target to be P 1 equipped with its toric logarithmic structure, and fix a genus g and degree d .The expected dimension of covers is 2d 2 C 2g.The contact order vector is given by A D .where the insertions are placed at the contact order 0 markings.The key point is that the class ev ?i .pt/i imposes a simple ramification condition at p i .This is a codimension 1 constraint, and since there are 2d 2 C 2g ramification points, the count is finite and equal to the Hurwitz number.
Let A 0 be the vector of length 2d which is the vector A with the final 0 entries removed.Since the virtual class pulls back along the forgetful morphism M g;A .P 1 / !M g;A 0 .P 1 /; the Hurwitz number can be computed by an integral on M g;A 0 .P 1 /.By fixing the first ramification point to be at the point 1 in P 1 , we can further reduce this to an integral on R g;A 0 .P 1 /.The main result of [17,Section 3] is an explicit piecewise polynomial class, called the branch polynomial, which when integrated against the double ramification class logDR g .A 0 / in the logarithmic Chow ring of M g;2d becomes equal to the Hurwitz number.
Note that in the paper above, the double ramification expression is used to construct analogues of the double Hurwitz numbers that are related to pluricanonical divisors in the same way that the ordinary numbers are related to principal divisors.The same could be done here to construct "pluricanonical" analogues of the Severi degrees.

1. 4 .
Logarithmic tori.We introduce the protagonist of our story.Definition 1.4.1.The logarithmic algebraic torus G log is the functor LogSch !AbGrp whose value on a logarithmic scheme S is G log .S / D H 0 .S; M gp S /: The tropical torus G trop is the functor defined by G trop .S / D H 0 .S; M gp S /: It will arise for us in the following fashion.Fix S an atomic logarithmic scheme [8, Section 2].It admits a strict map S !A to an Artin fan associated to a cone .Then each family of .g;n/ stable curves C =S produces a moduli map !M trop g;n or equivalently a tropical curve C trop over .Remark 1.4.3 (Piecewise linear functions).Let us use the tropical curve to describe sections of M gp C there is a component of C dual to V , and on the interior of this component, M gp C is constant with value M gp S .Write ˛.v/ for the value of ˛on the interior of this component.If E is an edge between V and W , near E we can write ˛D ˛.V / C mx; where x is the image of x 2 M C;E in M C;E and m 2 Z. Restricting to W , we find ˛.W / D ˛.V / C mt : Definition 1.5.1.The moduli space of prestable logarithmic maps to G r log , denoted M .G r log /, is the fibered category over logarithmic schemes whose fiber over a logarithmic scheme S is 0where is a log curve C ! S of type .g;n/ and s 1 ; : : : ; s r in H 0 .C; M gp C / with contact profile A.

Definition 1 . 6 . 1 .
Let M rub .G r log / be the moduli space of prestable logarithmic maps to rubber G r log analogously to M .G r log /, but with the following change: for a family of curves W C ! S , replace the r sections of M gp C by r sections on S of ?.M gp C /=M gp S .Let R .G r log / be the moduli space of stable logarithmic maps to rubber G n log as the subcategory of M rub .G r log / where the curves are stable.The spaces M rub .G r trop / are defined similarly using ?.M gp C / and M gp S , and the space R .G r trop / is the locus where the curve is stable and the maps are balanced, see Definition 1.5.1.

Definition 2 . 4 . 1 .
The consolidated rubber evaluation space Ev rub .G r log / is Ev rub .G r log / WD Ev .G r log /=G r log ;

2 Z
1; : : : ; 1; 1; : : : ; 1; 0; : : : 0/; where d entries are 1, another d are 1, and 2d 2 C 2g are 0. The Hurwitz number is H g .d/ D 1 .dŠ/ Therefore M rub .G r log / is representable.Similarly R .G r log / and R .G r trop / are schematically representable, and carry similar relationships with the moduli of curves from the fourth point.1.7.Comparing rigid and rubber.The space M rub .G r log / should be thought of as the moduli space of logarithmic stable maps to G r log upto shifts by G r D 1; : : : ; r: log .This suggests the existence of a map ˜W M .G r log / !M rub .G r log /: Precisely, a map from a curve W C ! S to G r log is given by r sections in ?.M gp C /. Sending each section s to the corresponding section N s of ?.M gp C /=M gp S , we obtain the map above.As G r We show that ı is a G log -torsor.The map S !Div g;A .O/ determines a log curve W C ! S together with a section s of H 0 .S; ?.M From the map ı above, and in keeping with the notation of Definition 1.7.3, we have the functor that assigns to an étale open U ! S F Therefore we can find a cover by étale open sets U such that F ı .U / is nonempty.The result follows.
ı Á gp S !?.M gp C / !? .M gp C /M gp S !0: ı .U / D ¹s 0 2 H 0 .C jU ; M gp C jU / j U .s 0 / D sº;where is the second arrow in the preceding short exact sequence.We have a natural action of the group sheafG log on F ı .If F ı .U / is nonempty, it is a coset in H 0 .C jU ; M gp C jU /.This implies the action of H 0 .U; M gp U / on F ı .U / is simply transitive.The section s locally comes from a section of ? .M gpC /.
trop / B: A point of B determines a curve together with global section of its characteristic monoid, i.e. an element ˛in H 0 .C; M that the vertical arrows are flat.The initial choice of Ev .X / may have to be refined.If this is done is pulled back.The morphism p is a subdivision.Now observe that M is precisely one of the spaces considered in Section 3.3.2.It is therefore equipped with a natural virtual fundamental class OEM vir .By Lemma 3.3.1,we have OEM "Fix a toric variety X and discrete data .Choose a cohomology class in some subdivision of Ev .X/.Noting our convention is to suppress subdivisions, by Proposition 2.vir D p ? OEM .X / vir :