Khovanskii Bases of Cox-Nagata Rings and Tropical Geometry MarthaBernal,DanielCorey,MariaDonten-Bury,NaokiFujita,andGeorgMerz 7 1 0 2 n a J 2 1 Abstract The Cox ring of a delPezzo surface of degree3 has a distinguished set of27 minimalgenerators.We investigateconditionsunderwhichthe initialforms ] G of these generatorsgeneratethe initialalgebraofthis Cox ring.SturmfelsandXu A provide a classification in the case of degree 4 del Pezzo surfaces by subdividing . the tropical Grassmannian TGr(2,Q5). After providing the necessary background h on Cox-Nagata rings and Khovanskiibases, we review the classification obtained t a by Sturmfels and Xu. Then we describe our classification problem in the degree m 3 case and its connections to tropical geometry. In particular, we show that two [ naturalcandidates,TGr(3,Q6)andtheNarukifan,areinsufficienttocarryoutthe classification. 1 v 5 3 4 3 0 . 1 0 7 MarthaBernal 1 Unidad Acade´mica de Matema´ticas UAZ, Calzada Solidaridad, Zacatecas, Mexico, e-mail: : [email protected] v i DanielCorey X YaleUniversity,DepartmentofMathematics,e-mail:[email protected] r a MariaDonten-Bury University of Warsaw, Institute of Mathematics, Banacha 2, 02-097 Warszawa, Poland, e-mail: [email protected] NaokiFujita Department of Mathematics, Tokyo Institute of Technology, 2-12-1 Oh-okayama, Meguro-ku, Tokyo152-8551,Japan,e-mail:[email protected] GeorgMerz Mathematisches Institut, Georg-August Universita¨t Go¨ttingen Bunsenstraße 3-5, D-37073 Go¨ttingen,Germany,e-mail:[email protected] 1 2 MarthaBernal,DanielCorey,MariaDonten-Bury,NaokiFujita,andGeorgMerz 1 Introduction ThestartingpointforthischapteristhefollowingproblemproposedbySturmfels and Xu as [26, Problem 5.4]: determine all equivalence classes of 3-dimensional sagbisubspacesofk6.Inthenextfewparagraphsweexplainitsstatementindetail andgiveanoutlineofthechapter. Letusbeginwithclarifyingtwoimportantaspectsofournotation.First,instead of the names sagbi bases resp. sagbi subspaces (where sagbi, first used in [20], standsfor“subalgebraanaloguetoGro¨bnerbasesforideals”)wewillusethename Khovanskiibasesresp. Khovanskiisubspaces.Thisnewnamewasintroducedina muchmoregeneralsettinginarecentarticle[12]. Second,wemakesomeassumptionsonthefieldk.Weusuallytakektobethe fieldofrationalfunctionsQ(t),buttoformulateandworkonthisproblemonemay consider any other field with a nontrivialvaluation. The residue field of k for the consideredvaluationwillbedenotedbyk. ThefundamentalobjectsforthischapterareKhovanskiibasesandmonericsets. WerepeattheirdefinitionsafterSturmfelsandXu;see[26,Sect.3]formoredetails andcommentsontheirproperties.Byval: k×→Zwedenoteavaluationmapofk. Ifk=F(t)forsomefieldF,weusethefollowingvaluation:val(p)∈Zistheunique integerw suchthatt−w p(t)takesanonzerovalueatt=0.Then,for f ∈k[x ,...,x ] 1 n wecancomputeitsinitialformin(f). Ifw istheminimumofvalforcoefficients 0 ofallmonomialsin f,then in(f)=(t−w 0f)|t=0∈k[x1,...,xn]. Thatis,in(f)identifiesallmonomialsof f whosecoefficientshavesmallestvalua- tion. Definition1.1.We calla subsetF ⊂k[x ,...,x ] monericif in(f) is a monomial 1 n forall f ∈F. For a k-subalgebraU ⊆k[x ,...,x ] we define the initial algebrain(U) as the 1 n k-subalgebrageneratedbyin(f)for f ∈U. Definition1.2.WesaythatasubsetF ⊂U isaKhovanskiibasisofak-subalgebra U ⊆k[x ,...,x ]if 1 n • F ismoneric,and • theinitialalgebrain(U)isgeneratedby{in(f)|f ∈F}asak-algebra. We are interested in Khovanskii bases of Cox-Nagata rings, which will be de- scribed in Section 2. After they are introduced, we will be able to explain how a 3-dimensional subspace of k6 determines a basis, possibly a Khovanskii basis, of the Cox ring of a del Pezzo surface of degree 3. We say that such a subspace ismoneric(resp.Khovanskii)ifthecorrespondingbasisismoneric(resp.Khovan- skii),seeDefinition2.4.Welookatmonericsubspacesuptoanequivalencerelation whichrespectsthepropertyofbeingaKhovanskiisubspace,seeDefinition2.5. KhovanskiiBasesofCox-NagataRingsandTropicalGeometry 3 We suggest that the reader treats this text as an introduction to the concept of Khovanskiibasesandrelatedresearchproblems.Forus,understandingthegeomet- ricmotivationandconnectionswasasimportantassolvingthecombinatorialclas- sificationproblemitself.Thisisthereasonwhy,besidespresentingourapproachto answeringthemainquestion,wealsospendsignificantamountoftimeonexploring itsbackground. In Section 2 we define the Cox ring and explain its construction for del Pezzo surfaces.WealsointroducetheNagata’saction,whichprovidesalinkbetweenlin- ear subspacesof kn and choices of initial formsof generatorsof Cox rings of del Pezzosurfaces(i.e.candidatesformonericorKhovanskiibasesoftheCoxring). Section3isdedicatedtoexplainingthegeometricconsequenceofaKhovanskii basisintermsofdegenerations.Roughlyspeaking,aKhovanskiibasisofa(finitely generated)subalgebraU of the polynomialringyields a degenerationof Spec(U) to a toric variety. We show that we obtain even more if we choose a Khovanskii basisoftheCoxringCox(X)ofavarietyX:wedonotonlyobtainatoricdegenera- tionofSpec(Cox(X)),butalsotoricdegenerationsofX withrespecttoallpossible embeddings. In Section 4 we explain and give examples for the problem which motivated SturmfelsandXutostudyKhovanskiibasesofCox-Nagatarings.Itturnsoutthat aKhovanskiibasisallowsustocomputetheHilbertfunctionofadelPezzosurface with respect to a specific embedding by counting lattice points in dilations of a rationalconvexpolytope. Finally, Sections 5 and 6 describe our first attempts to classify 3-dimensional Khovanskiisubspacesofk6. Firstwe describetwotropicalvarietieswhichweex- pecttoberelatedtotheproblem:thetropicalGrassmannianTGr(3,6)andthetrop- icalmodulispaceofdelPezzosurfacesofdegree3.Thenweexplainhowwetried to use them as parametrizing spaces for moneric and Khovanskii subspaces. The conclusionisthatneitherofthesemodelshasthecombinatorialstructuresuitableto playthisrole. 2 Cox-NagataRings Let G be a linear group acting on a polynomial ring R over a field k. Hilbert’s fourteenth problem asks whether the ring of invariants RG is a finitely generated K-algebra.TheanswerisaffirmativewhenthegroupGisreductiveandalsowhen G=G .Nagataconsideredtheactionofacodimension3subspaceG⊂Cn acting a onR=C[x ,...,x ,y ,...,y ]via 1 n 1 n x 7→x andy 7→y +l x, i i i i i i where(l ,...,l )∈G.HeprovedthattheringofinvariantsRGisnotfinitelygener- 1 n atedforn=16,see[17].MukairealizedtheringofinvariantsRG asacertainRees algebraandassuch,itis isomorphicto the Coxringofa blow-up([16]). Mukai’s 4 MarthaBernal,DanielCorey,MariaDonten-Bury,NaokiFujita,andGeorgMerz description of RG providesconditions for it to be finitely generated and a way of computingitsgenerators,atleastforcodimG≤3. In this section we review Mukai’s description of RG. We recall the definition ofCoxringsandstudywith somemoredetailtheisomorphismbetweenRG when codimG=3 and the Cox ring of the blow-up of P2 at n points in general posi- tion. Nextwe specialize to the blow-upof six pointsandgive a descriptionofthe invariantsthatgenerateRG. 2.1 Nagata’saction LetR=k[x ,...,x ,y ,...,y ]beanalgebrawithaZn-gradingviasettingdeg(x)= 1 n 1 n i deg(y)=e,wheree ,...,e isastandardbasisofZn.LetG⊂Cnbeasubspaceof i i 1 n codimensionrgivenbytheequations a t +...+a t =···=a t +...+a t =0. 111 1nn r11 rnn We considerNagata’sactionofGonR.Asx isinvariantforeveryi=1,...,n,we i canextendtheactiontothelocalization y y R =R[x−1,...,x−1]=k[x±1,...,x±1, 1,..., n]. x 1 n 1 n x x 1 n The grading on R extends naturally to a grading on R with deg(x−1)=−e. x i i Now, l =(l ,...,l )∈G acts on R by x 7→x and yi 7→ yi +l . Let y′ = yi. 1 n x i i xi xi i i xi Thenl ∈G acts onk[x±1,...,x±1,y′,...,y′] by x 7→x andy′ 7→y′+l . A direct 1 n 1 n i i i i i computationshowsthattheinvariantringRGisgeneratedoverk[x±1,...,x±1]bythe x 1 n linearpolynomials l′:=a y′ +···+a y′, 1≤i≤r. i i1 1 in n Letx =(cid:213) n x and 0 j=1 j y y l =x ·l′= x a 1 +···+a n . (1) i 0 i 0 i1x inx (cid:0) (cid:1)(cid:0) 1 n(cid:1) We define the algebra U := k[l ,...,l ] ⊂ R ∩RG. Let V be the k-vector space 1 r x spanned by l ,...,l . ThenU is a Z-graded ring andV is its degree one part. We 1 r also letV ⊂V be the polynomialsinV thatdo nothavey (cid:213) x asa monomial i i i6=j j andI ⊂U theidealgeneratedbyV.Thenwehavethefollowing: i i Proposition2.1.TheinvariantalgebraRGistheextendedmulti-Reesalgebra U[x ,...,x ]+ (cid:229) Id1∩···∩Idn x−d1···x−dn ⊂U[x±1,...,x±1]. 1 n 1 n 1 n 1 n d∈Zn(cid:0) (cid:1) Proof. Aproofisfoundin[16]orinthebook[1],section 4.3.4. ⊓⊔ KhovanskiiBasesofCox-NagataRingsandTropicalGeometry 5 2.2 Cox Rings TheCoxringofasmoothprojectivevarietyX overthefieldk,withfinitelygener- ateddivisorclassgroupCl(X),isthering Cox(X)= H0(X,O (a D +···+a D )), X 1 1 r r (a1,..M.,ar)∈Zr where D ,...,D is a fixed basis of Cl(X)≃Zr. This ring has the structure of a 1 r k-algebra.WhenitisfinitelygeneratedthevarietyX iscalledaMoriDreamSpace. ThisisthecaseforsmoothdelPezzosurfacesofdegree1≤d≤9,forwhichgen- eratorsandrelationsamongthemareknown. WeletAbeanr×nmatrixwithentriesinksuchthatGisthekernelofA.We denotebya(i)thei-thcolumnvectorofAandassumethattheyarepairwiselinearly independent.DenotebyX thedelPezzosurfaceresultingfromtheblow-upofPr−1 G atndifferentpointswithhomogeneouscoordinatesa(i).ThedelPezzosurfaceX G isdeterminedbyGonlyuptoisomorphism:anisomorphismofP2 asalinearmap leavestherowspaceofA,andthereforealsothekernelG,invariantandinducesan isomorphismofthecorrespondingdelPezzosurfaces.ThePicardgroupPic(X )is G isomorphictoZn+1andisgeneratedbythepropertransformofthehyperplaneclass H andtheclassesoftheexceptionaldivisorsE fori=1,...,n.ThustheCoxringof i X is: G Cox(X )= H0(X ,O(d H+d E +···+d E )). G G 0 1 1 n n (d0,...,Mdn)∈Zn+1 GivenadivisorclassD=d H+d E +···+d E ,thecorrespondinghomogeneous 0 1 1 n n partCox(X ) is thespaceH0 X ,O(d H+···+d E ) .Ifd ≥0thenD isthe G D G 0 n n 0 classofthepropertransformof(cid:0)adegreed0hypersurfaceth(cid:1)athasmultiplicity−diin thepointa(i).ThuswecanidentifyH0(X ,O(d H+···+d E ))withthespaceof G 0 n n homogeneouspolynomialsofdegreed ink[z]=k[z ,···,z ]thathavemultiplicity 0 1 r at least −d at a(i). Let I′ be the vanishingideal in k[z] of the pointa(i). Then the i i lattervectorspaceisprecisely I′ −d1∩···∩ I′ −dn (2) 1 n d0 (cid:0)(cid:0) (cid:1) (cid:0) (cid:1) (cid:1) where(Ii′)−di =k[z]if−di≤0.Ifd0<0thenH0(XG,O(D))=0. Letusconsiderthemap Cox(X ) ≃H0(X ,O(D))−→RG G D G d givenby g(z ,...,z )7→g(l ,...,l )xd1...xdn, 1 r 1 r 1 n where d = (d +d ,...,d +d ), and l,1≤i≤r are as in (1). Recall that l = 0 1 0 n i i x l′ where l′ ∈ RG are the invariants in R of degree 0 ∈ Zn. As g is homoge- 0 i i x x 6 MarthaBernal,DanielCorey,MariaDonten-Bury,NaokiFujita,andGeorgMerz neous of degree d , then g(l ,...,l ) = xd0g(l′,...,l′) is an invariant of degree 0 1 r 0 1 r (d0,...,d0)∈Zn. Thus g(l1,...,lr)xd11...xdnn is indeed an element of RG of degree (d +d ,...,d +d ).Nowwenoticethat 0 1 0 n RG=U[x±1,...,x±1]∩R=k[l ,...,l ][x±1,...,x±1]∩R. 1 n 1 r 1 n Givend∈Zn,anyhomogeneouselement f ∈RGadmitsapresentationoftheform d f = (cid:229) h (l ,...,l )xv1...xvn v 1 r 1 n v∈Zn where the h are homogeneousof degree d−v. On the other hand, the l are ho- v i mogeneous of degree (1,...,1)∈Zn and therefore d−v=(d ,...,d ) for some 0 0 d ≥0.Thus,h (z ,...,z )∈k[z ,...,z ] .Moreover,furthercalculationsshowthat hv0(l1,...,lr)xv11v...1xvnn beirngapol1ynomriadl0inRimpliesthathv(l1,...,lr)∈(Ii)−vi and thereforehv(z1,...,zr)∈(Ii′)−vi.Thus,givend∈Znfixed,wehaveanisomorphism Cox(X ) −→(RG) , g(z)7→g(l ,...,l )xd1···xdn (3) G D d 1 r 1 n D=(d0,M...,dn)∈Zn, d=(d0+d1,...,d0+dn) where d =(d +d ,...,d +d ). Thisand the previouspropositionprovethe fol- 0 1 0 n lowing: Proposition2.2.Cox(X )isisomorphictoRG. G WeshouldobservethattheidealsI′ in(2)donotchangeifwerescalethecolumns i ofA,yettheimageofapolynomialgundertheisomorphism(3)canbedifferent. 2.3 TheCox ring ofa delPezzo surface In[2]itwasproventhattheCoxringofadelPezzosurfaceofdegreeatleast2is generatedbytheglobalsectionsovertheexceptionalcurves.Anexceptionalcurve is one with self-intersection −1. Such a curve has only one global section (up to scalarmultiplication).Weusethisknowledgeandtheisomorphismoftheprevious parttocomputeasetofgeneratorsforRG. Example2.3.Before we moveto the case of delPezzo surfacesof degree3, most important for us, let us say what the Cox ring of a del Pezzo surface of degree 4 lookslike.ThisisasketchofasolutiontoProblem6onSurfacesin[24]. We need to identify all exceptional curves on the blow-up of P2 in 5 points P,...,P ingeneralposition.First,thereare5exceptionaldivisorsoftheblow-up, 1 5 E ,...,E .ThenonechecksthatstricttransformsoflinesthroughtwopointsP,P 1 5 i j areexceptionalcurves.Asdivisors,theyarelinearlyequivalenttoH−E −E .Fi- i j nally,thereisoneconicthroughall5chosenpoints,anditsstricttransformalsois KhovanskiiBasesofCox-NagataRingsandTropicalGeometry 7 anexceptionalcurve,linearlyequivalentto2H−E −E −E −E −E .Thuswe 1 2 3 4 5 have16generatorsoftheCoxringintotal. Relationsbetweenthemcome,roughlyspeaking,fromthepossibilityofdecom- posingadivisorclassassumsoftheonesgivenaboveinafewdifferentways.For instance,2H−E −E −E −E canbewrittenas: 1 2 3 4 (H−E −E )+(H−E −E )=(H−E −E )+(H−E −E )= 1 2 3 4 1 3 2 4 =(H−E −E )+(H−E −E ). 1 4 2 3 ThisleadtorelationsofcorrespondingsectionswhichgeneratetheCoxring.Agood explanationof these computations(also for del Pezzo surfaces of smaller degree) canbefoundintheMScthesisofJ.C.Ottem,[19].Itisworthnotingthatdifferent choicesofpointsgivedifferentrelations,buttheCoxringsareisomorphic. Now, let G and A be as before with r =3, n=6 and suppose that the points a(i)∈Pr−1 areingeneralposition,thatis,nothreeofthemlieonalineandnosix on a conic. Then X is a del Pezzo surface of degree 3 and it has 27 exceptional G curves,determinedin a verysimilarway as inExample2.3. Theseare theclasses of: • theexceptionaldivisorsE,1≤i≤6, i • the propertransformsof lines which passthroughpairs of the blown-uppoints L ,1≤i< j≤6,and ij • thepropertransformsofconicsthroughfiveofthesepoints,Q with1≤i≤6. i TheclassesinPic(X )ofthesecurvesareE,H−E −E and2H−(cid:229) E .This G i i j j6=i j meansthatCox(X)isgeneratedbytheimagesunder(3)oftheuniquepolynomials gink[z ,...,z ]havingtheprescribedmultiplicityontheblown-uppoints.Nowwe 1 r computetheseimagesexplicitly.Forsimplicitywewilldenote[6]={1,...,6}. Let us start with the exceptionaldivisors E. We have that the only monomials i ofdegree0 in k[z] arethe non-zeroconstantsandtheyallbelongto (I′)−1=k[z]. i Thus,by(3)weget (I′)−1 =k[z] ≃Cox(X ) ≃(RG) , i 0 0 G Ei ei (cid:0) (cid:1) wheree ∈Z6isthei-thstandardbasisvector,andthisisomorphismmaps17→1·x. i i Thustheelements{x |1≤i≤6}aregeneratorsofRG. i For each class of the form H−E −E , there is a polynomialof degreeone in i j I′∩I′,namely,theequationoftheuniquelinethroughthepointsa(i) anda(j).This i j is (a a −a a )z +(a a −a a )z +(a a −a a )z . 2j 3i 2i 3j 1 1i 3j 1j 3i 2 1j 2i 1i 2j 3 (cid:0) (cid:1) TheimageofthispolynomialinRGis g(l ,...,l )·(x x )−1=− (cid:229) p y ( (cid:213) x ), 1 3 1 2 ijk k s k6=i,j s∈/{i,j,k} wherethe p arethePlu¨ckercoordinatesofA,andithasdegree(cid:229) e ∈Z6. ijk k6=i,j k 8 MarthaBernal,DanielCorey,MariaDonten-Bury,NaokiFujita,andGeorgMerz Finally,fortheclass2H−(cid:229) E thereisalsoa uniquepolynomialofdegree j6=m j 2 in ∩ I′: the defining polynomial of the unique conic through the five points j6=m j differentfroma(m).Adirectcomputationshowsthattheimageofthisconichasthe form (cid:229) (cid:213) (cid:229) (cid:213) G =(x ) p yy p x +(y )· (u −v)y x m m ([6]\i,j,m) i j ijk k m i i i k i<j,i,j∈[6]\m k∈[6]\{i,j,m} i∈[6] k6=i whereu −v isabinomialofdegree4inthePlu¨ckercoordinatesofA.Thisconic i i generatorhasdegreee +(cid:229) e m i∈[6] i It is worth noting that even when it is difficultto write the exactexpression of the polynomialsG , its computationis straightforward.Also, we observethatthe m generatorsofRG aredetermineduptoscalarmultiplebyGsincethePlu¨ckercoor- dinatesofthematrixAare.Yet,asobservedafterProposition2.2,RGisnotitselfan invariantoftheisomorphismclassofX . G 2.4 Monericand Khovanskiisubspaces. The precedingparagraphsshow howa codimension3 vectorsubspaceG⊂kn, or a matrix containing its basis, gives a minimal generating set of the Cox ring of a del Pezzo surface of degree 9−n. Having covered this, we can finally introduce Khovanskiiandmonericsubspaces. Definition2.4.Wesaythatacodimension3subspaceG⊂kn isKhovanskii(resp. moneric)ifthecorrespondingminimalgeneratingsetoftheCoxringofadelPezzo surfaceofdegree9−nisaKhovanskii(resp.moneric)basisofRG. We would like to consider moneric and Khovanskii bases up to the following equivalencerelation: Definition2.5.Codimension3subspacesG,G′⊂knwillbecalledequivalentifthe correspondinginitialalgebrasoftheCoxringofadelPezzosurfaceareequal. NotethatifGandG′ determinethesameinitialtermsoftheminimalgenerating setofcorrespondingCoxringsthentheyareequivalent. 3 Khovanskii BasisandDegeneration ofthe CoxRing Degeneration of varieties is a powerfultool in algebraic geometry, used on many different occasions. The idea behind it is to introduce a notion of a “limit” of a family of algebraicvarieties. However,since the Zariski topologyon an algebraic varietyisnotwellbehavedinthissense(itisforexamplealmostneverHausdorff), itturnsoutthatabetterreplacementforanarbitraryfamilyofvarietiesisthenotion KhovanskiiBasesofCox-NagataRingsandTropicalGeometry 9 ofaflatfamily.Thisnotionhasthedesirablefeaturethatlimitpointsexistandare unique if we parametrize over a one dimensional variety. It also ensures that the pointsin the family, includingthe limit pointhave the same Hilbertfunction,and thussharemanyinvariantssuchase.g.thedegreeandthegenus.Degenerationsthus motivate the following approach: to compute properties of a given variety X first degeneratethevarietytoamoreaccessiblevarietyX′andthendothecomputations onthisvariety.ThisideacanberealizedinthenotionofaKhovanskiibasis. 3.1 Toricdegenerations Thefollowingdefinitionmakesprecisewhatwemeanbyadegenerationofavariety. Definition3.1.Let (k◦,m) be a discrete valuation ring and X be a variety over k = Quot(k◦). A degeneration of the variety X is a flat family X˜ → Spec(k◦) such that X˜ ×k◦Spec(k)∼=X. It is called a toric degeneration if the special fiber X˜×k◦Spec(k◦/m)isatoricvariety. In this section we provide a method for degeneratinga variety with respect to all possible embeddingsat once. The idea is to degenerate the Cox ring of the given varietywhichcontainsinformationaboutallpossibleembeddingsofthevariety.In order to talk about degenerationsof a projective variety with respect to a specific embedding,weneedtotakethechoiceofaveryamplelinebundleintoaccount. Definition3.2.Let(k◦,m)beadiscretevaluationringandX beaprojectivevariety overk,togetherwithaveryamplelinebundleL.AfamilyX˜ →Spec(k◦)together withalinebundleL˜ iscalledatoricdegenerationof Xwithrespecttotheembedding givenbyLifitisatoricdegeneration,L˜ isflatoverSpec(k◦),wehaveL|X˜×Spec(k)∼= LandthelinebundleL isample. |X˜×Spec(k◦/m) NotethatintheabovedefinitionwedidnotassumethatL isveryam- |X˜×Spec(k◦/m) ple.However,if we considerthe Veroneseembeddingof theembeddedvarietyX, we may assume that X as well as the special fiber are embeddedin the same PN. MoreconcretelybyreplacingLwithahighenoughmultipleL⊗k,wecanmakesure thatL isalsoveryample. |X˜×Spec(k◦/m) 3.2 Degenerations ofdelPezzo surfaces viathe Cox Ring Letk=F(t)fora fieldF ofcharacteristic0.We oftenassumeF =Q.As insec- tion2,givenn∈{1,...,8}wecanassociatetoamatrixA∈Mat (3,n)whichhas k maximalrank,withkernelG,thevarietyX .ThevarietyX istheblow-upofP2at G G thepointsrepresentedbyA.Proposition2.2givesusthefollowingidentity Cox(X )≃k[x ,...,x ,y ,...,y ]G=:RG. G 1 n 1 n 10 MarthaBernal,DanielCorey,MariaDonten-Bury,NaokiFujita,andGeorgMerz Byvaryingthevariablet wecaninterpretX asafamilyofdelPezzosurfacesover G F and Cox(X ) as the correspondingfamily of Cox rings. Note however that the G onlypropertywe areusingin thissectionaboutthe varietyX isthatitsCoxring G R isasubalgebraofapolynomialringk[x ,...,x ]. G 1 r Letk◦bethecorrespondingvaluationringofk,i.e.thesetofallelementshaving nonnegativevaluation. Theorem3.3.LetR⊂k[x ,...,x ]beanalgebra.AfiniteKhovanskiibasisF ofR 1 n inducesatoricdegenerationofSpec(R). Proof. LetF beafiniteKhovanskiibasis.Letusdenotefor f ∈F bytrop(f)(1) theminimumofthevaluationofthecoefficientsof f.Considerthek◦-algebra RG :={ttrop(f)(1)f | f ∈RG}⊂k◦[x ,...,x ]. k◦ 1 n WeclaimthatSpec(RG)→Spec(k◦)isatoricdegenerationofSpec(RG). k◦ ItisaflatmorphismsinceRG isatorsionfreemoduleoverthediscretevaluation k◦ ringk◦.Now,thegeneralfiberisgivenby Spec(RGk◦⊗k◦k)∼=Spec(RG), andthespecialfiberis Spec(RGk◦⊗k◦k◦/(t))∼=Spec(in(RG)). The last thing to prove is that the algebra in(RG) is an affine semigroup algebra. k◦ Butthisfollowseasilyfromthefactthatitisafinitelygeneratedalgebragenerated bymonomials. ⊓⊔ AsaconsequenceoftheabovetheoremweconcludethatafiniteKhovanskiibasis F ofRGinducesatoricdegenerationofSpec(RG).Nowwewanttoshowhowthis toricdegenerationgivesatoricdegenerationofX withrespecttoanyembedding. G Forthispurposethefollowinglemmaishelpful. Lemma3.4.LetF beafiniteKhovanskiibasisofRG. LetLbea very ampleline bundleonX andT := T := H0(X,L⊗q)⊂RGbeitsgradedsectionring. G q q∈N0 Thenin(T)isfinitelygeLnerated.L Proof. Let f ,...,f ∈RGbehomogenouselementswhichformaKhovanskiibasis 1 w ofRG.Foreachb ∈Nw0 considerthesetofallpolynomials fb :=(cid:213) wi=1 fibi suchthat thereisanon-negativeinteger p∈Nforwhichwehave w (cid:229) b ·deg(f)=p·deg(L). (4) i i i=1 By a slight abuse of notation,we use deg forthe functionwhich assigns to a sec- tionaswellastoadivisorthecorrespondingintegervectorundertheisomorphism