Functorial Tropicalization of Logarithmic Schemes

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References hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This ahmtc ujc Classification Subject Mathematics > g Y lsial seeg [ e.g. (see Classically . se[ (see 0 hs oyerlgoer upiigyotncrepnst h algebraic the to corresponds often surprisingly geometry polyhedral whose , ...... 15 oodlembedding toroidal n [ and ] 31 n [ and ] Y 34 osatcoefficients constant togydpnso h hsnembedding chosen the on depends strongly )o the or ]) 32 37 40 piay;1G2 20M14. 14G22; (primary(; 14T05 )t osrc aua rpclzto a associated map tropicalization natural a construct to ]) 22 atnUlirsch Martin ,[ ], Introduction Y Abstract ,[ ], 29 Contents umte xlsvl oteLno ahmtclSociety Mathematical London the to exclusively Submitted noasial oi variety toric suitable a into 23 ntesneo [ of sense the in ,ad[ and ], oi degenerations toric ...... ,ad[ and ], X hti oal ffiietp vratrivially a over type finite of locally is that 41 24 ...... ) nodrt en h tropicalization the define to order in ]), ouispaces moduli ) omnfaueo hs varieties these of feature common A ]). Y ...... plhda hdw nw as known shadow” ”polyhedral a 33 ]. rsn nteGross-Siebert the in arising M X g,n doi:10.1112/0000/000000 n,i eea,the general, in and, , fstable of , Y 7 n [ and ] → X n -marked . 8 25 23 18 30 32 )and ]) 7 2 9 Author Manuscript Hom( | 1.1. suggestions several of for University referee(s) the article. anonymous this and the of carried Jerusalem, readability to University, been the due improved have Hebrew significantly are research of that thanks this hospitality Lorscheid, Particular of the Oliver Parts Regensburg. Dhruv enjoying Temkin. Huszar, Rabinoff, Michael while Joseph Alana and out Nicaise, Talpo, Johannes Gross, Jeffrey to Molcho, Fantini, Mattia Andreas order Samouil Lorenzo Ranganathan, heavily Marcus, Gibney, in with to fans Steffen discussions fans Angela as Artin Maclagan, from Kato to Diane well Giansiracusa, profited related of as also Noah ideas use Payne, author whose and the The Sam Wise, work. author Jonathan and this to the Gubler influenced and to Walter maps, suggested to tropicalization originally define due who also Thuillier, are Amaury Thanks encouragement. and Acknowledgements Kajiwara well- of the sense with the construction in our varieties of toric [ comparison of) (see main (subvarieties Payne a our and prove for with and tropicalization concerned map projective embedded tropicalization is the the known finally, construct of 7 , we subsets 6 Section open in Section torus-invariant results. In the structures examples. using explicit logarithmic in differences as used their line of functors explains analytification theory and non-Archimedean text different the two this the to introduces originally [ 5 introduction as (see Section theory. quick fan, Kato-Fontaine-Illusie Kato of a Section a sense gives In of the spaces. notion 4 monoidal the Section locally construction, complexes. our of [ of theory in heart the introduces technical and proposed 2 the cones Section available. introduce and made we that monoids been 3 developments first of has further notions article discuss basic this after and the or results, parallel our in appeared of have statements precise provide [ background, (see algebraic variety toric of a into space [ maps (see stable moduli covers logarithmic admissible the of [ of Deligne-Knudsen-Mumford space (see curves the moduli skeleton from the the coming as structure such to toroidal [ the isomorphic (see to compactification is respect the curves with curves tropical stable [ sense In of the spaces. (in schön variety moduli a of for criterion a derive we techniques [ these of Using below). [ 1.2 Thuillier’s recovers and of it skeleton non-Archimedean that the and onto morphisms logarithmic to 34 of 2 Page . | let , Let support constant his for Abramovich Dan to gratitude his express to like would author The historical the of overview an give we 1 Section In article: this of outline short a give now us Let geometry tropical the in lie article this in developed techniques the of applications main The 46 itrclbcgon:Toiaiaino oi oi aite,adtria embeddings toroidal and varieties, toric tori, of Tropicalization background: Historical N, below). 1.3 Corollary (see tropicalization faithful a admit to ]) k N Z hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This eafil hti noe iha(osbytiil o-rhmda bouevalue absolute non-Archimedean trivial) (possibly a with endowed is that field a be eafiieygnrtdfe bla ru fdimension of group abelian free generated finitely a be swl as well as ) 11 32 o h aeo genus of case the for ] 29 ,adcneti ihteter f(xedd ainlplhda cone polyhedral rational (extended) of theory the with it connect and ], n [ and ] h 1 15 ,. ., 1. ,a naceyia eut h uhr hwta h ouispace moduli the that show authors the result, archetypical an as ], i n [ and ] 41 vriwadsaeeto h anresults main the of statement and Overview o h ult arn between pairing duality the for 1.3. Corollary of proof a and ]) 34 ) iia eut aeapae o te ouispaces, moduli other for appeared have results Similar ]). g n [ and 0 = X ATNULIRSCH MARTIN 31 1.1 Theorems (see case smooth logarithmicallly the in )adsoshwKt asntrlyaiei this in arise naturally fans Kato how shows and ]) 50 45 for ] ]). 12 g ) h ouisaeo egtdstable weighted of space moduli the ]), ≥ ) n h ouisaeo rational of space moduli the and 0), N 47 and togdfrainretraction deformation strong ] n M n write and , . M o t dual its for Author Manuscript n [ and hc sas nw sthe as known also is which map moment analytic Archimedean map tropicalization the and structure, monoid additive natural the with endowed natrsivratoe ffiesubset affine i open torus-invariant a on [ Thuillier all for sends trop( image The [ continuous Following natural space complex. analytic polyhedral non-Archimedean rational the a [ of and structure the with endowed torus k uhta h diagram the that such group from all for o cone a for subset affine open torus-invariant eemndb ,adtie osrcino hc a efudi [ in found be can which of construction detailed a ∆, by determined [ to reader the in refer ∆ we a fan varieties to polyhedral map rational tropicalization a above by the of extension continuous scmuaieadteiaeof image the and commutative is ∆ suiul eemndb h condition the by determined uniquely is , ...Spoethat Suppose 1.1.3. 1.1.1. ...Kjwr [ Kajiwara 1.1.2. h ooano h rpclzto a saprilcompactification partial a is map tropicalization the of codomain The sgvnby given is 48 X 27 T hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This x T m i s etos11ad1.2]). and 1.1 Sections , ,oeo h aywy fdfiigTrop( defining of ways many the of one ], ∈ Spec = i ∈ nothe onto ∈ 47 U on σ S M Tropicalization σ an eto ]cntut lsl eae togdfrainretraction deformation strong related closely a constructs 2] Section , σ ∈ X rma lentv on fve,oemyas hn ftrop of think also may one view, of point alternative an From . where , oteeeettrop element the to ,tecompactification the ∆, k i UCOILTOIAIAINO OAIHI SCHEMES LOGARITHMIC OF TROPICALIZATION FUNCTORIAL x [ y[ By . M o-rhmda skeleton non-Archimedean fapoint a of ) rpclzto map tropicalization ustTrop( subset a ] χ 29 47 m eto ]ad neednl,Pye[ Payne independently, and, 1] Section , rpsto .]teei aua embedding natural a is there 2.9] Proposition , | eoe h hrce in character the denotes . | sapoesta soitst lsdsubset closed a to associates that process a is aoia compactifcation canonical stetiilaslt au,i.e. value, absolute trivial the is σ x Hom( = 13 ∈ U h S N trop ∆ trop( T n [ and ] σ trop Y ( R ( an se[ (see X ⊆ X Spec = x trop Y o tnadntto n eea akrudo toric on background general and notation standard For . X in ) y   ) ie yaseminorm a by given , ∆ in ) an an i p S ∆ x U ∈ N ( σ ) x 18 σ : 29 m , : N Hom( ( : blw soitdto associated below) 5 Section (see , −−−−→ −−−−→ )( X σ U Spec = R N T trop R ]. s n [ and ] sgvnb Hom( by given is ) k S σ i ≥ i an an R p = ) [ = S 0 ∆ steclosure the is (∆) = ( ∆ −→ ) X σ S −→ −→ N ⊆ − of ] σ − of ) N , k 41 ⊗ S k log Y X Hom( R log [ [ R N N S X ( M y   sb aigi ob h rjcinof projection the be to it taking by is ) hti eemndby determined is that ) R i X eak33 swl s[ as well as 3.3] Remark , (∆) X σ R R ftecone the of

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σ x x X , T tropicalization R | a trcvariety -toric o cone a for , | ) | . | S o all for 1 = , x 41 σ f∆in ∆ of ∆ : σ , eto ]dfieanatural a define 3] Section , k R Hom( = [ ,where ), M ] Y 44 → i σ ftesltalgebraic split the of ∆ m eto ] o a For 3]. Section , a 17 of X R ∈ N S, : ∈ ∈ S swl s[ as well as ] Y R Y = ,teiaeof image the ∆, R extending 10 R k ∆.Nt that Note (∆). M N ( hc a be can which , ∗ ≥ X = X into S eto 4.1] Section , nti case this In . R ∆ . 34 of 3 Page 0 σ ∆ defined (∆) ) ∆ of (∆) ). R = , sa as → {∞} ∪ N σ R N ∨ | R i a via . ∩ non- | (∆) 26 N M on is R ] Author Manuscript tutr fa of structure fields. [ closed by algebraically but over étale neighborhoods, of instead formal ae h nue diagram induced the makes [ of of below). 4.3 Section (see embedding association The commute. over type finite of hold: to have theorems two following the precise, be To properties: trop two map following tropicalization continuous extension natural canonical a construct to and associate we article 1.2. self-intersection. without embeddings toroidal denote [ to of end Σ the complex in cone polyhedral rational the has the onto torus algebraic [ retraction deformation big Thuillier strong a actions a of obtains torus and embedding formal varieties open toric Using for the variety. construction to toric isomorphic a étale locally into is that embedding 34 of 4 Page n[ In ups o that now Suppose hoe 1.2. Theorem 1.1. Theorem morphisms. trop logarithmic case to smooth respect logarithmically with the functorial In is – map tropicalization The – Let that now Suppose 1.1.4. X 47 oself-intersection no ..teei homeomorphism a is there i.e. , rpclzto flgrtmcschemes logarithmic of Tropicalization eto ]frtebscter ftria medns n[ In embeddings. toroidal of theory basic the for 3] Section , 1 X baoih aoao n an xli how explain Payne and Caporaso, Abramovich, ] hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This o-rhmda skeleton non-Archimedean eafieadstrtdlgrtmcshm oal ffiietp over type finite of locally scheme logarithmic saturated and fine a be 33 eeaie xeddcn complex cone extended generalized eto .] oeta te uhr loueteadjectives the use also authors other that Note 2.1]. Section , k morphism A If nue morphism a induces X Σ X X slgrtmclysot over smooth logarithmically is ntetriooyo [ of terminology the in X o u ento ob esnbew eur trop require we reasonable be to definition our For . slgrtmclysot,then smooth, logarithmically is a f 7→ eeaie oecomplex cone generalized Σ( S X f ( f 0 : X ) S , X → J ) f ( sfntra in functorial is ( p X i X ATNULIRSCH MARTIN X X → X X p X y   0 i : of ) Σ( ) : X i soitdt h oodlebdiga constructed as embedding toroidal the to associated Σ satria medn,ie noe n dense and open an i.e. embedding, toroidal a is X f X 0 X −−−−→ −−−−→ i ) X trop ffieadstrtdlgrtmcshmslocally schemes logarithmic saturated and fine of 33 trop −→ : X ∼ J eoesTulirsrtato map. retraction Thuillier’s recovers −→ erfrt [ to refer We . ∼ Σ 16 X ,then ], i X X X S 0 eto ]bt prahsaeequivalent are approaches both 2] Section , Σ X ( → X Σ X f Σ i k y   . X y[ By . ) X Σ Then . trop Σ Σ( Σ 0 uhta h diagram the that such X trop X f X 0 Σ ) stecnnclcmatfiainof compactification canonical the is X xadn nbt [ both on expanding , fgnrlzdcn opee that complexes cone generalized of X 47 S 47 33 X X X ( rpsto .5,if 3.15], Proposition , X eto ]i bet ithis lift to able is 3] Section , a eto noteskeleton the onto section a has hpe ]adtebeginning the and 2] Chapter , : a h tutr fatoroidal a of structure the has a eedwdwt the with endowed be can ) X 33 i → h uhr okwith work authors the ] Σ X from simple X ofll the fulfill to 1 X n [ and ] k i X nthis In . or noits into 0 , strict → 33 X ], Author Manuscript ftelgrtmcsrcuefrom structure logarithmic the of r w ieetwy otropicalize Trop to tropicalization ways different two are 1.4. one). multiplicity oegnrlyit h quotient the into generally more monoid a from map of intersection the if only and that such section continuous unique a has map tropicalization Kajiwara-Payne the of Y [ of Trop results of Trop further then the trivial, Moreover, complex. is structure logarithmic the over smooth logarithmically uhthat such [ the define may One subvariety. ealdcmaio fteetocss hs osdrtostgte ihteaoetwo above of the principle with the together of considerations incarnation These following cases. the two to these lead of theorems comparison detailed a 1.3. commutative. is hti endb sending by by defined is that cone map polyhedral tropicalization rational continuous natural the of compactification canonical morphism X for 6 0 ,[ ], Let u praht h osrcino trop of construction the to approach Our oolr 1.3. Corollary [ In Let ntegoa aew rce ntoses soitdt aik oaihi scheme logarithmic Zariski a to Associated steps. two in proceed we case global the In = ihu monodromy without p 28 endb oec-ap n tpnvi [ in Stepanov and Popescu-Pampu by defined ∈ Y rpclzn subvarieties Tropicalizing h anie forconstruction our of idea main The hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This 49 Y X ,ad[ and ], P X ∩ hoe .]w aesonta,if that, shown have we 1.1] Theorem , ( . eacoe uvreyo a of subvariety closed a be X ealgrtmcshm oal ffiietp over type finite of locally scheme logarithmic a be J Y Y φ Y xadn nTvlvster of theory Tevelev’s on expanding , X osuypoete of properties study to ) ∩ ◦ : 14 T trop ( P X, UCOILTOIAIAINO OAIHI SCHEMES LOGARITHMIC OF TROPICALIZATION FUNCTORIAL 6= ]). notemlilctv oodo nalgebra an of monoid multiplicative the into ∅ ∆ O ∆ hnterestriction the Then . X ( : ups that Suppose Y Y ) aoe;teohri oendow to is other the above); 1.1.2 Section (see ) → an blw hr sa setal unique essentially an is there below) 4.2 Section (see x F → ∈ k X Y X S n if and trop noa into tropicalization ( ihevery with i A/A Y Trop otehmmrhs trop homomorphism the to X J ) ∆ Y Y Y stedfrainrtato notetria kltnif skeleton toroidal the onto retraction deformation the is ∗ | Y trop p aofan Kato Y n ocnie t mg in image its consider to and n set and ) Y huh fa ata meddcmatfiainof compactification embedded partial a as of thought , sapoe n c¨nsbait fa of schön subvariety and proper a is : T X − − 7−→ an h rti ocnie h lsia Kajiwara-Payne classical the consider to is first the : Trop o-rval nescsteoe locus open the intersects non-trivially trcvariety -toric ( Y α : 49 trop = ) T Y : ∆ X X hwta n a s h oyerlgeometry polyhedral the use can one that show ] obtin -orbit an log ( X F soitdto associated i Y rpclcompactifications tropical X sagoa eso fthe of version global a is ( −→ X ) Y −→

α ..alclymnia pc hti covered is that space monoidal locally a i.e. , Spec = X −→ ) 43 ( X ∩ p Trop ( σ ) saZrsilgrtmcshm htis that scheme logarithmic Zariski a is eto ] i morphism a Fix 6]. Section , X Y Σ

X Y P x X i an snnepyadirdcbe(..has (i.e. irreducible and non-empty is ∆ ) A n assume and locristesrcueo cone a of structure the carries also ( . h set The . σ Y Y P α ) atfltropicalization faithful ( Hom( = ihrsetto respect with k x ) n let and A ∈ Σ σ σ ffiietp over type finite of X P P Y P, w give we 7 Section In . Hom( = Hom( = ∩ se[ (see Y R oa tropicalization local Y T ≥ ihtepullback the with T ⊆ 0 6= n hr sa is there and ) trcvariety -toric X 46 X X ∅ P, P, characteristic 0 hnthere Then . 34 of 5 Page ]). ysetting by eaclosed a be of R R α ≥ ≥ X : see.g. (see 0 0 P sthe is ) given ) where k → (or X A Author Manuscript aik oaihi ceelclyo nt yeover type finite of of tropicalization locally scheme logarithmic Zariski to associated ree leshms hc n a hn fa nercmn ftecaatrsi morphism characteristic the of enrichment an as of think on can map φ one induced which schemes, the blue as ordered arises article, this in schemes blue X Let monodromy. theory without schemes Giansiracusa’s X logarithmic of and geometry Giansiracusa tropical T the both and generalizes schemes tropical that of tropicalization for framework change base numbers. the tropical consider and of scheme semi-ring semiring of category the ( ”analytifying” space monoidal in sharp Giansiracusa and Giansiracusa by developed in been has construction, our [ to spirit in similar is Trop and homeomorphism natural a is there that M relation equivalence the where the with analogy in theory Gromov-Witten logarithmic an to [ of in applications part outlined been tropical by have that motivated Siebert and is Gross M. of ideas 1.5. Σ complex cone monodromy. generalized a define of trop still may presence we the cases to these due is that morphism characteristic [ Stepanov the of trop map ”analytification” both tropicalization the that as case defined special formally then is of complex set The below). Spec 3.1 patches affine by 34 of 6 Page 20 X ≥ aual rs stestof set the as arise naturally ) i n[ In that varieties, toric of subvarieties of tropicalization the to approach alternative An 1.5.2. by inspired much very is schemes logarithmic of tropicalization the to approach Our 1.5.1. scheme logarithmic every not general, In X,x N 0 .Let ]. sdi u construction. our in used and X R etesmrn fnnngtv rpclnmes oshi,i atclr hw htboth that shows particular, in Lorscheid, numbers. tropical non-negative of semiring the be opeet,apiain,adfrhrdevelopments further and applications, Complements, hn∆dfie oi variety toric a defines ∆ Then . whenever , 36 : X Σ hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This ,bsdo i hoyo lerns(e [ (see blueprints of theory his on based ], Σ GS N i X 43 X → ( eafe ntl eeae bla ru n osdrartoa oyerlfn∆ fan polyhedral rational a consider and group abelian generated finitely free a be admr generally more (and X n,a ersi,sol etogto sa”o-rhmda nltcspace” analytic ”non-Archimedean a as of thought be should heuristic, a as and, ]. se[ (see F Σ hrfr aual are h tutr fagnrlzdcn complex. cone generalized a of structure the carries naturally therefore ) F X X X h aua otnostoiaiainmap tropicalization continuous natural The . x X 0 36 ytkn oiisoe l tit´tl oesb oaihi cee without schemes logarithmic étale by covers strict all over colimits taking by ycnieigthe considering by saseilzto of specialization a is xlddmanifold exploded sgvna h space the as given is hoe ].I hslnug h rpclzto a trop map tropicalization the language this In H]). Theorem , R P X X, X Trop ≥ o n n auae monoids saturated and fine for 0 Spec = sntigbttelcltoiaiainmpo oec-ap and Popescu-Pampu of map tropicalization local the but nothing is O vle points -valued X GS ( = ) monodromy ∼ T ( X Y ≥ A sidcdb h ul ftegnrzto maps generization the of duals the by induced is i 0 = ) X, vle onso eti utbecoe so-called chosen suitable certain of points -valued and n h rpclzto Trop tropicalization the and trop X Trop O ATNULIRSCH MARTIN R ∆ X ntesneo [ of sense the in x ≥ x F X G vrtefield the over ∈ / 0 F in X X GS X O vle onsof points -valued : X .In ). 4.2 Section (see structure logarithmic the in X ∗ Hom( X T X Spec = disacaatrsi opim phenomenon a morphism, characteristic a admits ( ( R X un u ob oi aofn nta of Instead fan. 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X swl satoiaiainmap tropicalization a as well as ∆ ∆ h uhr f[ of authors the , X ∼ X × ( 32 Y F , 1 o lsdsubset closed a for ) eto ]adSection and 9] Section , eafieadsaturated and fine a be T where , Σ X = T X 20 σ eoe the denotes sdefined as , φ P X M okin work ] nthe In . n the and ordered X,x Y 0 → of Author Manuscript all X P ideal with maximal unique a together contains numbers by real denoted is non-negative that The monoid otherwise. a form noted addition unless additively, written 2.1. homeomorphism natural a is There heuristic: mere a than more much is morphism for particuar, In that monodromy. stacks have T fan algebraic that Artin logarithmic an structures of is logarithmic category there with the scheme deal in logarithmic fans to every Kato suitable of more theory is the of incarnation an Nishinou- classical the of [ explanation (see moduli-theoretic theorem a correspondence [ with of Siebert us e.g. provides space (see which tropical agree moduli rational C]), invariants certain the Theorem that Gromov-Witten deduce to (descendant) can we identities variety, algebraic toric these and a into Applying maps stable counterparts. logarithmic algebraic rational their with products X R trop map tropicalization Payne If commute. Σ Σ the enrich can one approach on his cycles using algebraic on particular, expanding operation In lattice, an a article. to by this tropicalization generated in of space vector process developed a theory into the embedding weak on a admit that complexes eoe the denotes [ In field. npriua,gvseeyrtoa oyerlcn ope swl siscanonical its as well as Σ complex cone polyhedral rational every gives compactification particular, in ≥ X X trcvrey h ri fan Artin the variety, -toric 0 The A [ In [ In 1.5.4. intersection tropical natural certain identify to us allows also approach his cases, ”good” In An Let 1.5.3. − A → . 0 p rmtetplgclspace topological the from aigtediagram the making i p = monoid Monoids 51 ∈ ideal hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This in localization R X P hoe .]w hwta enn trop defining that show we 1.1] Theorem , 21 ≥ P h opeeto rm da in ideal prime a of complement The . hti ito h hrceitcmrhs oti aeoy o xml,if example, For category. this to morphism characteristic the of lift a is that 0 I .Goshsdvlpdavrino rpclitreto hoyo etne)cone (extended) on theory intersection tropical of version a developed has Gross A. ] {∞} ∪ sasbood reuvlnl,if equivalently, or submonoid, a is namonoid a in X ffiodtorus affinoid P 4 X sa is sacmuaiesmgopwt niett lmn.Almniswl be will monoids All element. identity an with semigroup commutative a is UCOILTOIAIAINO OAIHI SCHEMES LOGARITHMIC OF TROPICALIZATION FUNCTORIAL n [ and ] aoial h tutr fannAcieenaayi stack. analytic non-Archimedean a of structure the canonically Σ eaZrsilgrtmcshm hti oaihial mohoe h base the over smooth logarithmically is that scheme logarithmic Zariski a be famonoid a of ysetting by T trcvrey hssaeetgnrlzst h atta h Kajiwara- the that fact the to generalizes statement this variety, -toric 5 as e [ see (also ] P 2. sasubset a is of A a ood,cns n oodlspaces monoidal and cones, Monoids,

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neligtennAcieenaayi stack analytic non-Archimedean the underlying φ = an 38 I P ∞ X i ⊆ { ]). − → p o all for P P − 51 N uhthat such ∗ R s R nideal An . ) h uhr eeo h hoyof theory the develop authors the ]), X µ | p ∆ sasakquotient stack a is (∆) ∼ ≥ p A X 1 P i 0 X ∈ a + X t oodsrcuentrlyetnsto extends naturally structure monoid Its . srfre oa a as to referred is ∈ ste”nltﬁain ftecharacteristic the of ”analytiﬁcation” the as ,s P, p n nesnilyuiu titmorphism strict unique essentially an and 2 R ∈ trop p ≥ p 0 ∈ + p . in Σ S I led implies already X X } P ⊆ X , scalled is I S o all for X T sgvnby given is htnntiilyintersect non-trivially that . S face 1 prime 21 p ⊗ ∈ X eto ]ad[ and 5] Section , of N p P an 1 hsprocedure, This . P vr monoid Every . fiscomplement its if ∈ . T p ◦ µ or 34 of 7 Page X where , ri fans Artin p : A 2

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sa is 45 for T → P ◦ , , Author Manuscript Teeeit hr submonoid sharp a exists There (ii) h ulsbaeoyo oi ood by monoids toric of subcategory full the be to said nt by units that auae bla ugopin subgroup abelian saturated a called p group the ideal prime a of complement n If P f ˚ by the [ Gubler that [ and 1.2] Section where cone polyhedral ( 2.2. monoid u that and that subgroup abelian Q free generated finitely where 34 of 8 Page Teei oi submonoid toric a is There (i) ,N σ, ∈ tors · ∈ ∈ S em 2.1. Lemma monoid A nve f() emyassume may we (i), of view In osdrnwtefunctor the now Consider A write we notation, of abuse slight a In o h ovnec fterae epoiepof fteetowl-nw statements. well-known . 2.1 two Lemma these of of proofs Proof provide we reader the of convenience the For h ainlplhda oe( cone polyhedral rational the q σ P eaieinterior relative P Hom( t steset the is morphism A . ∈ titycne ainlplhda cone polyhedral rational convex strictly P Q p ossigo ntl eeae reaeingroup abelian free generated finitely a of consisting ) ainlplhda cones polyhedral Rational gp ∗ ∈ = p v + P Set . sstrtd Set saturated. is = h property the , i P P/P − oso element torsion P hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This ∗ P q ∈ ,N N, tors Q tors P s o some for ∗ .Dnt h ugopo oso lmnsin elements torsion of subgroup the Denote 0. = sharp M n n auae monoid saturated and fine A . gp ⊕ P eoe neuvlnecaso ar ( pairs of class equivalence an denotes ∗ P 27 since , . n ehv ˜ have we and N P uhthat such = 0 = uhthat such ) scalled is al ainlplhda cones polyhedral rational calls ] · ∗ if , σ Let P P ( 27 oeeyelement every So . f ,s p, ⊆ − ∩ o nelement an for f pedxA o h seta akrudo hs oin.Nt hereby Note notions. these on background essential the for A] Appendix , n P 1 fartoa oyerlcone polyhedral rational a of N n P Q ) P : ∗ ∈ n R · ∼ ( Since . if , eafieadstrtdmonoid. saturated and fine a be .W eoetectgr ffieadstrtdby saturated and fine of category the denote We 0. = = ,N σ, q N · P fine = σ e ( p h bla group abelian The 0 = > p { n = osntcnanaynntiillna usae.W ee o[ to refer We subspaces. linear non-trivial any contain not does N p f ∈ 0 0 p s , fi sfiieygnrtdadtecnnclhmmrhs into homomorphism canonical the and generated finitely is it if , earayhave already we , ) ( σ · = − p P P σ ⊗ → p ∈ 0 p ) on ) Q in p q ∩ o some for 0 = P R o some for P e ∃ ⇔ = σ ⊆ | ( H i.e. , − ..afiieitreto fhl spaces half of intersection finite a i.e. , Q P σ P ,q p, fs ˜ of f i p σ for P 0 N , t Every . N , = 0 edenote we − − ∈ P ∈ ∗ . P p N t ∈ P Q/P ATNULIRSCH MARTIN o h ro fpr i) Given (ii). part of proof the for 0 = P q u ∈ Mon ∈ 0 P P of frtoa oyerlcnsi ie ya element an by given is cones polyhedral rational of ) uhthat such ie by given ) u P ∈ ∈ Thus . tor S P ewrite we Hom( = Q n P e e } ∈ P Q P P ∗ uhthat such ∈ p a euiul rte as written uniquely be can sijcie ti adt be to said is It injective. is o h oi monoid toric the for of N op uhthat such hsimplies this , − sfe,adw a n subgroup a find can we and free, is n ∈ ssi obe to said is Q n some and N R o hr:a short: (or Q htascae oafieadstrtdmonoid saturated and fine a to associates that q Mon S ∈ >

P = h P one nerlplhda cones polyhedral integral pointed 0 − ∈ ,v u, N P uhthat such P a euiul rte s˜ as written uniquely be can ,s p, 1 P P = led implies already P > gp P σ f gp = i . 0 ..teitro of interior the i.e. , since , P ≥ i , ) ˜ p ti alda called is it ; by o h localization the for Z P P ∈ sfiieygnrtd ow a n a find can we So generated. finitely is + ˜ ⊕ ) n ⊕ P P 0 = s P

∈ toric p P 0 ainlplhda cone polyhedral rational P tors × . P + P Q , N N = tors P S ⊕ t > = sstrtd Therefore saturated. is hspoe at(i). part proves This . if , P = 0 n titycne rational convex strictly a and by ne h qiaec relation equivalence the under P . Q P/P e ⊕ led implies already p ∗ p P 0 . P ⊕ unit P + ∈ tors tors Q tors ∗ P s . tors saturated fteeis there if , p + nelement An . ;aymonoid any 0; = S n h ugopof subgroup the and σ + and . t − oeta hereby that Note . nissa in span its in u 1 P p q with P + ∈ fs n if and o h sharp the for t Q Q − f whenever if, edenote We . q ih˜ with q p sapair a is ) ∈ Mon of uhthat such ∈ ∈ p P S P Q P Q ∈ p ∗ sthe is using such such P P ∈ N and and sa is 18 R Q is is e , , Author Manuscript simdaethat immediate is ocekta ( that check to homomorphism a induces [ Lemma Gordon’s By Hom( cones. polyhedral rational of category the and monoids morphism ha fmonoids of sheaf a 2.3. τ morphism a that immediate is It 3.1. subcategory. this onto ( association the and all for ideals monoids maximal of unique homomorphism the map local continuous a a is is spaces f monoidal locally of to want we If morphism. face of class of notation face of proper) abuse necessarily by slight a cones In sharp sharp. of are category cones the our denote of to all going that therefore assume are to we going are we article this ( cones and o h ulcategory dual the for x on multiplication on monoids be of to said is a as to referred be simply ∗ → ∈ rpsto 2.2. Proposition Proof. A eoetectgr flclymnia pcsby spaces monoidal locally of category the Denote A polyhedral rational to corresponds monoids toric sharp of category the (ii) 2.1 Lemma By n[ In O X Y oal oodlspace monoidal locally σ aemorphism face oodlspaces Monoidal aofans Kato N, 32 hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This h category The . O → sa as ,N σ, eto ]K aoitoue h oino a of notion the introduces Kato K. 9] Section , Z σ and ) P σ osdrtefntr( functor the Consider X htare that ) rprfc morphism face proper ( Hom( = f fsevso ood uhta h nue homomorphism induced the that such monoids of sheaves of ) strict . X ) : ∨ UCOILTOIAIAINO OAIHI SCHEMES LOGARITHMIC OF TROPICALIZATION FUNCTORIAL ( l cee r mlctytogto smnia pcswt epc to respect with spaces monoidal as of thought implicitly are schemes All . σ S sa nes to inverse an is O O P σ fteidcdmorphism induced the if , X τ X, P, N , 3. X SMS sitga,strtd n oso-re morphism A torsion-free. and saturated, integral, is sh sharp oodlsae( space monoidal A . 18 → ie w oal oodlsae ( spaces monoidal locally two Given . R h functor The P O aofn,cn opee,adterextensions their and complexes, cone fans, Kato − eto rpsto ]temonoid the 1] Proposition 1 Section , ≥ ) σ X f cone 0 σ → σ m Mon ) fsapmnia pcsi uladfihu uctgr of subcategory faithful and full a is spaces monoidal sharp of = ) ..toecones those i.e. , ∨ samrhs fcnsta nue nioopimot (not a onto isomorphism an induces that cones of morphism a is sapi ( pair a is ∨ oeta eepiil lo uoopim facone a of automorphisms allow explicitly we that Note . 7→ Y,f : ( = σ S ( X n rte as written and Q σ x op u . f ) 0 N , . v ) σ ( = ∈ → ∨ and ftecategory the of : ∈ Q (i). 2.1 Lemma using Q σ Hom( htsns( sends that X, S X, M n htteassociation the that and ) → σ nue neuvlnebtentectgr ftoric of category the between equivalence an induces τ m R O O n hsascaini ucoilin functorial is association this and X,x

ob smrhct rprfc of face proper a to isomorphic be to P h X P X ,v u, ossigo oooia space topological a of consisting ) gp with ) X, fﬁeadstrtdmonoids saturated and ﬁne of σ in x , ≥ i σ hs pnin span whose R ∈ O O f ) ,N σ, ihu xlctrfrnet h lattice the to reference explicit without X − X X,x

f o all for 0 u 1 ssi obe to said is ) O sh hsmasthat means This . O ( : p X X in ) and LMS Y ) − = ≥ O → → Mon fan O RPC o all for 0 O v Y morphism A . X Y,y X, htsre sagoercmodel geometric a as serves that X ∈ / oehrwt morphism a with together N O fsapmnis(lose[ see (also monoids sharp of σ O respectively. sa smrhs fsheaves of isomorphism an is S R RPC X

∗ to f X σ seulto equal is . 7→ p ensartato functor retraction a defines n ( and ) sfiieygnrtdadit and generated finitely is S sharp ∈ σ σ nojc in object an ; P = ( f f x †

if , σ sfntra in functorial is ) ( f f Y, f m ⊆ ∨ x P † : Y,f : ∩ O ( ( : O f X N ,N σ, and N X O Y M ti o easy now is It . ( X,x ∗ R x P ,amorphism a ), σ Y,f → ) Throughout . oehrwith together where , ) erfrto refer we , ) 34 of 9 Page ) R Q ( ⊆ o all for 0 = Y → x RPC . ) nue a induces m in σ O → ( X,x σ nthe in LMS LMS 0 M N , N f f will 19 X,x for . . † = 0 ) , : Author Manuscript Smlry if Similarly, (ii) ha ensa nes oSpec. to inverse an defines sheaf fan Kato affine an association the by determined pcaiainb narrow. an by specialization element fans. Kato affine of category the and monoids in orbits Spec association morphism A of stalk the consequently, and, Hom SMS monoid [ a to associates that schemes, blocks. of building analogues fundamental as the of as thought monoids be allow should we rings, objects of These instead details). where, further for 3.5] Section 34 of 10 Page h pnsets open the If (i) 32 xmls3.3. Examples Spec on order partial a defines relation specialization the that Note Proof. 3.2. Proposition 3.1. Remark functor a is there particular, In sastSpec set a As rpsto .]tesetu Spec spectrum the 9.2] Proposition , monoid nti itr eidct h tl ftesrcuesefa ahpit If point. each at sheaf structure the of stalk the indicate Spec we picture this In P → P, P sntrlyhmoopi oΞ( to homeomorphic naturally is ∅ hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This O = Sets R X n nqemxmlelement maximal unique a and X ie hr monoid sharp a Given ≥ P R p 0 N ( . ≥ X 7→ D . otis nadto to addition in contains, φ 0 htascae oa to associates that ) hnSpec then , ( P : φ P f . Q − = ) oi monoid toric A seult e fpieiel of ideals prime of set to equal is = F 1 → ( p N h dniySpec identity the { n h nue opim ntesrcuesheaves. structure the on morphisms induced the and ) P then , p h functor The nue morphism a induces , ∈ R Spec ≥ P F 0 O A hr oodlsaeSpec space monoidal sharp a ossso h w rm ideals prime two the of consists P 1 F P P | Spec = / f Spec ehave we , sharp at Spec ensatrcvariety toric a defines ∈ p ∅ ATNULIRSCH MARTIN D p O O } ∈ and : X P ( F, F 0 ensa qiaec ewe h aeoyo sharp of category the between equivalence an defines Mon f N oodlsae( space monoidal for 0 Spec P m ( p ) suiul eemndb ersnigtefunctor the representing by determined uniquely is F ,testo eei onso the of points generic of set the ), P ossso h w rm ideals prime two the of consists 7−→ = R O f = ) = > Spec op P P ∈ 0 p P lotepieideal prime the also , P φ F −→ P /P sgvnby given is f # P − otkn lblscin ftestructure the of sections global taking So . h tutr sheaf structure The . P /P p (Spec ∗ : R P SMS n t oooyi h n eeae by generated one the is topology its and Spec f . ∗ ≥ N ∗ 0 ntefloigpcue eindicate we pictures following the In . X P P X, P = ) P → O Spec = aldthe called , ∅ X P/P Spec and h e fhomomorphisms of set the ) ∗ P Q k R = [ {∞} P > ihauiu minimal unique a with hti ie ythe by given is that P O 0 .Tean aofan Kato affine The ]. ntemonoid the in F n,cnesl,for conversely, and, spectrum . ∅ on T and P Spec = F = Spec = N R > of ≥ 0 k 0 P nthe in [ then , P R By . P gp ≥ 0 is ]- . Author Manuscript Suppose (iii) Let (iv) Cnie he copies three Consider (ii) auae.Dnt h aeoyo lclyfieadstrtd aofn by fans Kato saturated) and fine (locally of category the Denote saturated. is fan Kato subset monoids. [ Kato Gvntocopies two Given (i) xmls3.5. Examples every that assume to going are we otherwise, noted unless article, this of remainder the In K. 3.4. rings, Definition of category the of dual the extends that schemes of category the with analogy In a h olwn orpoints: four following the has epciey leteean aswt epc oteisomorphisms the to respect with fans affine these Glue respectively. fan Kato the defines hsvsaiainimdaeygnrlzsto generalizes immediately visualization This N functions. of sheaf their in Spec lies underlying difference spaces topological the that Note > hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This 32 U 0 P i × nrdcstectgr of category the introduces ] smrhct Spec to isomorphic etemni eeae by generated monoid the be N oal n n saturated and fine locally and , P = UCOILTOIAIAINO OAIHI SCHEMES LOGARITHMIC OF TROPICALIZATION FUNCTORIAL N N 2 2 h spectrum The . A { − U 0 aofan Kato 0 and F } U P . 1 0 . , P U U i 1 1 o oemonoids some for and , fSpec of F ∅ D D D N , sasapmnia pc htamt oeigb open by covering a admits that space monoidal sharp a is U U U P F aofans Kato ..ta emycos the choose may we that i.e. , 2 1 0 N N 0 A 0 ( ( ( U − p p p 2 ,q r q, p, 2 N 2 1 0 N Spec = ) ) ) fSpec of ecngu hmoe h eei point generic the over them glue can we , ' ' ' · p Spec Spec Spec , ujc oterelation the to subject P h xed h ulo h aeoyo sharp of category the of dual the extends the 0 N − N P 2 N N N P 2 N i ossso h orelements four the of consists = ' ' ' N . ihcoordinates with · N r n Spec and D D D and , k N U U U N P N o l oiieintegers positive all for 0 2 1 2 ( ( ( q q q N 0 2 1 P ) ) ) { − R ≥ P 0 0 } i p r h ae h crucial the same; the are . saoet efieand fine be to above as p + 0 q , r 2 = 0 , Fans p q 1 hnSpec Then . q , 34 of 11 Page . 1 ∅ k and , , . {∅} N × This . p N 2 > q , P 0 2 , Author Manuscript Gvnfu copies four Given (iii) aspeieycrepn otegnrcpit ftetrsobt fteetrcvreis(see varieties toric these of orbits torus the varieties of toric points the above). generic to 3.1 the associated Remark to naturally are correspond that precisely fans fans Kato the be to A out turn here described 34 of 12 Page n h notation The , p h aofnotie hswywl edntdby denoted be will way this obtained fan Kato The integers all integers Glueing nimdaegnrlzto fti osrcinyed h aofans Kato the yields construction this of generalization immediate An fan Kato the obtain to order in P 4 n q , n ( and , 4 hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This epciey ecngu hs ffieKt asvateisomorphisms the via fans Kato affine these glue can we respectively, k k . P oiso Spec of copies 1 + 1 k ) F . n A oe npriua,ta h neligtplgclsae fteeKato these of spaces topological underlying the that particular, in Note, . n , U F P 1 n , and , U 2 , U N N N 3 F 2 2 and , N N N D D D D ( 2 P k U U U U 1 F ) 4 3 2 1 na nlgu anr ie iet aofans Kato to rise gives manner, analogous an in n P ( ( ( ( ATNULIRSCH MARTIN 2 p p p p U . blw h aofans Kato The below: 4.11 Example in explained is 2 1 4 3 4 ) ) ) ) fSpec of ' ' ' ' Spec Spec Spec Spec N N N N 0 N N N N N 2 0 2 ' ' ' ' ihgenerators with D D D D F U U U U P 1 4 3 2 1 ( ( ( ( × q q q q 4 3 2 1 P ) ) ) ) 1 N N . . 2 N N N 2 2 p 1 q , F 1 ( , P 1 p ) 2 k q , o l positive all for 2 , p 3 q , F 3 P and , k for Author Manuscript Asubset A (iii) Gvnapoe face proper a Given (ii) ecnasm that assume in can cone we polyhedral rational a is s subset a of is, face a is complexes cone polyhedral rational of category the and fans Kato that of such category the the between all for commute pnan subsets affine open preimage the ipythat imply 3.2 Proposition image the to ihafml fmorphisms of family a with s morphism a is, space maps continuous topological a of [ see (also 5] Definition 2.1 called Section so objects, geometric 3.2. Temaps The (i) 3.7. Proposition of Proof eoetectgr fcn opee ihpiecewise with complexes cone of category the Denote 3.6. Definition n[ In rpsto 3.7. Proposition en the Define oecomplexes Cone hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This 32 .Kt nrdcdtento faKt a nodrt leriemc more much algebraize to order in fan Kato a of notion the introduced Kato K. ] A σ euto map reduction u A U | ⊆ ( φ f hs npriua,alw st endow to us allows particular, in This, . R of α UCOILTOIAIAINO OAIHI SCHEMES LOGARITHMIC OF TROPICALIZATION FUNCTORIAL α : > φ Σ . Σ r netv n nueabijection a induce and injective are 0 | α F Σ ftemxmlideal maximal the of ) U → Q | : | A σ scoe,i n nyi the if only and if closed, is scoe fadol fispreimages its if only and if closed is Spec = = hr sa equivalence an is There α Σ ainlplhda oecomplex cone polyhedral rational | Σ τ 0 | → P nti aeoyi ie yacniuu map continuous a by given is category this in | of p | RPCC Σ oehrwt olcino ainlplhda cones polyhedral rational of collection a with together r σ o oepieideal prime some for Σ Let σ ainlplhda oecomplexes cone polyhedral rational F α : P | α | | → | then , Σ uhta h olwn rprishold: properties following the that such σ σ Σ = of F V F U F σ 1 F | | F = eaKt a.Fra pnan subset affine open an For fan. Kato a be = and β 0 eto 2.1]). Section , = ( = and 2.2 Proposition with together observations These . R with r Fans r τ F − F ≥ σ − σ y   Fans sas ebro h aiy( family the of member a also is ( α 1 0 β 0 1 ( R F ( Hom = ) ( R G V U α R −−−−→ | −−−−→ | β ≥ ≥ 7−→ −−→ Hom( = ) > Hom( = ) 0 ˚ σ 0 ∼ = φ φ ) 0 .Gvna pnan subset affine open an Given ). r equivalent. are α α β 0 → β nSpec in −→ | Σ RPCC ∼ ( α p F F A uhta h diagrams the that such ) Σ in Σ y   Spec ∩ Σ ysnigamorphism a sending by 0 | | Q, | | P, f r P | R . − R Z R 1 ≥ n ehv that have we and R o hr:a short: (or Σ | ( lna opim by morphisms -linear ≥ 0 ≥ Σ U ≥ φ . 0 0 F 0 α − ) ) ) F , | 1 ⊆ ( ihtewa oooy That topology. weak the with A Hom( r lsdin closed are ) . ntetriooyo [ of terminology the in P, | f oecomplex cone | σ R : α ≥ | ). Σ 0 u V U scoe o all for closed is ) | → | : Spec = Spec = Spec RPCC 34 of 13 Page σ α Σ 0 o all for R | consists ) together ≥ P Q σ 0 That . α in of → and α 33 U F . F , , , Author Manuscript htntrlyetnstepiecewise the extends naturally that as well as r A U subset of affine face open a an for Moreover, 2]. compactification canonical the is subset affine open an for that observe We 3.7. Proposition of of proof the before right map morphism reduction a sending by morphism a sending by Spec morphisms of set the [ in defined as of associated scheme space analytic non-Archimedean 3.3. 34 of 14 Page n hsipistecniut of continuity the implies this and the as known : rpsto 3.10. Proposition Proof. 3.9. Proposition the Define 3.8. Definition fan Kato a Consider ⊆ F of Σ F Σ h preimage the F aoia extensions Canonical F → hscaatrzto of characterization This . X hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This scoe fadol if only and if closed is σ F oal ffiietp over type finite of locally eol edt osdra ffieKt fan Kato affine an consider to need only We U santi-continuous. is = euto map reduction aoia xeso of extension canonical 1 r eto 2.2]. Section , r − : 1 Σ ( F U Let n so and ) → h tutr map structure The u F morphism A ρ F u r F − : h eut n[ in results The . − ρ R Σ : 1 Spec F santrletnino h euto map reduction the of extension natural a is y   1 eaKt a.The fan. Kato a be ≥ F Spec D 0 D −−−−→ −−−−→ ( Σ A → ( f R Σ( f F σ Σ R ) ∩ ≥ f ρ ) F f U F ≥ r 0 steclmto all of colimit the is σ ) swl steat-otniyof anti-continuity the as well as k = . − 0 Σ = Z ftecone the of Σ( → U losu oedwi ihthe with it endow to us allows 1 f sntΣ not is → F ATNULIRSCH MARTIN lna map -linear ( Σ scoe o l pnan subsets affine open all for closed is f U : steset the is ) G u y   F V u G ) F ρ r F Hom( = ) ρ ∈ ∈ : notepoint the onto Spec = : : → ρ Σ 47 otepoint the to Hom( Σ Σ Hom( F : u F F F G ugs otiko Σ of think to suggest ] Σ u ahrits rather but , −→ 7−→ F → → F xeddcn complex cone extended σ fKt asidcsacniuu map continuous a induces fans Kato of nfc,tecretaaou of analogue correct the fact, In . P, U Q P, → Σ( F F P, F r Σ f Σ F Hom( = R y   R of ( F f F G R ◦ R ≥ ≥ ) ≥ 0 u ≥ U F −−−−→ −−−−→ 0 σ scniuu n h euto map reduction the and continuous is : ) 0 ) 0

Σ( h xeddcone extended the , U Σ u u )

of ) u Spec = u f ( F ( f R ae vraloe ffiesubsets affine open all over taken ( {∞} f P, ) f → > ) 0 = ) R R aoia compactification canonical 0 6= ≥ Σ ) Σ ≥ G y   0 ∈ G ∞ P F 0 r ∈ vle onson points -valued sdfie n[ in defined as )

Given . F

n ae h diagrams the makes and r F ektopology: weak . n the and oehrb htthe that hereby Note . F r soitdto associated steaaou fa of analogue the as : Σ U f F Spec = ∈ σ → tutr map structure P V ehave we , F = U 47 sdefined as F r X Spec = P subset A Section , − htis that , . F i 1 ( V o a for (also Σ is ) F P , Author Manuscript frtoa oyerlcones. polyhedral rational of x morphism A and morphism to equivalent morphism for that observe where r[ or 3.4. association The commute. lsdsbes If subsets. closed as well as f ecntikof think can we of ( ∈ U rpsto 3.12 Proposition Proof. Let morphism a that 2.3 Section from Recall If subsets affine open on that Note oolr 3.11. Corollary Proof. F 44 ) F G seult h e fmrhssSpec morphisms of set the to equal is , F taicto fetne oecomplexes cone extended of Stratification ⊆ ti stecs fadonly and if case the is this 3.10 Proposition By . hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This rpsto .] Write 3.4]. Proposition , F f containing Spec = # V eaKt a.Tecollection The fan. Kato a be sahomomorphism a is h map the Σ( eol edt hwtecmuaiiyo h w igas oaheeti we this achieve To diagrams. two the of commutativity the show to need only We eoefor Denote f − f f f 1 P ) O : nuigisomorphisms inducing : Σ F UCOILTOIAIAINO OAIHI SCHEMES LOGARITHMIC OF TROPICALIZATION FUNCTORIAL u G san,w a omlydsrb hssrtfiainfloig[ following stratification this describe formally can we affine, is F Σ x F ∈ → O → F and sirdcbe h nqeoe stratum open unique the irreducible, is Σ( sthe as Σ → G f F ([ x morphism A sgvnby given is ) F y 44 Σ ssrc,i n nyi tidcsa isomorphism an induces it if only and if strict, is ∈ Hom(Spec = r ρ epcieya hi aia onsby points maximal their as respectively Σ G G sa smrhs fmni hae on sheaves monoid of isomorphism an is F rpsto 3.4). Proposition ] G f aoia compactification canonical U aseeycn in cone every maps ◦ ◦ 7→ and Hom( = Σ( Σ( Q Hom( σ Σ( f f → o h oeHom( cone the for ) τ y ) f G ≺ U ( ) = P ( f σ u P, u sfntra in functorial is R P, N Spec = = ) ρ f : = ) inducing R − ≥ f ( = F ( R ( = R x x 1 0 † / ≥ ( ehave we ) ) r ≥ → ρ : 0 Spec x f G u f ∈ 0 ) G O R for ) ) ( ◦ G 7−→ −→ ( ◦ f P G,f G f ≥ ' f u u 0 hr santrlstratification natural a is There ◦ fKt asi titi n nyi h induced the if only and if strict is fans Kato of : τ )( ◦ and f )( Σ h nqemnmlan pnsbesin subsets open affine minimal unique the Σ ( → F u x x Hom( u . R −−→ u F {∞} V = ) ) ∼ = ) ∈ > ◦ → F P, O → smrhclyot oein cone a onto isomorphically Hom( = V 0 f f F ..tecn ope Σ complex cone the i.e. , ( = ) N . # G fΣ of R ( = ) Spec = Q, f ensasrtfiainof stratification a defines R ≥ F,x , ( fKt asis fans Kato of 0 loidcsisomorphisms induces also σ R swl as well as ) f F ρ ) f ≥ − . o all for ◦ Q, 0 ◦ 1 r ) Q ( ρ F R η F )( ,where ), ≥ of U )( 0 u x ) u Spec = F ) ) ∈ . F and N F R n oeta hsis this that note and . η strict ( U σ G P stegnrcpoint generic the is Hom( = ) ' epcieywith respectively and F ftenatural the if , V nti sense this In . 41 34 of 15 Page Σ o l points all for V F Σ eto 3] Section , G Spec = ylocally by P, . R ). Q F . Author Manuscript Suppose (iii) Suppose (ii) orsodn to corresponding scheme of subset closed locally u a with write identified we is where 3.12, Proposition from induced preimages in N of 34 of 16 Page nHom( in If (i) R ∈ eak3.13. Remark xmls3.14. Examples of stratification the identify may We N N / σ R complex cone extended the R R R Span ( ≥ ≥ ( otehomomorphism the to F σ σ 0 0 ) Hom( = ) (1 P, X hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This {∞} × = ylclycoe ust oemrhcto homeomorphic subsets closed locally by τ , ρ R Spec = ersne by represented , F ) w oisof copies two 0), − ≥ A 1 P F 1 0 ( ). x Spec = , = stemni eeae by generated monoid the is ftepoints the of ) ∞ × {∞} X P, F A i A fw hn of think we If R 2 ⊆ ffiietp over type finite of ) Spec = . N X R then , ≥ an 0 u h e fbuddsmnrson seminorms bounded of set the , R and , ∈ N ≥ p p Σ 0 x 2 7−→ 7−→ Σ N n h point the and , Then . F F in R { = N ( Hom( = soitdto associated F ∞ R Σ ( ATNULIRSCH MARTIN , ihtestratification the with τ A ∞ h ∞ h σ G ∞ ≺ ,p u, ,p u, 1 Hom( = ) Σ σ = ) F k } σ/τ i i P then , Σ . R = Σ F gp ,q r q, p, ≥ F σ/τ Σ −−→ 0 , ysniga lmn in element an sending by { h taaaegvnby given are strata The . ∼ A Hom( = R else if else if ( F 2 P, ∞ ) p p h homomorphism the , o h mg of image the for = Σ Σ Spec = ujc oterelation the to subject , R N ∈ ∈ F F ∞ R steaaou of analogue the as ) R τ τ ≥ 2 Hom( = ) / ⊥ ⊥ P, } 0 Span ∩ ∩ tinfinity. at n h taaaegvnby given are strata the and R P P P ≥ ossso h cone the of consists 0 τ ) A P, ie yascaigto associating by given , ⊆ below). 5 Section (see R Hom( σ ≥ 0 in stecoe subset closed the is ) N P, R R σ/τ / R ≥ X p Span ie ythe by given ) 0 an + and ersne by represented r o naffine an for τ 2 = R Here . {∞} ≥ 0 q (1 Then . . , R [ )+ 2) u σ/τ ≥ 2 ] 0 ∈ , Author Manuscript If (iii) If (ii) that | a allowed. that are Recall of self-intersection. face has a that embedding toroidal morphism a of structure combinatorial 3.5. Let (i) Σ xmls3.15. Examples frapoe face proper a for – | 3.16 Definition n[ In oehrwt rsnaina oii fadarmo aemrhssin morphisms face of diagram a of colimit a as presentation a with together R {−∞} h taaof strata The follows. ( eeaie oecomplexes cone Generalized 1 2 hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This ∞ F F he oisof copies three , h uhr eeo h oino a of notion the develop authors the ] F , = σ = ∞ hsfc edntb rprfc of face proper a be not need face This . . τ = F F ) → P

P F 2 1 , P t soitdetne oecomplex cone extended associated its , × σ 1 P UCOILTOIAIAINO OAIHI SCHEMES LOGARITHMIC OF TROPICALIZATION FUNCTORIAL hntesrt fteetne oecmlxaegvnby given are complex cone extended the of strata the Then . ( 1 samrhs frtoa oyerlcnsta nue nioopimonto isomorphism an induces that cones polyhedral rational of morphism a is ∞ t soitdetne oecomplex cone extended associated its , Σ , ([ −∞ P τ 1 1 × R eto .) A 2.6). Section ] facone a of P ) 1 tiﬁiyadtrepoints three and inﬁnity at

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tinfinity. at Σ . {∞} × Σ F P = 2 ossso h pnstratum open the of consists Σ P , τ , 1 × satplgclspace topological a is Σ −}× {−∞} → P nodrt eciethe describe to order in 1 σ a evsaie as visualized be can sas nΣ and Σ, in also is R 34 of 17 Page , R , R RPC {∞} {−∞} × and , such face , Author Manuscript X xcl hsosrainta oiae h olwn definition. following the motivates that observation this exactly X osrcin htaentrlyascae otrcvreist oegnrlshms The schemes. general combinatorial more nice the to of varieties variety many toric generalize toric to to a us associated of allows naturally that geometry are way a that in constructions schemes of category the 4.1. generality. full in treatment [ and 2-4] eeaie oecmlxscnnclyetnst otnosmap continuous a to extends canonically complexes cone generalized all of diagram the take V may subsets we affine since complex, cone 34 of 18 Page quotient h oii fthe of colimit the morphism a every for iue1. Figure A (i) Zar eiiin4.1. Definition n ftemi betvso oaihi emtyi h es fK ao[ Kato K. of sense the in geometry logarithmic of objectives main the of One h anrfrnefrlgrtmcgoer s[ is geometry logarithmic for reference main The xml 3.17. Example cone a of automorphism every for – xml 3.18. Example opimΣ morphism A fcus,w a omtecnncletnino eeaie oecmlxΣb taking by Σ complex cone generalized a of extension canonical the form may we course, Of Spec = Spec = X nue uoopimof automorphism induced oaihi tutrsadcharts and structures Logarithmic τ rteétale site the or r-oaihi structure pre-logarithmic n morphism a and hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This σ/G σ Q 2 k nΣ hr sacone a is there Σ, in eto n ]frsotacut fti hoyadt [ to and theory this of accounts short for 3] and 2 Section , [ h utetof quotient The P sa pnan ustof subset affine open an is σ san,temni ftrsivratCrirdvsr sgvnby given is divisors Cartier torus-invariant of monoid the affine, is ] 1). Figure (see complex cone generalized a is U → Spec = → σ σ 0 α fcones. of Let Σ Let Let nta fthe of instead 0 X fgnrlzdcn opee sacniuu map continuous a is complexes cone generalized of F et P G ρ X . 4. eaKt a.Te h oecmlxΣ complex cone the Then fan. Kato a be : of eafiiegopatn nacone a on acting group finite a be R M eashm n eoeby denote and scheme a be X ≥ 2 oaihi tutrsadKt fans Kato and structures Logarithmic F τ 3.18. Example by complex cone 0 O → σ ihbgtorus big with on 0 sas nΣ. in also is ythe by once y(rpr aemorphisms face (proper) by connected nΣ in X X σ τ α 0 fsevso ood on monoids of sheaves of ATNULIRSCH MARTIN U uhta h composition the that such sapi ( pair a is σ .I hscs morphism a case this In 1). Figure (see Z . 2 nΣta evsapoe face proper a leaves that Σ in -operation T ,ρ M, 31 sgvre ythe by governed is .W lorfrterae o[ to reader the refer also We ]. ( ossigo ha fmonoids of sheaf a of consisting ) ,y x, σ X U ) τ 7→ Hom( = ihrteascae aik site Zariski associated the either σ ( ,x y, X yatmrhss hnthe Then automorphisms. by σ . F ) → = P, sagnrlzdextended generalized a is f Σ T F R : ivratdvsr.If divisors. -invariant ( ≥ Σ → R 19 τ | σ 0 Σ ae vropen over taken ) → ≥ V Σ of | → | eto ]fra for 7] Section , 0 0 , sageneralized a is ) → 31 Σ σ atr through factors 0 . f st enlarge to is ] σ nain the invariant Σ U : 30 0 P/P | Σ whenever uhthat such Section , → ∗ M tis It . Σ 0 on of Author Manuscript Apelgrtmcsrcue( structure pre-logarithmic A (ii) on omnt ovnetysprs h eeec to reference the suppress conveniently to common oaihi ceejs y( by just scheme logarithmic a ensalgrtmcsrcueon structure logarithmic a defines ( as written be will structure iioillgrtmcsrcueascae otetrcboundary toric the to associated space structure vector logarithmic real divisorial the in ∆ fan X ntectgr fsevso ood on monoids of sheaves of category the in oaihi tutr soitdto associated structure logarithmic structures pre-logarithmic of category the to structures on logarithmic of category the from functor structure ( structure logarithmic Zariski a identify to going are ( structure logarithmic Zariski every to A] on ( structure étale logarithmic an restricting and o hc h ducinmorphism adjunction the which for on structures logarithmic Zariski to Let (i) Setting . h rpe( triple The xml 4.3 Example by Denote xml 4.2 Example eiiin4.4. Definition morphism a Given oaypelgrtmcsrcue( structure pre-logarithmic any To π X T X isomorphism structures. ρ structures ∗ ivratoe ffiesubset affine open -invariant Zar h sheaf The . 0 y[ By . hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This ◦ γ M o on (or = M 39 ρ and a π A . hoe .]tefunctor the A.1] Theorem , ,M ρ M, X, oehrwt morphism a with together ( : X ,ρ M, UCOILTOIAIAINO OAIHI SCHEMES LOGARITHMIC OF TROPICALIZATION FUNCTORIAL X Trcvreis.Let varieties). (Toric M Dvsra oaihi tutrs.Let structures). logarithmic (Divisorial ρ M opimo oaihi structures logarithmic of morphism et − et a ,w a ( say we ), 0 1 ) → O f sdfie ob h uhu of pushout the be to defined is scle a called is ) epelgrtmcsrcue on structures pre-logarithmic be → X ∗ : X Y ( O ' Zar M → 0 M ρ , X ∗ h aua opim fsts sepandi [ in explained As sites. of morphisms natural the X ,M ρ M, X, N X M . 0 ,M X, samorphism a is ) R ρ , fshms the schemes, of X D ,ρ M, U X = oaihi scheme logarithmic X Zar σ f = π α . ). N − Spec = ∗ or ) ,ρ M, sa is ) − k ssi ob a be to said is ) 1 π n h aeoyo eaelgrtmcsrcue on structures étale logarithmic of category the and ∗ O 1 ρ M ⊗ f X ∗ y   ( ( ⊕ ( X ∗ O X ,s a, O ∈ ,ρ M, R on ) ,ρ M, ea be π → X M S X ∗ aik rs.´tl)lgrtmcscheme étale) logarithmic (resp. Zariski ∗ ihteidcdmorphism induced the with soitdt h ohrce lattice cocharacter the to associated σ K ) ) f X nue neuvlnebtentectgr of category the between equivalence an induces epciey ntelte aetelogarithmic the case latter the In respectively. → ) −→ 7−→ ecnascaeispullback its associate can we ) − [ −−−−→ T

X ,ρ M, S → f 1 trcvreydfie yartoa polyhedral rational a by defined variety -toric nes image inverse σ M O | ⊆ ρ X o cone a for ] γ ecncnnclyascaealogarithmic a associate canonically can we k aχ ( X a ,ρ M, − r( or [ from ) : S D s ,ρ M, O → M hti don otentrlforgetful natural the to adjoint is that σ ftelgrtmcsrcuei defined is structure logarithmic the If . on M ] ,ρ M, O ∈ oaihi structure logarithmic sa smrhs.Fo o nwe on now From isomorphism. an is ) → X X on ) Y D A . X X ∗ M . rmti oainadt denote to and notation this from ) samrhs fpre-logarithmic of morphism a is et

eadvsro omlscheme normal a on divisor a be 0 σ X f to fmni hae uhthat such sheaves monoid of opimo pre-logarithmic of morphism n∆. in ∗ Zar X M X − Zar ihisplbc to pullback its with of T M sgvnby given is M ensa adjoint an defines sdfie ob the be to defined is a O → if , π ∗ 39 34 of 19 Page ( ρ ,ρ M, N X Appendix , nue an induces ti very is It . . of to ) T The . X X X et π et et . ∗ Author Manuscript Let (ii) nqesrc morphism strict unique a morphism morphisms. strict a is There monodromy. no has automatically scheme logarithmic small since a strict, So is (1.6)]. that Lemma spaces monoidal sharp of morphism a ( morphism strict a is connected. and non-empty is isomorphism an is points of locus M to associated P 34 of 20 Page nioopim morphism A isomorphism. an ftemrhssaesrc,te ti ie yedwn h ceetertcfie product fiber scheme-theoretic 4.2. both) the (or endowing either by If given schemes. is logarithmic it saturated then and strict, fine X are of morphisms category the the of in exists product morphisms two Given scheme. logarithmic structure M neighborhoods X U nohrwrs ie nte titmorphism strict another given words, other In 4.7. Proposition scheme logarithmic Zariski small a For 4.6. Definition eiiin4.5. Definition morphism A The term the article, this Throughout scheme logarithmic A X × → n morphism a and i flgrtmcsrcue on structures logarithmic of morphism a of consists fall If . Z = oaihi cee ihu monodromy without schemes Logarithmic Spec Y hrceitcsheaf characteristic M X hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This ihtelgrtmcsrcueascae opr to associated structure logarithmic the with X M P Z /M and i [ P is a ecoe ob ntl eeae fie auae,ec) esytelogarithmic the say we etc.), saturated, (fine, generated finitely be to chosen be can α X sstrict. is ] ∗ x U oeet(n,strtd etc.) saturated, (fine, coherent f : Y ' i P ∈ : of X X M elgrtmcshms A schemes. logarithmic be X α O → X X A aik oaihi scheme logarithmic Zariski A → X, hr h etito map restriction the where : / ups that Suppose P uhta o every for that such chart O Y M X X X F X ∗ falgrtmcsrcue( structure logarithmic a of φ f X flgrtmcshmsi adt be to said is schemes logarithmic of where , X O → on X ) : ssi obe to said is → falgrtmcsrcue( structure logarithmic a of X → P : X X ( X X F X, → φ F esyaZrsilgrtmcscheme logarithmic Zariski a say We . O → X . uhthat such aigtediagram the making P noaKt fan Kato a into M Y oaihi scheme logarithmic Γ( X : X ( ATNULIRSCH MARTIN X fshmstgte ihamorphism a with together schemes of X, X, X eoe h osatsefdfie by defined sheaf constant the denotes ) X saZrsilgrtmcshm ihu monodromy. without scheme logarithmic Zariski a is M → ensacati n nyi h nue morphism induced the if only and if chart a defines X M h isomorphism the quasi-coherent U X X F → P i . M ) X ) X hr samonoid a is there −−→ opimo oaihi schemes logarithmic of morphism Z −→ X ∼ noaKt fan Kato a into φ → and sioopi otelgrtmcstructure logarithmic the to isomorphic is : M X M F Spec ( X, . M X,x Y X ssi ohave to said is X # X M ilawy enafieadsaturated and fine a mean always will → oal it oacatof chart a to lifts locally P M ρ , M if , X X X Z P X X ) ρ , X on ) → flgrtmcshms h fiber the schemes, logarithmic of ⊕ X X = M F strict F on ) P disacvrn yétale by covering a admits X Z M X i noaKt fan Kato a into pr n chart a and hti nta mn such among initial is that X,x sdfie ob h sheaf the be to defined is if , omonodromy no Y # X X M → sgvnb monoid a by given is f is Y [ Γ( small . f : [ f P X, : ∗ M f on M β fteclosed the if , ∗ M i Y M f X F X : X → Y ( induces ) : hr is there , fthere if , τ P X . y[ by → i M ) U → X M i 32 → Y X is , Author Manuscript over Spec ( subset swap affine the open an that once) F eoti iga ffu aofans Kato four of diagram a obtain We X of cover open an choose X union a is all of colimit a is it of since property, diagram universal the desired fan, Kato some into fan Kato subsets open small Spec to isomorphic is point closed ( F o all for [ in appeared originally has that 4.8 Example ( composition the denote also we notation of abuse slight a In commute. f association The commutative. is X, F V X em 4.9. Lemma Fnly on Finally, – Smlry on Similarly, – On – o osdragnrlZrsilgrtmcshm ihu oorm.Cos oe by cover a Choose monodromy. without scheme logarithmic Zariski general a consider Now 4.7. Proposition of Proof o vr aik oaihi ceei ihu oorm,a hw ytefollowing the by shown as monodromy, without is scheme logarithmic Zariski every Not Proof. 4.8 Example 2 { − { − sasrc morphism. strict a is M by Spec = ( y( by N h chart the k a ,q p, q hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This 4 X 2 ihu oorm nue morphism a induces monodromy without x } φ a , → ) X a U ∈ and } → F 1 3 ( a h morphism The k a , C a , n ee oi sthe as it to refer and X by V p N 3 [ k Spec = ) 1 F y a , 3 4 3 1 U ∪ a , ) sa pnan ustof subset affine open an is n hrfr hr sa nue morphism induced an is there therefore and fw r ogu hs oran aofn,w n gigaon h circle the around (going find we fans, Kato affine four those glue to try we If . V N y , 4 noaKt fan Kato a into 1 ' ( ) C 4 V 4 UCOILTOIAIAINO OAIHI SCHEMES LOGARITHMIC OF TROPICALIZATION FUNCTORIAL q ([ morphism A 2 ) and 7→ 1 2 → = ) ] ( 25 U / ' b egu h oaihi tutrsaogteidentification the along structures logarithmic the glue we ftocpe of copies two of ( U 2 ( ( k y xml .) Let B.1). Example ] b , q ( a k i X 1 [ V b ecnie h oaihi structure logarithmic the consider we ) h intersections The . 4 x [ y 4 1 2 x a , 1 2 b , b , { − F 1 x , ie y( by given ) epciey edfiealgrtmcsrcueon structure logarithmic a define We respectively. x , 3 3 3 f ij n h eutn uteti o aofan. Kato a not is quotient resulting the and ) ). b , 2 p 2 ] U nue optbefml fmorphisms of family compatible a induces } / ] 4 / f ij n eoetetocnetdcmoet of components connected two the denote and , ( ,wieon while ), ups rtthat first Suppose ( x x : 1 ysallgrtmcschemes logarithmic small by hrceitcmorphism characteristic f 1 F ( X x x X, P 2 h mletoe ustcontaining subset open smallest the , 7→ ( 2 hti ie y( by given is that ) f, x → P etk h oaihi structure logarithmic the take we ) ( F b ( M f X, 1 1 Y, f N hrfr mligteuieslpoet sabove. as property universal the implying therefore V [ b , Y k F ) 2 F X etn ahohri w nodes two in other each meeting y   O O 2 φ U and of ) F flgrtmcschemes logarithmic of sfntra in functorial is V b , Y X U X ( X U 2 p ) ij 3 F ) ) i egu ln h identification the along glue we b , ic hr sasrc opim( morphism strict a is there Since . ealgrtmcshm hs neligscheme underlying whose scheme logarithmic a be U Spec = 25 F = −−−−→ −−−−→ ( 4 U p ) pedxB]. Appendix , φ φ F i U ) f φ X 7→ Y X i ol aet eietfidwt tefalong itself with identified be to have would le ogv aofan Kato a give to glues F X ∩ y : U N U 1 b F F F ssal ie titmorphism strict a Given small. is 1 i a 4 j y   X y X Y , 1 and 2 f b f a o esal u ecnagain can we but small, be not may a , F 2 of F → F . U n,if and, 2 ( X a , f V F q ) F k Y . 3 V . Spec = M a , k : uhta h diagram the that such X o vr open every For . F U 4 f X ) ( and q sasrc opim then morphism, strict a is 7→ ) → soitdt h chart the to associated N X, x Y F 4 1 a f , 1 Y p M x O [ oal ffiietype finite of locally x F φ F : and X 2 a X U hti functorial. is that V ( N X 2 M 1 x ( ) p 3 hc a the has which , sisunique its as ) sfollows: as Spec = ) → Y,f N U ' q 34 of 21 Page X, soitdto associated 3 Set . ( ( ( V N p ' x X, ) k M ) 3 ∩ N ⊆ → ie by given M X N U 3 U U ) 3 M X ( ( given and , → i p q ) = ) = ) X,x the φ → F : Author Manuscript netbeover invertible tit´tl opimw have we étale morphism strict γ of chart γ splitting a [ in as field, base [ the by of equivalent closure are algebraic definitions an these over charts formal using that δ oooy h hr oodlspace monoidal sharp the topology, in ideal x embedding open the Then M part well-behaved x scheme certain logarithmic of A existence following: the of terms in smoothness logarithmic for charts. check to how [ on in and schemes logarithmic 4.3. morphism strict all a for that particular, in Note, 34 of 22 Page F morphism characteristic morphism induced the mohover smooth ideal h hrceitcfan characteristic the h tutr ha sgvnby given is sheaf structure the X : ∈ ∈ : : 4.10. Corollary of Proof y[ By ups that Suppose words, other In 4.10. Corollary For [ In Write xml 4.11 Example X,x U U U sgvnb edn point a sending by given is X X γ P I → oaihial mohschemes smooth Logarithmically → → 31 − = Y ( x hr sa étale neighborhood an is there tors ednt by denote we 32 ,x M, 1 O X Spec Spec X ∈ hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This .Kt enstento of notion the defines Kato K. ] ( O Spec = M T rpsto 82 n 1.) ehv ht if that, have we (10.1)] and (8.2) Proposition , X,x 0 X,x X swl sa étale morphism an as well as a re netbeon invertible order has = ) U P in ) o h ou fpoints of locus the for k = k k mligthat implying o aoscriterion. Kato’s of 4.10 Corollary immediate following the have We . = eeae by generated X [ [ swtotmonodromy. without is U P P δ ˜ O P X k 0 h nue opimSpec morphism induced the , − ssrc n ehave we and strict is ] ] noe ihtetiillgrtmcstructure logarithmic trivial the with endowed , X,x tors X 1 = Trcvreis.Let varieties). (Toric → salgrtmcshm hti oaihial mohover smooth logarithmically is that scheme logarithmic a is M m δ snra n o vr point every for and normal is o npriua,aZrsilgrtmcscheme logarithmic Zariski a particular, in So, . Spec X F − ⊕ X,ξ γ Let X 1 because , P ( ˜ φ : X a dnie ihtesto eei onso the of points generic of set the with identified can h nqemxmlieli h oa ring local the in ideal maximal unique the X U k X 0 with X M [ y[ By f se[ (see ) P 31 Ξ( → ed point a sends 0 ˜ F X,x s´tl.Mroe,w lohv that have also we étale. Moreover, is ] ealgrtmcshm hti oaihial mohover smooth logarithmically is that scheme logarithmic a be , δ → M hoe .](lose[ see (also 3.5] Theorem , X x sstrict. is Spec − 32 P X ˜ = ) 1 ∆ ∈ − P X ( rpsto 83 n hoe (4.1)] Theorem and (8.3) Proposition , x X 47 X ic h re of order the Since . 2.1 Lemma in as toric being 16 tors X slgrtmclysot over smooth logarithmically is = M sa is k in Ξ( swl sachart a as well as , 0 δ eto .].Tria mednsmyas edefined be also may embeddings Toroidal 2.1]). Section , X,x [ M eto 2]. Section , = ) ∗ otegnrcpito h unique the of point generic the to ATNULIRSCH MARTIN ξ P : X X sawy apdinto mapped always is γ ∆ ∈ γ sétale. is ] U oodlembedding toroidal Set . X ) − /M oaihial mohmorphism smooth logarithmically : U , hr h oaihi tutr stiil ..where i.e. trivial, is structure logarithmic the where X → x 1 M U f 0 =

. ∈ ∆ I : ∗ Spec X → X k X ( X h hrceitcmorphism characteristic The . X [ M | n n auae monoid saturated and ﬁne a , P Ξ( ∆ eatrcvreywt i torus big with variety toric a be (∆) Z → X ] X otepoint the to k x noatrcvariety toric a into → ξ , ) [ Y P ∈ ˜ = ) Spec steKt a soitdto associated fan Kato the is gp nue isomorphisms induces X 30 ] P m hr sa eaeoe neighborhood ´etale open an is there U = hoe .] egvsacriterion a gives he 4.1]) Theorem , k X,ξ . [ O → P U ˜

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X { y   sgvnby given is 6 j = ξ sgvnb utpiaieseminorm multiplicative a by given is ψ ,where ), etos22ad3 n [ and 3] and 2.2 Sections , nan pnsubsets open affine on 0 r qiaet fteei common a is there if equivalent, are ) soitn oascheme a to associating L 7 X } ρ n [ and ] − an 1 X ( stecass aigtemaps the making coarsest the is U oe npriua,ta,if that, particular, in Note, . i X 8 ∩ r X o h eea theory. general the for ] x an eoe the denotes U ylclycoe subsets. closed locally by 7→ j φ ). stesto ar( pair of set the as ker : U Spec i X | ,φ R, ξ . of 0 | i x of K ngnrl i.e. general, In . X of where ) euto map reduction U 34 of 23 Page → ξ X yoe and open by X X 27 Spec( = witnas (written oal of locally , X an Section , htare that modulo k htis that R The . ,φ K, sa is A | X . ). | ) . Author Manuscript n a hn of point think Gauss may One the connects One intervals: connects half-open two contains henceforth | point Gauss on value absolute trivial 34 of 24 Page nltcsae( space analytic point Gauss the and ( functors analytification two line the affine between the differences using the illustrate we examples 5.3. ( functors two the case this In properness. in domain already analytic are an onto isomorphism ρ morphism a . | : r h nltcsae( space analytic The xml 5.2. Example every for Finally, – (Spec inclusion natural The ehv lim have We point closed every For – Frevery For – The – 5.1. Example Let X for

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othe to in . and . A ) i 1 Author Manuscript Σ ∞ hssaecmatfis( compactifies space This η ( k 6.1. geometry. tropical of applications many of heart very the at lies that points the while of point generic point X, X of . xml 5.3. Example oeta h tutr morphism structure the that Note osdra(n n auae)Zrsilgrtmcscheme logarithmic Zariski saturated) and (fine a Consider by 4.2 Section from morphism characteristic its denote and monodromy without O Σ = P rpclzto i aofn h aewtotmonodromy without case the – fans Kato via Tropicalization X hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This 1 a h euto morphism reduction The . ) F in → X P n its and F 1 X to , emyascaeto associate may we 3.3 , and 3.2 Sections in explained As . UCOILTOIAIAINO OAIHI SCHEMES LOGARITHMIC OF TROPICALIZATION FUNCTORIAL P | ∞ . 0 | 1 ∞ r l te points other All . aoia extension canonical h rjcieline projective The for η h point the > r A 6. 1 ( ) r apdto mapped are 1 G an osrcigtetolclzto map troplicalization the Constructing m yadn point a adding by ∞ ) G an r m in : ρ ( | P P . : | P Σ 1 a,r 1 ( 1 u oalohrpit n( in points other all to but , ) P X i ∞ spoe over proper is 1 0 for ) = → an ∞ η Σ P < r → F ti xcl hsdcooybetween dichotomy this exactly is It . 1 ( X P soitsol oteGuspoint Gauss the to only associates P . ∞ 1 r apdt h lsdpoint closed the to mapped are 1 1 ) an soitst seminorm a to associates otehl-pnitra connecting interval half-open the to ( = k n o( so and P 1 A ) i 1 X P oal ffiietp over type finite of locally 1 ) P an 1 ) η an n ( and X h eei point generic the the P | . | 1 ( 34 of 25 Page oecomplex cone a, ) G i 0 m h closed the r equal. are ) G | i a . ρ | m 1 in , and 0 φ η the P X to 1 r : , Author Manuscript Amorphism A (ii) omt.Tu ehv val have we Thus commute. rpclzto map tropicalization phism morphism a by induced being sgvnby given is x R category the in SMS point A ring follows: valuation as defined is 34 of 26 Page Tetoiaiainmpi eldfie n otnos tmkstediagrams the makes It continuous. and well-defined is map tropicalization The (i) ≥ ∈ ups htboth that Suppose Proof. 6.2. Proposition trop that Note 6.1. Definition 0 X omt.Teassociation The commute. diagram the morphism a induces monodromy commute. on en h on trop point the Define . i φ hr savlainrn xedn both extending Ω ring valuation a is there , R # hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This . ie w representatives two Given : P → R SMS X f extending A : sidcdby induced is Spec X nuigtecaatrsi morphism characteristic the inducing The hr val where , X → ( X, r Spec = R X X X rpclzto map tropicalization ≥ y   O k # 0 i X 0 trop fZrsilgrtmcshmslclyo nt yeover type finite of locally schemes logarithmic Zariski of X n hsntrlyidcsamorphism a induces naturally this and ( ◦ −−−−→ f x x ) α f val ) A : # ∈ −−−−→ −−−−→ 7→ (val = X φ ∈ X trop x # trop X X and eoe h opimidcdb h auto val valuation the by induced morphism the denotes φ Spec pcΩ Spec : Σ Σ( 7−→ X → X i naaoywt h nltcmorphism analytic the with analogy in Σ( X X α y   Spec f X ATNULIRSCH MARTIN i a erpeetdb opimSpec morphism a by represented be can 0 f ( X F : = f ) ) R X i X # : −→ Spec Σ X F ) p 0 sfntra in functorial is y   y   F X 0 X : X i fshmslclyo nt over finite of locally schemes of ◦ ) →− 7→ −−−−→ −−−−→ R Spec = r X i Σ i x F β Σ X ( X R α 7−→ −→ −−−−→ R −−−−→ −−−−→ soitdto associated n otrop so and X trop trop ≥ → log → x Hom( = 0 stecomposition the as ) trop Σ X X Σ P Spec X | 0 R X φ X ( # r ohan n hoeahomomor- a choose and affine both are X y   0 ( and X, and Σ ρ X Σ X, ( X hs aoia extension canonical whose β y   X p R X ( X f y   ) P, Σ( x O 0 0 O i | X . ) β R X f R X φ X ) swell-defined. is 0 : ) ≥ X ) Spec htmkstediagram the makes that 0 −−−−→ −−−−→ −−−−→ : ) trop φ Spec φ X X R X 0 x → A Σ : F X y   Spec → X X X ρ F k X Spec ftesm point same the of . f R i : → P R X Σ( hnthe Then . ( → i X, k f → X without ) O makes : ( X X o a for R in ) 0 → ) i Author Manuscript e aelgrtmcscheme étale logarithmic map as well as point R commutative. is extending maps reduction the for ρ diagram the that established have we once glueing, n h pair the and scheme logarithmic a by extension canonical the into 6.2. for omt,te hr sauiu opimΣ morphism unique a is there then commute, Σ morphism a étale is morphisms strict there all monodromy for without that scheme such Σ complex cone generalized Σ( set hereby we that Note map. tropicalization X

rpsto 6.3. Proposition of characterization following the from follows diagrams the of commutativity The nti eto ecntutaddsrb h rpclzto a soitdt general a to associated map tropicalization the describe and construct we section this In nohrwrs ie otnosmap continuous a given words, other In the of deﬁnition the and 3.10 Proposition 4.9, Lemma from follows immediately (ii) Part | Given : a x | ∈ rpclzto ntepeec fmonodromy of presence the in Tropicalization < η hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This g X 1 =

i k x hssosta ntean aetrop case aﬃne the in that shows This . ∈ h euto map reduction the , a ∈ (Σ Spec ∈ X X UCOILTOIAIAINO OAIHI SCHEMES LOGARITHMIC OF TROPICALIZATION FUNCTORIAL R i ,

R trop ( ersne yamorphism a by represented | φ a ( φ n h tutr morphism structure the and | X X 0 = X hr sagnrlzdcn complex cone generalized a is There ◦ ◦ ) ρ U X

r siiilaogalsc maps. such all among initial is X X ihu oorm hr sacmuaiediagram commutative a is there monodromy without ∈ )( hti oal ffiietp over type finite of locally is that Σ )( x Spec X x = ) = ) r = ( = of = ( = X R φ ρ Σ φ r φ sends F X φ hrfr ehave we Therefore . F X X trop X X X X X X U x U uhta o vr titsretv étale cover surjective strict every for that such ◦ ◦ y   x y   ◦ ◦ ( i i i val i x ( η X val x η x ) g ) s −−−−→ −−−−→ ) −−−−→ −−−−→ : # ) otepoint the to trop # trop trop trop val X val X ◦ τ ◦ x i φ − φ X : U → X U f − X : −→ 1 X X ρ Σ( = ) Spec 1 ( X X ◦ ( aigtediagram the making Σ {∞} i ◦ U Σ Σ Σ R x Σ y   y   x sends scniuu;tegnrlcs olw by follows case general the continuous; is X → U U Σ > → ( = R 0 ( = X ) ) f notecnniletnino a of extension canoncial the into Σ aigtediagram the making Σ x → F ρ r ( Σ F swl as well as ) x η F X X X s X otepoint the to k o h pca point special the for ) ◦ n otnostropicalization continuous a and ◦ . o nitga auto ring valuation integral an for trop trop U X X )( → )( Σ( x x X ) ) f x . = ) ( rmalogarithmic a from η g o h generic the for ) Σ( 34 of 27 Page f F η ). s U r = X → and a X R ∈ Author Manuscript morphism Whenever cone the commute. map continuous natural a induces that diagram complexes cone alized product fiber morphism étale strict schemes logarithmic small Σ all for commutes every Since diagram space. the Then topological above. of as Σ category complex the well. as in commutes (6.3) colimits diagram of the property scheme universal logarithmic the by map tropicalization continuous natural a define may we therefore étale morphism morphisms strict face all induced an morphism have strict we another is there Whenever map P commute. 34 of 28 Page group ( U U X ups o that now Suppose 1.1. Theorem of Proof Let [ By Proof. 0 , Γ( = → M τ 1 π U sdesired. as Σ em ..]teaayi space analytic the 6.1.3] Lemma , 1 0 U, : hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This ) ( σ X E → σ o vr titétale morphism strict every For E E M prtso h cntn! hrceitcmonoid characteristic (constant!) the on operates ) i E Hom( = U F 0 si h lsr fastratum a of closure the in is → U → U ,w aeacone a have we ), 0 = eacniuu a notecnncletnino eeaie cone generalized a of extension canonical the into map continuous a be Σ σ ssrc.Teeoeteei nqemrhs Σ( morphism unique a is there Therefore strict. is E X U X M hti nue yteseilzto map specialization the by induced is that × ihu oorm a ewitna oii fsaloe subsets, open small of colimit a as written be can monodromy without E X slgrtmcll moh o vr stratum every For smooth. logarithmicallly is U , 0 0 R U U Let → ≥ 0 → 0 sas ihu oorm,snetecmoiin( composition the since monodromy, without also is X .Dnt by Denote ). f X 0 : σ rmalgrtmcscheme logarithmic a from rmasallgrtmcscheme logarithmic small a from X U trop X σ trop trop Hom( = f → ( V i X i ATNULIRSCH MARTIN X X U y   → X y   X 0 i ) i U i X i 0 σ : −−−−→ −−−−→ : X eamrhs flgrtmcshms o every For schemes. logarithmic of morphism a be U U H −−−−→ V −−−−→ U trop E X trop P Σ trop en Σ Define . U i E i → τ τ 0 U → i X Y stetplgclclmto l such all of colimit topological the is btnteulto equal not (but , −→ X h mg of image the n hrfr eoti nqemorphism unique a obtain we therefore and X −→ R X U 0 ≥ Σ 0 rmasallgrtmcscheme logarithmic small a from Σ rmasallgrtmcscheme logarithmic small a from y   Σ σ Σ U Σ swl sacniuu tropicalization continuous a as well as ) y   U X X X Σ( Σ 0 X . f ) steclmto all of colimit the as π 1 M Σ( U ( E M 0 E f E nAut( in ) ) U ihu oorm,the monodromy, without E → 0 f : ehv rprface proper a have we ) . E ) Σ on M : of X Σ E E → X X 0 . M n hrfr on therefore and → Σ h fundamental the σ X E Σ U 0 Aut( = ) X ae over taken , aigthe making 0 U, fgener- of V M U U i U to with σ ) and E U → ). , Author Manuscript in that note and retraction embedding over toroidal type finite of Σ defines that morphism face of diagram the follows. of description claim complete a gives This over type finite of homeomorphism of 6.3. stratum closed unique a is there of strata logarithmic all over taken oolie3.13] Corollaire commute. trop p Σ σ on E X : Proof. Let 1.2. Theorem prove we section this In 6.4 Proposition o osdraZrsilgrtmcshm hti oaihial mohover smooth logarithmically is that scheme logarithmic Zariski a consider Now whenever Moreover, – of operation The – ups rtthat first Suppose X S X k : = P 0 H opimfo ml oaihi ceesc httecoe tau of stratum closed the σ that such scheme logarithmic into small a from morphism [ ( P i o-rhmda kltn flgrtmclysot schemes smooth logarithmically of skeletons Non-Archimedean E σ hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This X , → E ysending by n so and ] U → . rssti a n,since and, way this arises ) oigfo h pcaiainmap specialization the from coming E X Hom( = X p Let 0 . h titétale map strict the , i : X sgvnb sending by given is U i J x p UCOILTOIAIAINO OAIHI SCHEMES LOGARITHMIC OF TROPICALIZATION FUNCTORIAL J X → P X → u X k and U k hntegnrlzdcn ope of complex cone generalized the Then . ∈ X swl-end otnos n ufil trop fulfills and continuous, well-defined, is X , X suiul eemndb t etito osaloe subsets open small to restriction its by determined uniquely is ([ : X h oaihi scheme logarithmic the 4.10 , Corollary in seen as Then, . R Σ Σ 1 π 0 i ≥ Spec = p easrc eaemrhs rmasallgrtmcscheme logarithmic small a from étale morphism strict a be rpsto 6.2.6). Proposition ] 1 P X , → 0 ( ( nothe onto E .Teeaetocasso aemrhssi h iga defining diagram the in morphisms face of classes two are There ). x Hom( = E → ge,we etitdto restricted when agree, ) X nue rcsl h uoopim of automorphisms the precisely induces ) si h lsr fastratum a of closure the in is S h anrsl f[ of result main The . k ( J X [ X P p U X E ∈ skeleton aigtediagram the making ) P, ( nan oi ait.Hr h eomto ercinmap retraction deformation the Here variety. toric affine an ] x 0 P p X u of X ∈ → ∈ ) R a X P k p . : ≥ Σ [ X X U a χ P i k 0 Σ X p p [ i oteseminorm the to ) ] χ P nue h rprfc morphism face proper the induces h lsdsrtmof stratum closed the P −→ 7−→ lim = p oteseminorm the to S ] trop −→ 7−→ scmat h rpclzto a nue a induces map tropicalization the compact, is ( M −→ p Let X R max X E of ) X ≥ σ max R X 0 → E ealgrtmclysot ceelocally scheme smooth logarithmically a be ≥ | S /H 0 a X ealgrtmclysot ceelocally scheme smooth logarithmically a be 47 Σ M p ( | i a || X X E sta hr sasrn deformation strong a is there that is ] J E p χ P htdpnso h oodlstructure toroidal the on depends that ) X | 0 p e . en section a Define . | − ( x u X ( ) E p ) sgvna h colimit the as given is 0 U and smpe noadw have we and into mapped is P ◦ U J σ X 0 U → d vr element Every id. = led otie in contained already σ U J E X 0 sasrc étale strict a is = : 34 of 29 Page U Σ X σ X 0 P U k n othe so and smapped is 0 y[ By . → ensa defines U U → Then . X of σ i U 47 X of = , . Author Manuscript Spec complex [ in defined as map tropicalization extended Kajiwara-Payne the Consider varieties. toric on background and in ∆ fan polyhedral rational monodromy. without [ by of since homeomorphism, colimit a is This commutes. map continuous a is there that implies colimits of property Σ universal the and commute [ By monodromy. without i.e. topology, Zariski the diagrams in defined also is morphism claim. the yields this 1.1 Theorem by of stratum of closed skeleton the the 3.7] of image the étalethat morphism strict a Choose 34 of 30 Page S X ( Proof. 7.1. Proposition Let that assume Finally U → ) k −→ [ X ∼ S S σ hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This ( Σ = .W a aual identify naturally can We ]. S X S X U ti nuht hc h setoson assertions the check to enough is It ( X uhta h diagram the that such ) ( Z R → stecoueof closure the is ∆ eanra oi ait ihbgtorus big with variety toric normal a be (∆) 29 .Snetetoiaiainmpntrlyfcosas factors naturally map tropicalization the Since ). ) X ⇒ eto ]a ela [ as well as 1] Section , σ rmaZrsilgrtmcschemes logarithmic Zariski a from S U = 7. ( i U X u sdfie as defined is o oi variety toric a For o vr titétale morphism strict every for ) oprsnwt aiaaPyetropicalization Kajiwara-Payne with Comparison ∈ sa eaelgrtmcshm.Te o vr titsretv étale surjective strict every for Then scheme. étale logarithmic an is U Hom( N i ∆ R −−−−→ Hom( = in γ S γ σ : N i trop , ⊆ S X U X R R X y   X U ( (∆) an ) ATNULIRSCH MARTIN i → y   U i ∆

41 Z i i u M, ( = ) U ( i : Z −−−−→ −−−−→ n hr sacmuaiediagram commutative a is there and , 1.1.2. Section in to alluded and 3] Section , s −−−−→ −−−−→ trop trop X ) trop si h closed the in is R X −−−−→ noa affine an into p p ≥ an p trop .W ee o[ to refer We ). γ X U X ∆ = i 0 X 47 −→ ) Z X − N ∀ rpsto .1 h skeleton the 3.31] Proposition , S S 1 (∆) S s R Hom( Σ N Σ ( ( y   S (∆) ∈ y   ( X U X X X R U ⊆ ( T J S ) (∆) ) Z ehave we ) h oaihi scheme logarithmic the , X ivratoe ffiesubsets affine open -invariant σ U ) . P,

T → Hom( = T and 13 R trcvariety -toric T obtof -orbit ≥ X n [ and ] 0 Spec = γ ) rmalgrtmcscheme logarithmic a from Σ i , X nue homeomorphism a induces S 18 ∆ = σ 47 , Z k o tnadnotation standard for ] R rpsto .9 all 3.29] Proposition , [ y[ By . M ≥ h xeddcone extended the , Z 0 eemndb a by determined ] ) Spec = 47 Proposition , R S = ( k X [ U P sthe is ) such ] × U J X σ X U U = : Author Manuscript xeddcn complex cone extended Trop( Trop( hnteKt fan Kato the Then below. indicated that such fields base all (over 11.12] in in variety tropical trop identity Hom( and utpiiy2 n seult h -kltnof 1-skeleton the to equal is and 2) multiplicity Trop injective. not general in is morphism inclusion well. as [ true By is claim. the implies 7.1 Proposition Proof. oolr 7.3. Corollary Proof. 7.2. Corollary Let Consider 7.4. Example endow may One N N R R X Y, Y, (∆). ∆;oemyatraieycaatrz Trop( characterize alternatively may one (∆); hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This Y ( Y ∆) ∆) eacoe ustof subset closed a be eaieto relative ) i S Σ( If 7.1. Proposition of consequence immediate an is This . : ∩ σ Y Y ∆ , i Y ∆ R salgrtmcshm ihrsett h ulaklgrtmcstructure logarithmic pullback the to respect with scheme logarithmic a as ) ( → x . ≥ Trop( Trop spoe over proper is trop = ) UCOILTOIAIAINO OAIHI SCHEMES LOGARITHMIC OF TROPICALIZATION FUNCTORIAL 0 X stecoueof closure the is ) Y Let teei piecewise a is there 1.1 Theorem By . F ihtelgrtmcsrcueta sgvna h ulakof pullback the as given is that structure logarithmic the with Y Y Y, X ie lsdsubset closed a Given h lsdsubset closed The ( Σ Y Y sgvnb i oiso Spec of copies six by given is )i endt eteiaeudrtrop under image the be to defined is ∆) X sgvnb olpigec ftepiso oe ooe(ihtropical (with one to cones of pairs the of each collapsing by given is Y ) k ( eagnrccncta nescstetrcbudr of boundary toric the intersects that conic generic a be ossso i oisof copies six of consists x ) Trop = od o all for holds ) Y X spoe over proper is o ipiiyw assume we simplicity for ; k ehave we , X σ ( Y in this 7.4 Example following the in seen be can As ). N 46 R Y x rpsto .](over 2.3] Proposition , ( Y σ ∈ k Hom( = ) spoe over proper is i fadol fTrop( if only and if , U Y Σ = σ i P R of 2 Y ⊆ Y . ≥ Y, an U 0 N X )a h lsr ftrop of closure the as ∆) σ le tteoii.Tetoia variety tropical The origin. the at glued an S ehv h identity the have we , n hrfr Trop( therefore and le vrtegnrcpit n the and points generic the over glued σ Y . Z , P R ∩ lna morphism -linear 2 .Udrteeietfiain the identifications these Under ). T k 6= ∆ fadol if only and if ∅ Y, ftecoe subspace closed the of hogot t associated Its throughout. C ∆) n [ and ) ⊆ .S h converse the So ∆. Y, Σ( 27 ∆) ∆ i 34 of 31 Page Proposition , i ) Trop Trop ∗ : ⊆ M ( X Y Σ X .Then ∆. Y ∩ X X = i the via i → ( ( T ∗ P Y Y M ) Y 2 Σ = ) = ) an X an as X . Author Manuscript yed h claim. the yields 1.2 Theorem of intersection Trop the that if means only also this above, 7.3 of intersection the when precisely irreducible. of case strata logarithmic the the is between correspondence one-to-one a one). of intersection the if of subvariety 34 of 32 Page ssot.B [ By smooth. is nue from induced 8. 9. 7. 6. 5. 4. 3. 2. 1. 1.3 . Corollary of Proof Proof. that Suppose example. above the in made observation the generalize now us Let rpsto 7.5. Proposition ldmrG Berkovich. G. Vladimir ldmrG ekvc.Vnsigcce o omlshms II. schemes. formal for cycles Vanishing Berkovich. G. Vladimir ate ae,SmPye n oehRbnff oacieengoer,toiaiain n metrics Berkovich. and G. tropicalization, geometry, Vladimir stable Nonarchimedean Rabinoff. of Joseph space and theory. Payne, the Gromov-Witten Sam logarithmic Baker, of Matthew in Boundedness Invariance Wise. Wise. Jonathan Jonathan and and Abramovich Dan Marcus, Steffen Chen, Qile Abramovich, Dan a baoih ieCe,See acs atnUish n oahnWs.Seeosadfn of fans and Skeletons Wise. and Jonathan and Satriano, Ulirsch, Matthew Martin Marcus, Olsson, Steffen Martin Chen, Qile Huang, Abramovich, Yuhao Dan Gillam, Danny Chen, Qile Abramovich, Dan a baoih ui aoao n a an.Tetoiaiaino h ouisaeo curves. of space moduli the of tropicalization The Payne. Sam and Caporaso, Lucia Abramovich, Dan ul Math. Publ. Monographs and Surveys Mathematical curves. on maps. logarithmic [math] 2016. Publishing, International Springer In structures. logarithmic In moduli. and geometry Logarithmic Sun. Shenghao et ah (ALM) Math. Lect. Sci. Ann. hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This ue2013. June , h map The c om Supér. (4) Norm. Ec. ´ ler Geom. Algebr. X X 7)511(94,1993. (1994), (78):5–161 , ..asm httemlilcto map multiplication the that assume i.e. , en oaihial mohover smooth logarithmically being , 49 rpsto .]ti seuvln to equivalent is this 2.7] Proposition , ri:4806 [math] arXiv:1408.0869 Y Σ( ae –1 n.Pes oevle A 2013. MA, Somerville, Press, Int. 1–61. pages , ihevery with i h nue map induced The ) pcrlter n nltcgoer vrnnAcieenfields non-Archimedean over geometry analytic and theory Spectral tl ooooyfrnnAcieenaayi spaces. analytic non-Archimedean for cohomology Etale ´ Y : ()6–0,2016. 3(1):63–105, , Since Σ o-rhmda n rpclGeometry Tropical and Non-Archimedean ihevery with Y 84:6–0,2015. 48(4):765–809, , → Y Σ T spoe,w have we proper, is X obtin -orbit ∆ sa smrhs fadol fteebdiginduces embedding the if only and if isomorphism an is mrcnMteaia oit,Poiec,R,1990. RI, Providence, Society, Mathematical American . ( T ATNULIRSCH MARTIN µ Y obtin -orbit uut21.J u.Mt.Sc,t appear. to Soc., Math. Eur. J. 2014. August , : = ) Σ( References T i × X ) Σ : Y X o-mt n reuil ie a multiplicity has (i.e. irreducible and non-empty Y Σ w have we 7.5 Proposition By . X −→ Y ihevery with → snnepyadirdcbe n therefore and irreducible, and non-empty is adoko oui o.I Vol. moduli. of Handbook k X Y . Σ i X Σ = Y sa smrhs of isomorphism an is Y net Math. Invent. Y ihtelgrtmcstructure logarithmic the with , an T Y obtin -orbit iosSmoi,pgs287–336. pages Symposia, Simons , n,a xlie nCorollary in explained as and, n the and 2()3730 1996. 125(2):367–390, , X nt Hautes Inst. T snnepyand non-empty is obt of -orbits oue2 of 24 volume , Σ Y Σ arXiv:1306.1222 Y ' Y oue3 of 33 volume , fadonly and if Σ tdsSci. Etudes saschön a is ´ X X This . fand if Adv. Author Manuscript 14. 13. 12. 11. 10. 15. 34. 33. 22. 21. 20. 19. 18. 16. 35. 32. 31. 24. 23. 17. 39. 37. 36. 30. 29. 28. 27. 26. 25. 40. 38. 46. 45. 44. 43. 42. 41. ai neiaCeo aha ¨bc,adAnteWre.Fihu rpclzto fteGrassmannian the of tropicalization Faithful Werner. Annette Häbich, Schenck. and Mathias K. Cueto, Angelica Henry Maria and Little, covers. B. admissible John rational of Cox, space of A. the spaces David Tropicalizing Moduli Ranganathan. Dhruv and Ranganathan. Markwig, Dhruv Hannah Cavalieri, and Renzo varieties. Markwig, toric Hannah of Hampe, geometry Simon Arithmetic Cavalieri, Mart´ınRenzo Sombra. and Philippon, Patrice Gil, Burgos José Ignacio .DlgeadD ufr.Teirdcblt ftesaeo uvso ie genus. given of curves of space the of irreducibility The Mumford. D. and Deligne P. inF nde.Tepoetvt ftemdl pc fsal uvs I h stacks The II. curves. stable of space moduli the of projectivity The Saint-Donat. Knudsen. B. F. Finn and Mumford, D. Knudsen, Faye Finn Kempf, G. I. data. degeneration logarithmic via symmetry Mirror Siebert. Bernd and e-print Gross arXiv Mark Embeddings. Toroidal of release. Tropicalizations on sixth Theory Intersection – Gross. varieties. theory Andreas tropical ring of Equations almost Giansiracusa. for Noah and Foundations Giansiracusa Jeffrey Ramero. Lorenzo and Gabber Ofer Fulton. William morphisms. toroidal on remarks Some Denef. Jan lvrLrced h emtyo lerns atI leri akrudadshm theory. scheme and background Algebraic I: Part blueprints: of geometry The Lorscheid. Oliver singularities. In Toric Kato. Fontaine-Illusie. Kazuya of structures Logarithmic Kato. Kazuya geometry. complex to geometry affine II. real data, From degeneration Siebert. logarithmic Bernd via symmetry and Mirror Gross Mark Siebert. Bernd and Gross Mark tropical and amoebas Non-Archimedean Lind. Douglas and Kapranov, Mikhail Einsiedler, Manfred atnC lsn oaihi emtyadagbacstacks. algebraic and geometry Logarithmic Olsson. C. Martin Sturmfels. Bernd and Maclagan tropicalization. Diane theoretic Scheme Lorscheid. Oliver theory. deformation smooth Log In Kato. Fumiharu geometry. toric Tropical Kajiwara. Takeshi tropicalizations. and Skeletons Werner. Annette and Rabinoff, Joseph Gubler, Walter In tropicalizations. to guide spaces. A analytic Gubler. Walter non-Archimedean for varieties Tropical Gubler. invariants. Walter Gromov-Witten Logarithmic Siebert. Bernd and Gross Mark rt akr xlddmanifolds. Exploded Parker. Brett curves. tropical and varieties toric of degenerations Toric Siebert. Bernd and Nishinou Takeo hu agnta.Mdl frtoa uvsi oi aite n o-rhmda geometry. non-Archimedean and varieties toric in tori. of curves subvarieties of rational Compactifications Tevelev. of Jenia Moduli intersections. tropical and Ranganathan. polygons, Dhruv Newton In geometry, analytic Tropical tropicalization. geometry. algebraic Local Rabinoff. complex Joseph Stepanov. to relations Dmitry and and spaces Popescu-Pampu analytic Patrick nonarchimedean tropicalizations. of all Topology of limit Payne. the Sam is Analytification Payne. 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