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Towards a tropical Hodge bundle BoLinandMartinUlirsch 7 1 0 2 n a J Abstract The moduli space Mtrop of tropical curves of genus g is a generalized 6 g 1 conecomplexthatparametrizesmetricvertex-weightedgraphsofgenusg.Foreach such graph Γ, the associated canonical linear system |K | has the structure of a Γ ] polyhedral complex. In this article we propose a tropical analogue of the Hodge G bundle on Mtrop and study its basic combinatorial properties. Our construction is g A illustratedwithexplicitcomputationsandexamples. . h t a m 1 Introduction [ 1 Letg≥2anddenotebyM themodulispaceofsmoothalgebraiccurvesofgenus g v g.TheHodgebundleΛ isavectorbundleonM whosefiberoverapoint[C]inM g g g 5 isthevectorspaceH0(C,ω )ofholomorphicdifferentialsonC.Onecanthinkof 8 C thetotalspaceofΛ asparametrizingpairs(C,ω)consistingofasmoothalgebraic 3 g 4 curveandadifferentialω onC.SinceforeverycurveCthecanonicallinearsystem 0 |K |canbeidentifiedwiththeprojectivizationP(cid:0)H0(C,ω )(cid:1),thetotalspaceofthe C C 1. projectivizationHg=P(Λg)ofΛgparametrizespairs(C,D)consistingofasmooth 0 algebraiccurveC andacanonicaldivisorDonC;itisreferredtoastheprojective 7 Hodgebundle.Letπ :C →M betheuniversalcurveonM .WemaydefineΛ g g g g 1 formallyasthepushforwardπ ω oftherelativedualizingsheafω onC overM . ∗ g g g g : v TheHodgebundleisoffundamentalimportancewhendescribingthegeometry i ofM .Forexample,itsChernclasses,theso-calledλ-classes,formanimportant X g collectionofelementsinthetautologicalringonM (see[31]foranintroductory g r a BoLin Department of Mathematics, University of California, Berkeley, Berkeley, CA 94720, e-mail: [email protected] MartinUlirsch FieldsInstituteforResearchinMathematicalSciences,UniversityofToronto,222CollegeStreet, Toronto,OntarioM5T3J1e-mail:[email protected] 1 2 BoLinandMartinUlirsch survey).TheHodgebundleadmitsanaturalstratificationbyprescribingcertainpole and zero orders (m ,...,m ) such that m +...+m =2g−2 and the study of 1 n 1 n naturalcompactificationsofthesecomponentshasrecentlyseenansurgefromboth theperspectiveofalgebraicgeometryaswellasfromTeichmu¨llertheory(seee.g. [4]). Intropicalgeometry,thenaturalanalogueofM isthemodulispaceMtrop that g g parametrizes isomorphism classes [Γ] of stable tropical curves Γ of genus g. In Section2belowwearegoingtorecalltheconstructionofthismodulispace.Inpar- ticular,wearegoingtoseehowthismodulispacenaturallyadmitsthestructureofa generalizedconecomplexwhoseconesareinanaturalorder-reversingone-to-one correspondencewiththeboundarystrataoftheDeligne-Mumfordcompactification M ofM (see[1]aswellasSection2belowfordetails). g g Wealsoreferthereaderto[19],[20],[24],and[25]forthetheoryingenusg=0 (with marked points), to [7], [10], [13], [16], and [32] for its connections to the tropicalTorellimap,aswellasto[1],[11],[12],and[30]forconnectionsofMtrop g (and some of its variants) to non-Archimedean analytic geometry and to [14] and [15]foranin-depthstudyofthetopologyofMtrop.We,inparticular,highlightthe g,n twosurveypapers[8]and[9]. LetΓ be a tropical curve. We denote by K the canonical divisor onΓ and by Γ Rat(Γ)thegroupofpiecewiseintegerlinearfunctionsonΓ (seeSection3belowfor details).Inthisnoteweproposetropicalanaloguesoftheaffineandtheprojective Hodgebundle,andstudytheirbasiccombinatorialproperties. Definition1.1.Asaset,thetropicalHodgebundleΛtropisgivenas g Λgtrop=(cid:8)([Γ],f)(cid:12)(cid:12)[Γ]∈Mgtropand f ∈Rat(Γ)suchthatKΓ +(f)≥0(cid:9) andtheprojectivetropicalHodgebundleHtropisgivenas g Hgtrop=(cid:8)([Γ],D)(cid:12)(cid:12)[Γ]∈MgtropandD∈|KΓ|(cid:9). (cid:0) (cid:1) (cid:0) (cid:1) The associations [Γ],f (cid:55)→[Γ] and [Γ],D (cid:55)→[Γ] define natural projection mapsΛtrop −→Mtrop and Htrop −→Mtrop, which, in a slight abuse of notation, g g g g wedenotebothbyπ . g In [18, 22, 26] the authors describe a structure of a polyhedral complex on the linear system |D| associated to a divisor D on a tropical curveΓ; we are going to reviewthisdescriptioninSection3below.Wealso,inparticular,highlightthefirst authors[23],wherehepresentsalgorithmsforcomputingthispolyhedralcomplex. Ourmainresultisthefollowing: Theorem1.2.Letg≥2. (i)The tropical Hodge bundle Λtrop and the projective tropical Hodge bundle g Htropcarrythestructureofageneralizedconecomplex. g (ii)ThedimensionsofΛtropandHtroparegivenby g g dimΛtrop=5g−4 and dimHtrop=5g−5 g g TowardsatropicalHodgebundle 3 respectively. (iii)There is a proper subdivision of Mtrop such that, for all [Γ] in the relative g interiorofaconeinthissubdivision,thecanonicallinearsystems |K |=π−1(cid:0)[Γ](cid:1) Γ g havethesamecombinatorialtype. We are going to refer to this subdivision of Mtrop as the wall-and-chamber de- g compositionofMtrop.Ingeneral,thegeneralizedconecomplexesΛtrop andHtrop g g g are not equi-dimensional. So Theorem 1.2 (ii) really states that the dimension of a maximal-dimensional cone inΛtrop (or Htrop) has dimension 5g−4 (or 5g−5 g g respectively). AsafirstexamplewereferthereadertoFigure1below,whichdepictstheface latticeofthetropicalHodgebundleinthecaseg=2. Fig.1:ThefacelatticeofHtrop.Thenumbersingreenarethepositiveh(v)andthe 2 numbersinblackdenotecoefficientsgreaterthan1inthedivisors. Letusgiveaquickoutlineofthecontentsofthiscontribution.InSection2we recall the construction of the moduli space Mtrop of stable tropical curves and in g Section3thepolyhedralstructureoflinearsystemsontropicalcurvesrespectively. InSection4weproveTheorem1.2bydescribingthepolyhedralstructuresofboth Λtrop and Htrop simultanously. Section 5 contains a selection of explicit (some- g g timespartial)calculationsofthepolyhedralstructureofHtropinsomesmallgenus g cases. Finally, in Section 6 we describe a natural tropicalization procedure for the 4 BoLinandMartinUlirsch projectivealgebraicHodgebundlevianon-Archimedeananalyticgeometryandex- hibitanaturalrealizabilityproblem. 2 Modulioftropicalcurves AtropicalcurveisafinitemetricgraphΓ (withafixedminimalmodelG)together withagenusfunctionh:V(G)→Z .ThegenusofΓ (orofG)isdefinedtobe ≥0 g(Γ)=g(G)=b (G)+ ∑ h(v) 1 v∈V(G) whereb (G)denotestheBettinumberofG.Intheabovesumoneshouldthinkof 1 the vertex-weight terms as the contributions of h(v) infinitesimally small loops at every vertex v. We say a tropical curve Γ (or the graph G) is stable, if for every vertexv∈V(G)wehave 2h(v)−2+|v|>0, (1) where|v|denotesthevalenceofGatv. Definition2.1.Asaset,themodulispaceMtropofstabletropicalcurvesofgenusg g isgivenas Mtrop=(cid:8)isomorphismclasses[Γ]ofstabletropicalcurvesofgenusg(cid:9). g Let us now recall from [1] the description of Mtrop as a generalized extended g conecomplex. Proposition2.2([1]Section4).ThemodulispaceMtrop carriesthestructureofa g generalizedrationalpolyhedralconecomplexthatisequi-dimensionalofdimension 3g−3. First, recall that a morphism τ →σ between rational polyhedral cones is said to be a face morphism, if it induces an isomorphism onto a face of σ. Note that we explicitly allow the class of face morphisms to include all isomorphisms. A generalized(rationalpolyhedral)conecomplexisatopologicalspaceΣ thatarises asacolimitofafinitediagramoffacemorphisms(see[1,Section2]and[29,Section 3.5]fordetails). In order to understand this structure on Mtrop, we observe that it is given as a g colimit Mgtrop=limM(cid:101)G, → ofrationalpolyhedralconesM(cid:101)GtakenoveracategoryJg.Letusgointosomemore detail: 1. ThecategoryJ isdefinedasfollows: g TowardsatropicalHodgebundle 5 • itsobjectsarestablevertex-weightedgraphs(G,h)ofgenusg,and • itsmorphismsaregeneratedbyweightededgecontractionsG→G/eforan edgeeofGaswellasbytheautomorphismsofall(G,h). Hereaweightededgecontractionc:G→G/eisanedgecontractionsuchthat foreveryvertexvinG/ewehave g(cid:0)c−1(v)(cid:1)=h(v). 2. Moreover,foreverygraphGwedenoteby M(cid:101)G=R≥E(0G) theparameterspaceofallpossibleedgelengthsonG. 3. The association G(cid:55)→M(cid:101)G defines a contravariant functor Jg →RPCZ from Jg to the category of rational polyhedral cones. It associates to a weighted edge contraction G→G/e the embedding of the corresponding face of M(cid:101)trop and G to an automorphism of G the automorphism of M(cid:101)G that permutes the entries correspondingly. Wenoteherebythatwehaveadecompositionintolocallyclosedsubsets Mtrop=(cid:71)RE(G)/Aut(G), g >0 G where the disjoint union is taken over the objects in J , i.e. over all isomorphism g classesofstablefinitevertex-weightedgraphsGofgenusg. Example2.3([13] Theorem 2.12). For a d-dimensional cone complex C, its f- vector is defined as (f ,f ,...,f ), where f is the number of i-dimensional cones 0 1 d i inC. The 12-dimensional moduli space Mtrop has 4555 cells; its f-vector is given 5 by f(Mtrop)=(1,3,11,34,100,239,492,784,1002,926,632,260,71). 5 Remark2.4.Earlierapproaches,suchas[7],[8],[10],[13],[16],and[32],usedto refer to the structure of a generalized cone complex as a stacky fan. Since there is acloselyrelated,butnotequivalent,notionofthesamenameinthetheoryoftoric stacks we prefer to follow the terminology of generalized cone complexes intro- ducedin[1]. 3 Linearsystemsontropicalcurves LetΓ beatropicalcurve.AdivisoronΓ isafiniteformalZ-linearsum D=∑a p , i i i 6 BoLinandMartinUlirsch over points p in Γ, i.e. D is an element in the free abelian group Div(Γ) on the i pointsofΓ.Thedegreedeg(D)ofadivisorD=∑iaipi isdefinedtobetheinteger ∑iai.WesayD=∑iaipiiseffective,ifai≥0foralli. A rational function onΓ is a continuous function f :Γ →R whose restriction toeveryedgeisapiecewiselinearintegralaffinefunction.Givenarationalfunction f onΓ asaboveandapoint p∈Γ,theorderord (f)of f at pisdefinedtobethe p sumoftheoutgoingslopesof f emanatingfrom p.Observethatord (f)isequalto p zeroforallbutfinitelymanypoints p∈Γ.Sowehaveamap (.):Rat(Γ)−→Div(Γ) f (cid:55)−→(f)=∑ord (f)·p. p p Divisorsoftheform(f)forafunction f ∈Rat(Γ)formasubgroupPDiv(Γ)of Div(Γ)andarereferredtoastheprincipaldivisorsonΓ.TwodivisorsDandD(cid:48)on Γ aresaidtobeequivalent(writtenasD∼D(cid:48)),ifD−D(cid:48)∈PDiv(Γ),i.e.ifthereis arationalfunction f ∈Rat(Γ)suchthatD+(f)=D(cid:48).Notethatthecontinuityof f impliesthatdeg(f)=0. Letusnowdefinethemainplayersofthisgame: Definition3.1.LetDbeadivisorofdegreenonatropicalcurveΓ. 1.DenotebyR(D)theset (cid:8) (cid:12) (cid:9) R(D)= f ∈Rat(Γ)(cid:12)D+(f)≥0 . For f ∈R(D),thedivisorD+(f)issupportedindeg(D+(f))=deg(D)=n points(countedwithmultitplicity).Wemaythereforedefine: (cid:8) (cid:12) S(D)= (f,p1,...,pn)(cid:12)f ∈Rat(Γ)and p1,...,pn∈Γ (cid:9) suchthatD+(f)=p +...+p ≥0 . 1 n 2.Thelinearsystem|D|associatedtoDistheset |D|=(cid:8)D(cid:48)∈Div(Γ)(cid:12)(cid:12)D(cid:48)≥0andD∼D(cid:48)(cid:9). Observe that R(D)=S(D)/S , where the symmetric group S acts on S(D) by n n permutationofthepoints p ,...,p .Moreover,theadditivegroupR=(R,+)op- 1 n erates on R(D) by adding a constant function and, taking the quotient under this operation,weobtainthat R(D)/R=|D|, since(f)=0ifandonlyif f isaconstantfunctiononΓ. ThespacesS(D),R(D),and|D|areknowntocarrythestructureofapolyhedral complex(seee.g.[26]or[18]).Thefollowingpropositionisamoredetailedversion of[18,Lemma1.9]. TowardsatropicalHodgebundle 7 Proposition3.2.Given a divisor D on a tropical curveΓ, the space S(D) has the structure of a polyhedral complex. Choose an orientation for each edge e of Γ, identifyingitwiththeopeninterval[0,l(e)].ThenthecellsofS(D)canbedescribed bythefollowing(discrete)data: (i)apartitionof{p ,...,p }intodisjointsubsetsP andP (indexedbyv∈V(G) 1 n e v andedgese∈E(G))thattellsusonwhichedge(oratwhichvertex)every p i islocated, (ii)atotalorderoneachP,and e (iii)theoutgoingslopem ∈Zof f atthestartingpointofe e suchthatforeveryvertexvtheequality #P =D(v)+ ∑ m + ∑ −(#P +m ) v e e e outwardedgesatv inwardedgesatv holds.Furthermore,thispolyhedralstructuredescendsfromS(D)toR(D)=S(D)/S n and|D|=R(D)/R. Proof. Setd =#P andd =#P.WeclaimthatthepointsinacellofS(D)canbe v v e e parametrizedbythefollowingtwotypesofcontinuousdata: • thevalue f(v)atavertexv,aswellas • thedistanced(pe)ofevery pe∈P from0∈e=[0,l(e)]. i i e Thedistancesd(pe)immediatelydeterminethe p.Inordertoreconstruct f (ifit i i exists)wewrite∑p∈Pep=∑jde,jxj forpoints0<x1<···<xr<l(e)one,where the positive integers d indicate the number of pe that are all located at the same e,j i pointx .Therationalfunction f isthendeterminedbytakingthevalue f(v)atthe j originofeveryedgee=[0,l(e)]andcontinuingitpiecewiselinearlywithslopem e untilwehitx ,atwhichpointwechangetheslopetod +m untilwehitx ,where 1 e,1 e 2 we change the slope to d +d +m , and so on until we hit the vertex v(cid:48) at the e,2 e,1 e endofe=[0,l(e)].So,bycontinuity,foreveryedgeweobtainthelinearcondition r k f(v(cid:48))= f(v)+m x +∑(cid:0)m +∑d (cid:1)(x −x )= e 1 e e,j k+1 k k=1 j=1 r (cid:0) (cid:1) = f(v)+m l(e)+∑d l(e)−x e e,i i i=1 on the parameters of a cell in S(D). This, together with the inequalities 0<x < 1 ···<x <l(e)determinesthepolyhedralstructureofacellinS(D).Notethatour r parametersarestilloverdeterminedinthesensethattheremaybenorationalfunc- tion f suchthatD+(f)=p +...+p ≥0andwhichalsofulfillsalloftheabove 1 n inequalities;inthiscaseweobtainanemptycell. TheconditionsonthecellsofS(D)arealldiscreteandthepointswithinonecell are all parametrized by the distances d(pe)∈(0,l(e)) and the values f(v) subject i tothesediscreteconditions.ThereforeS(D)isapolyhedralcomplexthatdoesnot dependonthechoiceoftheorientationofΓ. 8 BoLinandMartinUlirsch TheactionofS oneverycellisaffinelinearandthereforethepolyhedralstruc- n ture descends to R(D). Moreover, the additive group R acts on R(D) by adding a constanttoall f(v)andthereforethepolyhedralstructurealsodescendsto|D|. (cid:116)(cid:117) 4 StructureofthetropicalHodgebundle LetΓ beatropicalcurvewithafixedminimalmodelG.Asexplainedin[3,Section 5.2],thecanonicaldivisoronΓ isdefinedtobe K =K = ∑ (2h(v)+|v|−2)(v), Γ G v∈V(G) where|v|denotesthevalenceofthevertexv.Observethatdeg(K )=2g−2.The Γ h(v)-term in the sum should hereby be thought of as contributing h(v) infinitely small loops at the vertex v. In fact, given a semistable curveC whose dual graph is G, the canonical divisor is the multidegree of the dualizing sheaf onC (see [2, Remark3.1]).WerecallDefinition1.1fromtheintroduction. Definition4.1.Letg≥2.Asaset,thetropicalHodgebundleΛtropisdefinedtobe g Λgtrop=(cid:8)([Γ],f)(cid:12)(cid:12)[Γ]∈Mgtropand f ∈Rat(Γ)suchthatKΓ +(f)≥0(cid:9) andtheprojectivetropicalHodgebundleHtropas g Hgtrop=(cid:8)([Γ],D)(cid:12)(cid:12)[Γ]∈MgtropandD∈|KΓ|(cid:9) ThetropicalHodgebundlescomewithnaturalprojectionmaps Λtrop−→Mtrop and H −→Mtrop g g g g (cid:0) (cid:1) (cid:0) (cid:1) givenby [Γ],f (cid:55)→[Γ]and [Γ],D (cid:55)→[Γ],which,inabuseofnotation,weboth denotebyπ . g trop InordertounderstandthestructureofthetropicalHodgebundleΛ wecon- g siderthepullbackofΛgtropandHgtroptoM(cid:101)G,definedas (cid:8) (cid:12) (cid:9) Λ(cid:101)G= ([Γ],f)(cid:12)[Γ]∈M(cid:101)Gand f ∈Rat(Γ)suchthatKΓ +(f)≥0 and H(cid:102)G=(cid:8)([Γ],D)(cid:12)(cid:12)[Γ]∈M(cid:101)GandD∈|KΓ|(cid:9). InanalogywiththespaceS(D),asinSection3above,wealsoset (cid:8) (cid:12) S(cid:101)G= ([Γ],f,p1,...,p2g−2)(cid:12)[Γ]∈M(cid:101)Gand f ∈Rat(Γ) (cid:9) suchthatK +(f)=p +...+p ≥0 . Γ 1 2g−2 TowardsatropicalHodgebundle 9 Proposition4.2. (i)TheactionofS onS thatpermutesthepointsp ,...,p 2g−2 G 1 2g−2 inducesanaturalbijection Λ(cid:101)G(cid:39)S(cid:101)G/S2g−2. (ii)TheactionoftheadditivegroupR=(R,+)onΛ(cid:101)G,givenbyaddingconstant functionsto f,inducesanaturalbijection H(cid:102)G(cid:39)Λ(cid:101)G/R. Proof. TheprojectionsS(cid:101)G→M(cid:101)GandΛ(cid:101)G→M(cid:101)Garebothinvariantundertheaction of S and R. Therefore our claims follow from the respective identities on the 2g−2 fibers. (cid:116)(cid:117) LetusnowrecallTheorem1.2fromtheintroduction. Theorem4.3.Letg≥2. (i)The tropical Hodge bundlesΛtrop and Htrop canonically carry the structure g g ofageneralizedconecomplex. (ii)ThedimensionsofΛtropandHtroparegivenby g g dimΛtrop=5g−4 and dimHtrop=5g−5 g g respectively. (iii)ThereisapropersubdivisionofMtropsuchthat,forall[Γ]intherelativeinte- g riorofaconeinthissubdivision,thecanonicallinearsystems|K |=π−1(cid:0)[Γ](cid:1) Γ g havethesamecombinatorialtype. Proof (Proof of Theorem 1.2). Part (i): We are going to show that S(cid:101)G canonically carries the structure of a cone complex. Then, by Proposition 4.2 above, both H(cid:102)G andΛ(cid:101)Gcarrythestructureofageneralizedconecomplex. ChooseanorientationforeachedgeeofG,identifyingitwiththeclosedinter- val [0,l(e)]. As in Proposition 3.2 above, we can describe the cells of S(cid:101)G by the followingdiscretedata: (i) apartitionof{p ,...,p }intodisjointsubsetsP andP (indexedbyver- 1 2g−2 e v ticesv∈V(G)andedgese∈E(G))thattellsusonwhichedge(oratwhich vertex)each p islocated, i (ii) atotalorderoneachP,and e (iii) theintegerslopem of f atthestartingpointofe e suchthatforeveryvertexvtheequality d =2h(v)−2+|v|+ ∑ m + ∑ −(d +m ) v e e e outwardedgesatv inwardedgesatv holds,whered =#P andd =#P.Thecontinuousparametersdescribingallele- v v e e mentsinourcellaregivenby 10 BoLinandMartinUlirsch (i) thevalues f(v), (ii) thedistancesd(pe)of pefrom0∈[0,l(e)],and i i (iii) thelengthsl(e). In order to find the conditions on those parameters, we again write ∑ p= p∈Pe ∑de,jxj forx1<···<xr.Usingthisnotationwehave0<x1<...<xr <l(e)as conditionsonthed(pe)=x aswellasbythecontinuityof f: i i m x = f(x )−f(v) e 1 1 (m +d )(x −x )= f(x )−f(x ) e e,1 2 1 2 1 (m +d +d )(x −x )= f(x )−f(x ) e e,1 e,2 3 2 3 2 . . . r (cid:0)m +∑d (cid:1)(l(e)−x )= f(v(cid:48))−f(x ). e e,j r r j=1 Eliminating the non-parameters f(x ),...,f(x ) we can combine the system of 1 r equationsto r f(v(cid:48))= f(v)+(m +d )l(e)−∑d x . (2) e e e,j j j=1 Since these conditions are invariant under multiplying all parameters simultane- ouslybyelementsinR≥0,everynon-emptycellinS(cid:101)Ghasthestructureofarational polyhedralcone. Finally,thenaturalactionofAut(G)onS(cid:101)G,givenby φ·(cid:0)[Γ],f,p ,...,p (cid:1)=(cid:0)[φ(Γ)],f◦φ−1,φ(p ),...φ(p )(cid:1) 1 2g−2 1 2g−2 forφ ∈Aut(G)iscompatiblewithboththeS -andtheR-operation.Moreover, 2g−2 givenaweightededgecontractionG(cid:48)=G/eofG,thenaturalmapS(cid:101)G(cid:48) (cid:44)→S(cid:101)G iden- tifies S(cid:101)G(cid:48) with the subcomplex of S(cid:101)G given by the condition l(e)=0 in the above coordinates. Thereforewecanconcludethatboth Λgtrop=limΛ(cid:101)G and Hgtrop=limH(cid:102)G, −→ −→ where the limits are taken over the category J as in Section 2 above, carry the g structureofageneralizedconecomplex. Part(ii):Weneedtoshowthatthedimensionofamaximal-dimensionalconein H is5g−5.By[7,Proposition3.2.5(i)],wehavedimMtrop=3g−3and,by[23, g g Corollary7],thedimensionofthefiber|K |ofapoint[Γ]isatmostdeg(K )=2g− Γ Γ 2.ThisshowsthatthedimensionofHtropisatmost(3g−3)+(2g−2)=5g−5. g In addition we now exhibit a (5g−5)-dimensional cone in Htrop as follows: g ConsiderthetropicalcurveΓ asindicatedinFigure2andnotethatithas2g−2 max verticesand3g−3edges.