TTowards a tropical Hodge bundle
Bo Lin and Martin Ulirsch
Abstract
The moduli space M tropg of tropical curves of genus g is a generalizedcone complex that parametrizes metric vertex-weighted graphs of genus g . For eachsuch graph Γ , the associated canonical linear system | K Γ | has the structure of apolyhedral complex. In this article we propose a tropical analogue of the Hodgebundle on M tropg and study its basic combinatorial properties. Our construction isillustrated with explicit computations and examples. Let g ≥ M g the moduli space of smooth algebraic curves of genus g . The Hodge bundle Λ g is a vector bundle on M g whose fiber over a point [ C ] in M g is the vector space H ( C , ω C ) of holomorphic differentials on C . One can think ofthe total space of Λ g as parametrizing pairs ( C , ω ) consisting of a smooth algebraiccurve and a differential ω on C . Since for every curve C the canonical linear system | K C | can be identified with the projectivization P (cid:0) H ( C , ω C ) (cid:1) , the total space of theprojectivization H g = P ( Λ g ) of Λ g parametrizes pairs ( C , D ) consisting of a smoothalgebraic curve C and a canonical divisor D on C ; it is referred to as the projectiveHodge bundle . Let π : C g → M g be the universal curve on M g . We may define Λ g formally as the pushforward π ∗ ω g of the relative dualizing sheaf ω g on C g over M g .The Hodge bundle is of fundamental importance when describing the geometryof M g . For example, its Chern classes, the so-called λ -classes , form an importantcollection of elements in the tautological ring on M g (see [31] for an introductory Bo LinDepartment of Mathematics, University of California, Berkeley, Berkeley, CA 94720, e-mail: [email protected]
Martin UlirschFields Institute for Research in Mathematical Sciences, University of Toronto, 222 College Street,Toronto, Ontario M5T 3J1 e-mail: [email protected] a r X i v : . [ m a t h . AG ] J a n Bo Lin and Martin Ulirsch survey). The Hodge bundle admits a natural stratification by prescribing certain poleand zero orders ( m , . . . , m n ) such that m + . . . + m n = g − M g is the moduli space M tropg thatparametrizes isomorphism classes [ Γ ] of stable tropical curves Γ of genus g . InSection 2 below we are going to recall the construction of this moduli space. In par-ticular, we are going to see how this moduli space naturally admits the structure of ageneralized cone complex whose cones are in a natural order-reversing one-to-onecorrespondence with the boundary strata of the Deligne-Mumford compactification M g of M g (see [1] as well as Section 2 below for details).We also refer the reader to [19], [20], [24], and [25] for the theory in genus g = M tropg (and some of its variants) to non-Archimedean analytic geometry and to [14] and[15] for an in-depth study of the topology of M tropg , n . We, in particular, highlight thetwo survey papers [8] and [9].Let Γ be a tropical curve. We denote by K Γ the canonical divisor on Γ and byRat ( Γ ) the group of piecewise integer linear functions on Γ (see Section 3 below fordetails). In this note we propose tropical analogues of the affine and the projectiveHodge bundle, and study their basic combinatorial properties. Definition 1.1.
As a set, the tropical Hodge bundle Λ tropg is given as Λ tropg = (cid:8) ([ Γ ] , f ) (cid:12)(cid:12) [ Γ ] ∈ M tropg and f ∈ Rat ( Γ ) such that K Γ + ( f ) ≥ (cid:9) and the projective tropical Hodge bundle H tropg is given as H tropg = (cid:8) ([ Γ ] , D ) (cid:12)(cid:12) [ Γ ] ∈ M tropg and D ∈ | K Γ | (cid:9) . The associations (cid:0) [ Γ ] , f (cid:1) (cid:55)→ [ Γ ] and (cid:0) [ Γ ] , D (cid:1) (cid:55)→ [ Γ ] define natural projectionmaps Λ tropg −→ M tropg and H tropg −→ M tropg , which, in a slight abuse of notation,we denote both by π g .In [18, 22, 26] the authors describe a structure of a polyhedral complex on thelinear system | D | associated to a divisor D on a tropical curve Γ ; we are going toreview this description in Section 3 below. We also, in particular, highlight the firstauthors [23], where he presents algorithms for computing this polyhedral complex.Our main result is the following: Theorem 1.2.
Let g ≥ .(i) The tropical Hodge bundle Λ tropg and the projective tropical Hodge bundle H tropg carry the structure of a generalized cone complex.(ii) The dimensions of Λ tropg and H tropg are given by dim Λ tropg = g − and dim H tropg = g − owards a tropical Hodge bundle 3 respectively.(iii) There is a proper subdivision of M tropg such that, for all [ Γ ] in the relativeinterior of a cone in this subdivision, the canonical linear systems | K Γ | = π − g (cid:0) [ Γ ] (cid:1) have the same combinatorial type. We are going to refer to this subdivision of M tropg as the wall-and-chamber de-composition of M tropg . In general, the generalized cone complexes Λ tropg and H tropg are not equi-dimensional. So Theorem 1.2 (ii) really states that the dimension ofa maximal-dimensional cone in Λ tropg (or H tropg ) has dimension 5 g − g − g = H trop . The numbers in green are the positive h ( v ) and thenumbers in black denote coefficients greater than 1 in the divisors.Let us give a quick outline of the contents of this contribution. In Section 2 werecall the construction of the moduli space M tropg of stable tropical curves and inSection 3 the polyhedral structure of linear systems on tropical curves respectively.In Section 4 we prove Theorem 1.2 by describing the polyhedral structures of both Λ tropg and H tropg simultanously. Section 5 contains a selection of explicit (some-times partial) calculations of the polyhedral structure of H tropg in some small genuscases. Finally, in Section 6 we describe a natural tropicalization procedure for the Bo Lin and Martin Ulirsch projective algebraic Hodge bundle via non-Archimedean analytic geometry and ex-hibit a natural realizability problem . A tropical curve is a finite metric graph Γ (with a fixed minimal model G ) togetherwith a genus function h : V ( G ) → Z ≥ . The genus of Γ (or of G ) is defined to be g ( Γ ) = g ( G ) = b ( G ) + ∑ v ∈ V ( G ) h ( v ) where b ( G ) denotes the Betti number of G . In the above sum one should think ofthe vertex-weight terms as the contributions of h ( v ) infinitesimally small loops atevery vertex v . We say a tropical curve Γ (or the graph G ) is stable , if for everyvertex v ∈ V ( G ) we have 2 h ( v ) − + | v | > , (1)where | v | denotes the valence of G at v . Definition 2.1.
As a set, the moduli space M tropg of stable tropical curves of genus g is given as M tropg = (cid:8) isomorphism classes [ Γ ] of stable tropical curves of genus g (cid:9) . Let us now recall from [1] the description of M tropg as a generalized extendedcone complex. Proposition 2.2 ([1] Section 4).
The moduli space M tropg carries the structure of ageneralized rational polyhedral cone complex that is equi-dimensional of dimension g − . First, recall that a morphism τ → σ between rational polyhedral cones is saidto be a face morphism , if it induces an isomorphism onto a face of σ . Note thatwe explicitly allow the class of face morphisms to include all isomorphisms. A generalized (rational polyhedral) cone complex is a topological space Σ that arisesas a colimit of a finite diagram of face morphisms (see [1, Section 2] and [29, Section3.5] for details).In order to understand this structure on M tropg , we observe that it is given as acolimit M tropg = lim → (cid:101) M G , of rational polyhedral cones (cid:101) M G taken over a category J g . Let us go into some moredetail:1. The category J g is defined as follows: owards a tropical Hodge bundle 5 • its objects are stable vertex-weighted graphs ( G , h ) of genus g , and • its morphisms are generated by weighted edge contractions G → G / e for anedge e of G as well as by the automorphisms of all ( G , h ) .Here a weighted edge contraction c : G → G / e is an edge contraction such thatfor every vertex v in G / e we have g (cid:0) c − ( v ) (cid:1) = h ( v ) .
2. Moreover, for every graph G we denote by (cid:101) M G = R E ( G ) ≥ the parameter space of all possible edge lengths on G .3. The association G (cid:55)→ (cid:101) M G defines a contravariant functor J g → RPC Z from J g to the category of rational polyhedral cones. It associates to a weighted edgecontraction G → G / e the embedding of the corresponding face of (cid:101) M tropG andto an automorphism of G the automorphism of (cid:101) M G that permutes the entriescorrespondingly.We note hereby that we have a decomposition into locally closed subsets M tropg = (cid:71) G R E ( G ) > / Aut ( G ) , where the disjoint union is taken over the objects in J g , i.e. over all isomorphismclasses of stable finite vertex-weighted graphs G of genus g . Example 2.3 ([13] Theorem 2.12).
For a d -dimensional cone complex C , its f -vector is defined as ( f , f , . . . , f d ) , where f i is the number of i -dimensional conesin C . The 12-dimensional moduli space M trop has 4555 cells; its f -vector is givenby f ( M trop ) = ( , , , , , , , , , , , , ) . Remark 2.4.
Earlier approaches, such as [7], [8], [10], [13], [16], and [32], used torefer to the structure of a generalized cone complex as a stacky fan . Since there isa closely related, but not equivalent, notion of the same name in the theory of toricstacks we prefer to follow the terminology of generalized cone complexes intro-duced in [1].
Let Γ be a tropical curve. A divisor on Γ is a finite formal Z -linear sum D = ∑ i a i p i , Bo Lin and Martin Ulirsch over points p i in Γ , i.e. D is an element in the free abelian group Div ( Γ ) on thepoints of Γ . The degree deg ( D ) of a divisor D = ∑ i a i p i is defined to be the integer ∑ i a i . We say D = ∑ i a i p i is effective , if a i ≥ i .A rational function on Γ is a continuous function f : Γ → R whose restrictionto every edge is a piecewise linear integral affine function. Given a rational function f on Γ as above and a point p ∈ Γ , the order ord p ( f ) of f at p is defined to be thesum of the outgoing slopes of f emanating from p . Observe that ord p ( f ) is equal tozero for all but finitely many points p ∈ Γ . So we have a map ( . ) : Rat ( Γ ) −→ Div ( Γ ) f (cid:55)−→ ( f ) = ∑ p ord p ( f ) · p . Divisors of the form ( f ) for a function f ∈ Rat ( Γ ) form a subgroup PDiv ( Γ ) ofDiv ( Γ ) and are referred to as the principal divisors on Γ . Two divisors D and D (cid:48) on Γ are said to be equivalent (written as D ∼ D (cid:48) ), if D − D (cid:48) ∈ PDiv ( Γ ) , i.e. if there isa rational function f ∈ Rat ( Γ ) such that D + ( f ) = D (cid:48) . Note that the continuity of f implies that deg ( f ) = Definition 3.1.
Let D be a divisor of degree n on a tropical curve Γ .1. Denote by R ( D ) the set R ( D ) = (cid:8) f ∈ Rat ( Γ ) (cid:12)(cid:12) D + ( f ) ≥ (cid:9) . For f ∈ R ( D ) , the divisor D + ( f ) is supported in deg ( D + ( f )) = deg ( D ) = n points (counted with multitplicity). We may therefore define: S ( D ) = (cid:8) ( f , p , . . . , p n ) (cid:12)(cid:12) f ∈ Rat ( Γ ) and p , . . . , p n ∈ Γ such that D + ( f ) = p + . . . + p n ≥ (cid:9) .
2. The linear system | D | associated to D is the set | D | = (cid:8) D (cid:48) ∈ Div ( Γ ) (cid:12)(cid:12) D (cid:48) ≥ D ∼ D (cid:48) (cid:9) . Observe that R ( D ) = S ( D ) / S n , where the symmetric group S n acts on S ( D ) bypermutation of the points p , . . . , p n . Moreover, the additive group R = ( R , +) op-erates on R ( D ) by adding a constant function and, taking the quotient under thisoperation, we obtain that R ( D ) / R = | D | , since ( f ) = f is a constant function on Γ .The spaces S ( D ) , R ( D ) , and | D | are known to carry the structure of a polyhedralcomplex (see e.g. [26] or [18]). The following proposition is a more detailed versionof [18, Lemma 1.9]. owards a tropical Hodge bundle 7 Proposition 3.2.
Given a divisor D on a tropical curve Γ , the space S ( D ) has thestructure of a polyhedral complex. Choose an orientation for each edge e of Γ ,identifying it with the open interval [ , l ( e )] . Then the cells of S ( D ) can be describedby the following (discrete) data:(i) a partition of { p , . . . , p n } into disjoint subsets P e and P v (indexed by v ∈ V ( G ) and edges e ∈ E ( G ) ) that tells us on which edge (or at which vertex) every p i is located,(ii) a total order on each P e , and(iii) the outgoing slope m e ∈ Z of f at the starting point of esuch that for every vertex v the equality P v = D ( v ) + ∑ outward edges at v m e + ∑ inward edges at v − ( P e + m e ) holds. Furthermore, this polyhedral structure descends from S ( D ) to R ( D ) = S ( D ) / S n and | D | = R ( D ) / R .Proof. Set d v = P v and d e = P e . We claim that the points in a cell of S ( D ) can beparametrized by the following two types of continuous data: • the value f ( v ) at a vertex v , as well as • the distance d ( p ei ) of every p ei ∈ P e from 0 ∈ e = [ , l ( e )] .The distances d ( p ei ) immediately determine the p i . In order to reconstruct f (if itexists) we write ∑ p ∈ P e p = ∑ j d e , j x j for points 0 < x < · · · < x r < l ( e ) on e , wherethe positive integers d e , j indicate the number of p ei that are all located at the samepoint x j . The rational function f is then determined by taking the value f ( v ) at theorigin of every edge e = [ , l ( e )] and continuing it piecewise linearly with slope m e until we hit x , at which point we change the slope to d e , + m e until we hit x , wherewe change the slope to d e , + d e , + m e , and so on until we hit the vertex v (cid:48) at theend of e = [ , l ( e )] . So, by continuity, for every edge we obtain the linear condition f ( v (cid:48) ) = f ( v ) + m e x + r ∑ k = (cid:0) m e + k ∑ j = d e , j (cid:1) ( x k + − x k ) == f ( v ) + m e l ( e ) + r ∑ i = d e , i (cid:0) l ( e ) − x i (cid:1) on the parameters of a cell in S ( D ) . This, together with the inequalities 0 < x < · · · < x r < l ( e ) determines the polyhedral structure of a cell in S ( D ) . Note that ourparameters are still overdetermined in the sense that there may be no rational func-tion f such that D + ( f ) = p + . . . + p n ≥ S ( D ) are all discrete and the points within one cellare all parametrized by the distances d ( p ei ) ∈ ( , l ( e )) and the values f ( v ) subjectto these discrete conditions. Therefore S ( D ) is a polyhedral complex that does notdepend on the choice of the orientation of Γ . Bo Lin and Martin Ulirsch
The action of S n on every cell is affine linear and therefore the polyhedral struc-ture descends to R ( D ) . Moreover, the additive group R acts on R ( D ) by adding aconstant to all f ( v ) and therefore the polyhedral structure also descends to | D | . (cid:117)(cid:116) Let Γ be a tropical curve with a fixed minimal model G . As explained in [3, Section5.2], the canonical divisor on Γ is defined to be K Γ = K G = ∑ v ∈ V ( G ) ( h ( v ) + | v | − )( v ) , where | v | denotes the valence of the vertex v . Observe that deg ( K Γ ) = g −
2. The h ( v ) -term in the sum should hereby be thought of as contributing h ( v ) infinitelysmall loops at the vertex v . In fact, given a semistable curve C whose dual graphis G , the canonical divisor is the multidegree of the dualizing sheaf on C (see [2,Remark 3.1]). We recall Definition 1.1 from the introduction. Definition 4.1.
Let g ≥
2. As a set, the tropical Hodge bundle Λ tropg is defined to be Λ tropg = (cid:8) ([ Γ ] , f ) (cid:12)(cid:12) [ Γ ] ∈ M tropg and f ∈ Rat ( Γ ) such that K Γ + ( f ) ≥ (cid:9) and the projective tropical Hodge bundle H tropg as H tropg = (cid:8) ([ Γ ] , D ) (cid:12)(cid:12) [ Γ ] ∈ M tropg and D ∈ | K Γ | (cid:9) The tropical Hodge bundles come with natural projection maps Λ tropg −→ M tropg and H g −→ M tropg given by (cid:0) [ Γ ] , f (cid:1) (cid:55)→ [ Γ ] and (cid:0) [ Γ ] , D (cid:1) (cid:55)→ [ Γ ] , which, in abuse of notation, we bothdenote by π g .In order to understand the structure of the tropical Hodge bundle Λ tropg we con-sider the pullback of Λ tropg and H tropg to (cid:101) M G , defined as (cid:101) Λ G = (cid:8) ([ Γ ] , f ) (cid:12)(cid:12) [ Γ ] ∈ (cid:101) M G and f ∈ Rat ( Γ ) such that K Γ + ( f ) ≥ (cid:9) and (cid:102) H G = (cid:8) ([ Γ ] , D ) (cid:12)(cid:12) [ Γ ] ∈ (cid:101) M G and D ∈ | K Γ | (cid:9) . In analogy with the space S ( D ) , as in Section 3 above, we also set (cid:101) S G = (cid:8) ([ Γ ] , f , p , . . . , p g − ) (cid:12)(cid:12) [ Γ ] ∈ (cid:101) M G and f ∈ Rat ( Γ ) such that K Γ + ( f ) = p + . . . + p g − ≥ (cid:9) . owards a tropical Hodge bundle 9 Proposition 4.2. (i) The action of S g − on S G that permutes the points p , . . . , p g − induces a natural bijection (cid:101) Λ G (cid:39) (cid:101) S G / S g − . (ii) The action of the additive group R = ( R , +) on (cid:101) Λ G , given by adding constantfunctions to f , induces a natural bijection (cid:102) H G (cid:39) (cid:101) Λ G / R . Proof.
The projections (cid:101) S G → (cid:101) M G and (cid:101) Λ G → (cid:101) M G are both invariant under the actionof S g − and R . Therefore our claims follow from the respective identities on thefibers. (cid:117)(cid:116) Let us now recall Theorem 1.2 from the introduction.
Theorem 4.3.
Let g ≥ .(i) The tropical Hodge bundles Λ tropg and H tropg canonically carry the structureof a generalized cone complex.(ii) The dimensions of Λ tropg and H tropg are given by dim Λ tropg = g − and dim H tropg = g − respectively.(iii) There is a proper subdivision of M tropg such that, for all [ Γ ] in the relative inte-rior of a cone in this subdivision, the canonical linear systems | K Γ | = π − g (cid:0) [ Γ ] (cid:1) have the same combinatorial type.Proof (Proof of Theorem 1.2). Part (i): We are going to show that (cid:101) S G canonicallycarries the structure of a cone complex. Then, by Proposition 4.2 above, both (cid:102) H G and (cid:101) Λ G carry the structure of a generalized cone complex.Choose an orientation for each edge e of G , identifying it with the closed inter-val [ , l ( e )] . As in Proposition 3.2 above, we can describe the cells of (cid:101) S G by thefollowing discrete data:(i) a partition of { p , . . . , p g − } into disjoint subsets P e and P v (indexed by ver-tices v ∈ V ( G ) and edges e ∈ E ( G ) ) that tells us on which edge (or at whichvertex) each p i is located,(ii) a total order on each P e , and(iii) the integer slope m e of f at the starting point of e such that for every vertex v the equality d v = h ( v ) − + | v | + ∑ outward edges at v m e + ∑ inward edges at v − ( d e + m e ) holds, where d v = P v and d e = P e . The continuous parameters describing all ele-ments in our cell are given by (i) the values f ( v ) ,(ii) the distances d ( p ei ) of p ei from 0 ∈ [ , l ( e )] , and(iii) the lengths l ( e ) .In order to find the conditions on those parameters, we again write ∑ p ∈ P e p = ∑ d e , j x j for x < · · · < x r . Using this notation we have 0 < x < . . . < x r < l ( e ) asconditions on the d ( p ei ) = x i as well as by the continuity of f : m e x = f ( x ) − f ( v )( m e + d e , )( x − x ) = f ( x ) − f ( x )( m e + d e , + d e , )( x − x ) = f ( x ) − f ( x ) ... (cid:0) m e + r ∑ j = d e , j (cid:1) ( l ( e ) − x r ) = f ( v (cid:48) ) − f ( x r ) . Eliminating the non-parameters f ( x ) , . . . , f ( x r ) we can combine the system ofequations to f ( v (cid:48) ) = f ( v ) + ( m e + d e ) l ( e ) − r ∑ j = d e , j x j . (2)Since these conditions are invariant under multiplying all parameters simultane-ously by elements in R ≥ , every non-empty cell in (cid:101) S G has the structure of a rationalpolyhedral cone.Finally, the natural action of Aut ( G ) on (cid:101) S G , given by φ · (cid:0) [ Γ ] , f , p , . . . , p g − (cid:1) = (cid:0) [ φ ( Γ )] , f ◦ φ − , φ ( p ) , . . . φ ( p g − ) (cid:1) for φ ∈ Aut ( G ) is compatible with both the S g − - and the R -operation. Moreover,given a weighted edge contraction G (cid:48) = G / e of G , the natural map (cid:101) S G (cid:48) (cid:44) → (cid:101) S G iden-tifies (cid:101) S G (cid:48) with the subcomplex of (cid:101) S G given by the condition l ( e ) = Λ tropg = lim −→ (cid:101) Λ G and H tropg = lim −→ (cid:102) H G , where the limits are taken over the category J g as in Section 2 above, carry thestructure of a generalized cone complex.Part (ii): We need to show that the dimension of a maximal-dimensional cone in H g is 5 g −
5. By [7, Proposition 3.2.5 (i)], we have dim M tropg = g − | K Γ | of a point [ Γ ] is at most deg ( K Γ ) = g −
2. This shows that the dimension of H tropg is at most ( g − ) + ( g − ) = g − ( g − ) -dimensional cone in H tropg as follows:Consider the tropical curve Γ max as indicated in Figure 2 and note that it has 2 g − g − owards a tropical Hodge bundle 11 Fig. 2: The tropical curve Γ max with 2 g − g − Lemma 4.4.
Let Γ be a tropical curve with minimal model G = ( V , E ) . Let D be adivisor on Γ such that the support of D is contained in V . Then the combinatorialstructure of | D | is independent of the length of any loop or bridge in G. A proof of Lemma 4.4 is provided below. Lemma 4.4 implies that the combi-natorial structure of | K Γ max | is independent of the edge lengths, so we can choose ageneric chamber. We obtain a divisor D ∈ | K Γ max | as indicated in Figure 3.Fig. 3: The divisor D on Γ max (red).By [22, Proposition 13], the divisor D belongs to a ( g − ) -dimensional face in | K Γ max | . Thus there is a ( g − ) -dimensional cone in H tropg .Part (iii): We use the coordinates described in part (i) above. For every edge e of G with d e = P e = m e l ( e ) = f ( v ) − f ( v (cid:48) ) which is parametrizedby the l ( e ) . So suppose d e > x = d e ∑ rj = d e , j x j . Then (2) can be rewrittenas f ( v (cid:48) ) = f ( v ) + ( m e + d e ) l ( e ) − d e x . Eliminating x from this equation, subject to the condition 0 < x < l ( e ) we obtain ( − d e − m e ) l ( e ) > f ( v ) − f ( v (cid:48) ) m e l ( e ) < f ( v (cid:48) ) − f ( v ) and therefore we obtain that the images of cells in (cid:101) M G are polyhedra. Moreover,the combinatorial type of | K Γ | is independent under scaling all edge lengths with afactor in R > and thus all these polyhedra determine a subdivision of (cid:101) M G such thaton each relatively open cell of this subdivision, the corresponding | K Γ | has the sameset of cells, i.e., the combinatorial type of (cid:101) S G is constant. (cid:117)(cid:116) Proof (of Lemma 4.4).
Suppose e b ∈ E is a bridge in G . Let Γ and Γ be two tropicalcurves with minimal model G such that the length of e in Γ and Γ is l and c · l respectively (where l , c > e ∈ E − { e b } , the lengths of e in Γ and Γ arethe same. It suffices to show that the sets of cells in | D | Γ and | D | Γ are exactly thesame.In Γ , we view the bridge e b as the open interval ( , l ) . For any cell C in | D | Γ ,its data consists of an integer m e b and a partition of nonnegative integers d e b = ∑ rj = d e , j . Suppose a divisor D + Div ( f ) is ∑ rj = d e , j · x j on the bridge e b , where0 < x < · · · < x r ( e ) < l . Here the rational function f is unique up to a translation.So we may assume that the value of f is zero on the endpoint 0 of e b . Then on e b the function f is defined inductively as follows: • For 0 < x ≤ x it is given by f ( x ) = m e b · x . • Given x k < x < x k + for 1 ≤ k ≤ r ( e ) we have f ( x ) = f ( x k ) + (cid:0) m e b + k ∑ j = d e b , j (cid:1) · ( x − x k ) Now on Γ we also view the bridge e b as the open interval ( , c · l ) , with the sameorientation. We construct a rational function f on Γ . First, we define f on thebridge e b inductively as follows: • For 0 < x ≤ cx we define f ( x ) by f ( x ) = m e b · x , • and, given cx k < x < cx k + for 1 ≤ k ≤ r ( e ) , we set f ( x ) = f ( cx k ) + (cid:0) m e b + k ∑ j = d e b , j (cid:1) · ( x − cx k ) Since e b is a bridge in G , the graph G − e b consists of two connected components.We denote them by G and G , where G contains the endpoint 0 of e b and G contains the endpoint cl of e b . For convenience we let owards a tropical Hodge bundle 13 f ( l ) = m e b · x + ( m e b + d e b , ) · x + · · · + ( m e b + r ∑ j = d e b , j ) · ( l − x r ( e ) ) and f ( cl ) = c · f ( l ) . Then we define f on G − e b as follows: f ( x ) = (cid:40) f ( x ) , if x ∈ G ; f ( x ) + f ( cl ) − f ( l ) = f ( x ) + ( c − ) f ( l ) , if x ∈ G . By definition, f and f admit the same data on G i for i = ,
2. In addition, on thebridge e b , both functions admit the integer m e b and the same partition ∑ rj = d e b , j .So f corresponds to a cell C in | D | Γ that is exactly the same as C . For the samereason we can get the cell C from C (just note that l = ( c ) · ( cl ) ). So Lemma 4.4holds for bridges.Suppose e l is a loop in G . In this case almost the same proof works, except thatthe condition f ( l ) = G (cid:48) = G − e l is connected. We have f ( cl ) = f on e l in the same way as on e b , as well as f ( x ) = f ( x ) for all x ∈ G (cid:48) . Thus our claim also holds for loops. (cid:117)(cid:116) In this section we present some computational results on the polyhedral structure oftropical Hodge bundles of small genus. In order to describe all cones in Λ tropg wefirst list all cones in M tropg . Then for each cone, we compute its subdivision by thestructure of | K Γ | . It turns out that already the two cases g = g = Proposition 5.1.
Let Γ be a tropical curve in M trop . Then the combinatorial struc-ture of | K Γ | is uniquely determined by the minimal model G of Γ . In other words, itis independent of the edge lengths in Γ .Proof. There are 7 faces in M trop as in [13, Figure 4]. For 6 faces among them, alledges are loops or bridges, so the claim follows from Lemma 4.4. For the ”thetagraph” G θ , by explicit computation we know that the canonical linear system | K G θ | is always a one-dimensional polyhedral complex with three segments, as in Figure4. (cid:117)(cid:116) The face lattice of Λ trop is visualized in Figure 1 from the introduction. Remark 5.2.
The f -vector of Λ trop is ( , , , , , ) , which is consistent withTheorem 1.2 (ii). The unique 5-dimensional face consists of the ”dumbbell” graphand a triangular cell in | K Γ | . In other words, any divisor in this cell is of the form P + Q , where P and Q are distinct points in the interior of the bridge in the dumbbellgraph. Fig. 4: The polyhedral complex | K G θ | . Its f -vector is ( , ) .When g =
3, the counterpart of Proposition 5.1 is no longer true. One counterex-ample consists of the 6-dimensional cone C (cid:39) R > in M trop parametrizing tropicalcurves whose minimal model G is a complete metric graph K . The following propo-sition characterizes the open chambers of C regarding the structure of | K G | . It is aresult of explicit computations using the algorithm described in [23, Section 2.3]. Proposition 5.3. (i) There are open chambers in C. For all metrics in the samechamber, the canonical linear system | K G | has the same set of cells. In thatcase, the polyhedral complex | K G | always has vertices, edges, and two-dimensional faces ( triangles and quadrilaterals). However, thereare non-isomorphic combinatorial structures of | K G | .(ii) Let M = ( M , M , M , M , M , M ) be a metric. Consider the followingfour -subsets: { M , M , M } , { M , M , M } , { M , M , M } , { M , M , M } . (3) Then M belongs to an open chamber if and only if among the elements of each -subset in (3), the minimum is attained only once.Remark 5.4 (The structure of | K G | for a generic metric). If M belongs to an openchamber, the canonical linear system | K G | always has the 13 vertices in Figure 5.Among them, the 10 labeled vertices are all connected to an extra vertex that is thedivisor K G . The remaining 20 vertices come from 4 copies of a sub-structure (wecall a bat ) attached at D , D , D , and D . Note that some edges in Figure 5 aresubdivided by other vertices in the bats.The 4 distinct combinatorial types of | K G | come from different ways of attach-ing the bats. Since M belongs to an open chamber, the minimum of M , M , M appears only once. Suppose it is M , then the bat at D is attached along the edgestowards D and D , as in Figure 6. Figure 7 shows the divisors D i and D i j . owards a tropical Hodge bundle 15 Fig. 5: The main skeleton of | K G | . Anextra vertex is connected to the 10vertices in red. Fig. 6: Each bat consists of 5 ex-tra vertices (green). This bat appearswhen M < M , M . The divisor D . The divisor D . Fig. 7: Numbers are the coefficients of divisors and the (cid:52) symbols show the equalline segments.
Remark 5.5 (The boundary of the open chambers). The action of the permutationgroup S on the vertices of K induces 4 orbits among the 51 open chambers, withlengths 24, 12, 12, and 3. Each orbit corresponds to a combinatorial type of | K G | .Each open chamber is an open cone in C , defined by homogeneous linear inequal-ities involving M , M , M , M , M , and M . The inequalities are displayedas the covers in a lattice. For example, M covering M means that the inequality M > M holds. M M M M M M M M M M M M Fig. 8: Representatives of two chamber orbits of length 12. M M M M M M Fig. 9: Representatives of a chamberorbit of length 3. M M M M M M Fig. 10: Representatives of a chamberorbit of length 24.
Let k be an algebraically closed field carrying the trivial absolute value. In [1], ex-panding on earlier work (see e.g. [32]), the authors have constructed a natural (con-tinuous) tropicalization map trop M g : M ang −→ M tropg x (cid:55)−→ [ Γ x ] from the non-Archimedean analytic moduli space M ang onto M tropg . Let us recall theconstruction of Γ x : A point x ∈ M ang parametrizes an algebraic curve C over somenon-Archimedean extension K of k . Possibly after a finite extension K (cid:48) | K we canextend C to a stable model C → Spec R (cid:48) over the valuation ring R (cid:48) of K (cid:48) . Denote by G x the weighted dual graph of the special fiber C s of C , whose vertices correspondto the components of C s and in which we have an edge e between two vertices v and v (cid:48) for every node connecting the two corresponding components C v and C v (cid:48) . Thevertex weight function is given by h ( v ) = g ( (cid:101) C v ) , owards a tropical Hodge bundle 17 where (cid:101) C v denotes the normalization of C v . Finally, around every node p e in C s wecan find formal coordinates x and y of C such that xy = t for some element t in thebase. Then the edge length of e is given by l ( e ) = val ( t ) .Denote by H ang the non-Archimedean analytification of the total space of thealgebraic Hodge bundle H g . Proposition 6.1.
There is a natural tropicalization map trop H g : H ang → H tropg thatmakes the diagram H ang trop H g −−−−→ H tropg (cid:121) (cid:121) M ang trop M g −−−−→ M tropg commute. We expect that trop H g is also continuous, but refrain from investigating this ques-tion here, since such an investigation appears to be too technical for the nature of acontribution to this volume. Proof.
An element x ∈ H ang parametrizes a tuple ( C , K C ) consisting of a smoothprojective curve C over a non-Archimedean extension K of k together with a canon-ical divisor on C . Then we may associate to ( C , K C ) the point (cid:0) [ Γ x ] , τ ∗ ( K C ) (cid:1) , where τ ∗ : Div ( C K ) −→ Div ( Γ ) denotes the specialization map constructed in [5, Section 2.3] that is given by push-ing K C forward to the non-Archimedean skeleton of C . As shown in [5, Section 2.3]and the references therein, this is well-defined and the commutativity of the abovediagram is an immediate consequence of the definition. (cid:117)(cid:116) It is well-known that trop g : M ang → M tropg is surjective. By Theorem 1.2 (ii) wehave that dim C H g = g − < g − = dim H tropg and therefore the analogous statement for H g appears to be false. This gives rise tothe following problem. Problem 6.2.
Find a characterization of the locus trop H g ( H ang ) in H tropg , the so-called realizability locus in H tropg .In other words, given a (stable) tropical curve Γ of genus g together with a divisor D that is equivalent to K Γ , find algebraic and combinatorial conditions that ensurethat there is an algebraic curve C over a non-Archimedean field extension K of k together with a canonical divisor (cid:101) D on C such thattrop H g (cid:0) [ C ] , (cid:101) D (cid:1) = (cid:0) [ Γ ] , D (cid:1) . Since trop M g is surjective, we know already that every tropical curve Γ can belifted to a smooth algebraic curve C . In the case of Γ having integer edge lengths l ( e ) we can give a constructive approach to this problem: Consider a special fiber C s over k whose weighted dual graph is G , then apply logarithmically smooth deformationtheory to find a smoothing of C s to a stable family C → Spec R with deformationparameters l ( e ) at each node (see e.g. [21, Proposition 3.38]). If all l ( e ) =
1, wemay alternatively also proceed as in [5, Appendix B].Now let (cid:101) D be a divisor on C that specializes to the given canonical divisor D on Γ . Since deg (cid:101) D = deg D = g −
2, Clifford’s theorem (or alternatively Baker’sSpecialization Lemma [5, Corollary 2.11]) shows that the rank of (cid:101) D is at most g − D is smaller than g − (cid:101) D has rank g −
1, then, by Riemann-Roch, it is a canonical divisor. Sothe realizability problem reduces to finding a lift of the divisor D of rank g − Acknowledgements
This article was initiated during the Apprenticeship Weeks (22 August-2September 2016), led by Bernd Sturmfels, as part of the Combinatorial Algebraic GeometrySemester at the Fields Institute. Both authors would like to acknowledge his input. Thanks arealso due to the Max-Planck-Institute of Mathematics in the Sciences in Leipzig, Germany, for itshospitality. The second author, in particular, would like to thank Diane Maclagan for several dis-cussion related to the topic of this note, as well as the Fields Institute for Research in MathematicalSciences. Finally, many thanks are due to the anonymous referees for several helpful commentsand suggestions.
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