On moduli spaces of polarized Enriques surfaces

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Contents lists available atScienceDirect

Journal de Mathématiques Pures et Appliquées

www.elsevier.com/locate/matpur

On moduli spaces of polarized Enriques surfaces

Andreas Leopold Knutsen

DepartmentofMathematics,UniversityofBergen,Postboks7800,5020Bergen,Norway

a r t i c l e i n f o a b s t r a c t

Article history:

Received17March2020

Availableonline5November2020

MSC:

14C05 14C20 14D06 14J10 14J28

Keywords:

Enriquessurfaces Modulispaces Hilbertschemes Degenerations

Weprovethat,foranyg≥2,theétaledoublecoverρg:Eg→Eg fromthemoduli spaceEg of complexpolarizedgenus g Enriquessurfacesto themoduli spaceEg

ofnumerically polarized genusg Enriquessurfaces isdisconnectedprecisely over irreduciblecomponentsofEgparametrizing2-divisibleclasses,answeringaquestion ofGritsenko andHulek[13].Wecharacterize all irreduciblecomponentsof Eg in termsofanewinvariantoflinebundlesonEnriquessurfacesthatgeneralizestheφ- invariantintroducedbyCossec[8].Inparticular,wegetaone-to-onecorrespondence between the irreducible components of Eg and 11-tuples of integers satisfying particularconditions. This makes it possible, in principle, to list all irreducible componentsofEgforeachg≥2.

r és u m é

Ondémontreque, pourtout g ≥2, lerevêtement doubleétale ρg :Eg →Eg de l’espacedemodulesEgdessurfacesd’Enriquescomplexespolariséesdegenregvers l’espacedemodulesEgdessurfacesd’Enriquescomplexesnumériquementpolarisées degenregestnonconnexeexactementsurlescomposantesirréductiblesdeEg qui paramétrisentdesclasses2-divisibles,répondantainsiàunequestiondeGritsenko etHulek[13].OncaractérisetouteslescomposantesirréductiblesdeEgenutilisant unnouvelinvariant desﬁbrésendroitessurlessurfacesd’Enriquesqui généralise l’invariantφintroduitparCossec[8].Notammenton obtientunecorrespondance biunivoqueentre les composantes irréductibles deEg et les n-uplets denombres entierssatisfaisantdesconditionsparticulières.Celarendpossible,enprincipe,une classiﬁcationcomplètedetouteslescomposantesirréductiblesdeEgpurtoutg≥2.

1. Introduction

Foranyintegerg≥2,letEg (resp.,Eg)denotethemodulispaceofcomplexpolarized(resp.numerically polarized) Enriques surfaces (S,L) (resp.(S,[L]))of (sectional) genus g, that is, such thatL² = 2g−2.

E-mailaddress:[email protected].

https://doi.org/10.1016/j.matpur.2020.10.003

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(Thus, g is the arithmetic genus of all curves in the linear system |L|.) The moduli spaces Eg exist as quasi-projective varieties by Viehweg’s theory, cf. [28, Thm. 1.13]. The moduli spaces Eg exist by results of Gritsenkoand Hulek [13]; more precisely, foreach orbith of theaction of theorthogonal groupin the Enriqueslattice N:=U⊕E8(−1) (see[1,LemmaVIII.15.1]), there isanirreducible moduli spaceM^aEn,h

parametrizingisomorphismclassesofnumerically polarizedEnriques surfaces (S,[L]) with[L] intheorbit h⊂NNumS.ThespaceEg inournotationisthustheunionofallM^aEn,h wherehvariesoverallorbits withh²= 2g−2.Itfollowsfrom [13,Prop.4.1] thatthere isanétaledoublecoverρg:Eg→Eg identifying (S,L) and(S,L+KS).NotethatingeneralthespacesEg and Eg havemanyirreduciblecomponents.

Inthispaperweanswerthefollowingfundamentalquestions:

(1) GivenanirreduciblecomponentofEg,isitsinverseimagebyρgirreducibleornot(cf.[13,Question 4.2])?

(2) Howcanonedetermineall theirreducible componentsofEg?

Regarding the ﬁrst question, for each irreducible component E of Eg either ρ⁻¹_g E is irreducible or it consistsoftwodisjointcomponents,accordingtowhether(S,L) and(S,L+K_S) lieinthesamecomponent ofEg ornotfor(S,[L])∈E.Wewillprove:

Theorem 1.1.Let E be an irreducible component of Eg. Then ρ⁻_g¹(E) is reducible if and only if E parametrizespairs(S,[L])suchthat [L] is2-divisiblein NumS.

Weremarkthatamuchweakerversionofthistheorem wasobtainedin[4,Cor.1.5],withacompletely diﬀerentapproach.

Regardingquestion(2) above,onecanstartbyﬁxing anotherfundamental invariantinadditionto the genus,namelytheφ-invariant

φ(L) := min

E·L|E²= 0, E >0

∈Z+, (1)

introducedbyCossec [8], whichhasinterestinggeometricalinterpretations,cf.,e.g.,[9,18,15,26]. Thenone may,asin[4],considerthemodulispacesEg,φandEg,φparametrizingpairswithL²= 2g−2 andφ(L)=φ, whichingeneralstillhavemanydiﬀerentirreduciblecomponents.Alsorecallthatnotallpossiblepairs(g,φ) occur;for instance it is knownthatφ² 2g−2 by [9,Cor. 2.7.1],and that there are nocases satisfying φ²<2g−2< φ²+φ−2 by[17,Prop.1.4],butacompleteclassiﬁcationofallpossible pairs(g,φ) isstill missing.

Someirreducibilityresultshavebeenknowninlowgenusforawhile;forinstance E3,2,E4,2 andE6,3are irreducible, see[2],[10,§3] and [27].In[4] allirreduciblecomponentsofEg,φ weredeterminedforφ≤4 or g≤20 anddescribedintermsofdecompositionsofthelinebundlestheyparametrizeintoeffective,primitive isotropicdecompositions(that is,into effectiveclassesofsquarezerothatareindivisible inNumS),cf. §2 below.As asample, whichwas classicallyknown,E5,2 hasthree irreducible components,denotedby E5,2^(I), E5,2^(II)⁺andE5,2^(II)⁻ in[5],correspondingtothefollowingdecompositionsofLintoeffective,primitiveisotropic classes:

E5,2^(I) L∼2E₁+E_1,2, E₁·E_1,2= 1;

E5,2^(II)⁺ L∼2E1+ 2E2, E1·E2= 1;

E5,2^(II)⁻ L∼2E1+ 2E2+KS, E1·E2= 1

(where‘∼’denoteslinearequivalence).Thecomponentscanalsobedistinguishedbystudyingtheprojective models of its generalmembers,which is classical,cf. [9, Prop.4.1.2, Prop. 4.5.1, Thm.4.6.3, Prop.4.7.1,

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Thm.4.7.1].ThesecasesalsofurnishanicesampleofTheorem1.1:underρ₅:E5→E5,thetwocomponents E5,2^(II⁾⁺ andE5,2^(II)⁻ areidentiﬁed,whereasE5,2^(I)ismappedtwo-to-one ontooneirreduciblecomponentofE5.

To explain ourresultsand ouranswerto question(2), letLbe aneﬀective linebundle onan Enriques surfacesatisfyingL²>0.Set

εL =

0, ifL+KS is not 2-divisible in PicS,

1, ifL+KS is 2-divisible in PicS. (2)

Wewillprove(cf.Theorem5.7)thatthereexistunique nonnegativeintegersai,dependingonL,satisfying a1≥ · · · ≥a7 and a9+a10≥a0≥a9≥a10

suchthatLcanbewrittenas¹

L∼a₁E₁+· · ·+a₇E₇+a₉E₉+a₁₀E₁₀+a₀E_9,10+ε_LK_S, (3) foranisotropic10-sequence{E₁,. . . ,E₁₀}ofeffectivedivisors(cf.Definition2.1)and aneffectiveisotropic divisor E_9,10 ∼ ¹₃(E₁+· · ·+E₁₀)−E₉−E₁₀ (cf.Lemma2.2).Wecall (3) afundamental presentation of L andthecoefficientsa_i=a_i(L) andε_L thefundamental coefficientsof L(cf.Definitions5.1and5.8).We will provethatthe irreduciblecomponents of Eg are precisely the loci parametrizing pairs of genus g with thesame fundamental coefficients(cf. Theorem5.9).

As analternative description of the irreducible components of Eg we will introducea newfunction on the Enriques lattice N that generalizes the φ-function deﬁned in (1). On the set of ordered 10-tuples of integers one has an order relation by setting (a₁,. . . ,a₁₀) < (b₁,. . . ,b₁₀) if either 10

i=1a_i < 10 i=1b_i or 10

i=1a_i=10

i=1b_i andthereis ann∈ {1,. . . ,9}suchthata_i=b_i foralli∈ {1,. . . ,n−1}anda_n< b_n. Deﬁnition 1.2.Let Lbe aneﬀective line bundleonan Enriques surfaceS suchthatL² >0.The φ-vector associatedtoL,denotedbyφ(L)= (φ1(L),. . . ,φ10(L))∈Z¹⁰₊,istheminimalvalueofall(E1·L,. . . ,E10·L) undertheabovementionedorderrelation,where(E1,. . . ,E10) runsoverallisotropic10-sequencessatisfying 0< E1·L≤ · · · ≤E10·L.

Wesaythatanisotropic10-sequence{E1,. . . ,E10}computesφ(L) ifEi·L=φi(L) foralli∈ {1,. . . ,10}. Thus, theφ-vector functionmeasures the“lowest” intersectionnumbersofline bundles withrespect to entire isotropic10-sequences,generalizingCossec’sφ-function, since,asprovedinthenextresult,φ₁(L)= φ(L).Wewill provethefollowingproperties:

Theorem 1.3. Letφ(L)= (φ₁,. . . ,φ₁₀) betheφ-vector associatedto aneﬀectiveline bundleLwith L²>0 on anEnriquessurface S.Then

(a) 0< φ1≤ · · · ≤φ10; (b) 10

i=1φi isdivisibleby 3;

(c) φ1+· · ·+φ7≥2(φ8+φ9+φ10);

(d) L²= ¹₉10 i=1φi

2

−10 i=1φ²_i;

(e) φ₁,. . . ,φ₈ are the eight lowest intersection numbers with L achieved by numerically distinct eﬀective isotropicdivisors onS;in particularφ₁=φ(L);

(f) L isnumerically2-divisibleif andonlyif φ_i iseven foralli∈ {1,. . . ,10};

1 Thereasonforchoosingtowrite(3) withouttheterm“a8E8”isbecausethenoneautomaticallyhasE1·L≤ · · · ≤E10·L.

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(g) theisotropic10-sequencescomputingφ(L)are,uptonumericalequivalence,preciselytheonesappearing infundamental presentationsof L.

Conversely,forany Enriquessurface S and forany 10-tupleof integers (φ1,. . . ,φ10)satisfying (a)-(c), thereis an[L]∈NumS suchthat L²>0andφ(L)= (φ₁,. . . ,φ₁₀).

Inparticular,weremarkthatthesetofvalues(g(L),φ(L)) occurringasgenusandφ-vectorofpolarized Enriques surfaces (S,L) are completely determined,andthis aposteriori determinesallpossible values of pairs(g(L),φ(L)) byproperty (e).

Ouranswer toquestion(2)cannowbe summarizedas:

Theorem 1.4.The irreduciblecomponentsof Eg are in one-to-one correspondencewith the set of 11-tuples of integers(φ1,. . . ,φ10,ε)satisfying

(i) 0< φ₁≤ · · · ≤φ₁₀, (ii) 10

i=1φi isdivisibleby 3,

(iii) φ1+· · ·+φ7≥2(φ8+φ9+φ10),

(iv) ε∈ {0,1},withε= 0 occurringif atleastoneφi is odd, (v) 2g−2= ¹₉10

i=1φi

2

−10 i=1φ²_i.

Precisely, theirreduciblecomponent of Eg corresponding toaspeciﬁc (φ₁,. . . ,φ₁₀,ε)parametrizes allpairs (S,L) with φ(L)= (φ1,. . . ,φ10) and εL =ε,which are precisely thepairs with thefollowing fundamental presentation,settings:= ¹₃10

j=1φ_j: L∼

7

i=1

(φ8−φi)Ei+ (s−2φ8−φ9)E9+ (s−2φ8−φ10)E10+ (s−3φ8)E9,10+εKS,

foranisotropic 10-sequence{E₁,. . . ,E₁₀},withE_9,10∼¹₃(E₁+· · ·+E₁₀)−E₉−E₁₀. Weproposethefollowing notationfortheirreducible componentsofEg:

• Eg;φ1,...,φ10 correspondsto (φ1,. . . ,φ10) and= 0, ifat leastoneφi isodd.

• E_g;φ⁺ ₁_,...,φ₁₀ correspondsto (φ1,. . . ,φ10) and= 0, ifallφi areeven.

• Eg;φ⁻ 1,...,φ10 correspondsto (φ₁,. . . ,φ₁₀) and= 1, ifallφ_i areeven.

Thus, by property (f) in Theorem 1.3, the components Eg;φ1,...,φ10 parametrize pairs (S,L) with L not numerically 2-divisible, E_g;φ⁺ ₁_,...,φ₁₀ parametrize pairs (S,L) with L 2-divisible in PicS, and E_g;φ⁻ ₁_,...,φ₁₀ parametrizepairs(S,L) withL+K_S 2-divisibleinPicS.Inparticular,byTheorem1.1weobtainthatthe mapρ_g:Eg→Eg identiﬁesEg;φ⁺ 1,...,φ10 with Eg;φ⁻ 1,...,φ10 andistwo-to-oneon Eg;φ1,...,φ10.Weproposeto use thenotationEg;φ1,...,φ10 fortheimagesbyρ_g ofE_g;φ^± ₁_,...,φ₁₀ andEg;φ1,...,φ10.Asanapplicationofourresults weobtainaspeciﬁccomponentdominatingallothers:

Proposition1.5. Foranyg≥2andanyirreduciblecomponentEg;φ1,...,φ10ofEgthereisasurjectivemorphism fromtheirreduciblecomponent E621;30,31,32,33,34,35,36,37,38,39 toEg;φ1,...,φ10.

Itwouldbeinterestingtoknowwhetherthiscomponentisthesameasthedominatingcomponentfound byGritsenkoand Hulek[13],cf. Question5.11.

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Thepaperis organizedas follows.In§2we recallandimprovesomeresultsfrom [4] oneﬀectivedecom- positionsoflinebundlesonEnriquessurfacesintoisotropicdivisors.In§3westudyreduciblesurfacesthat areatransversalunionofarationalsurfaceandasurfacebirationaltothesecondsymmetricproductofan elliptic curve, considered ﬁrst in[6]. Ourmain resultis Theorem 3.10, whichsays thatprojective models in P^g⁻¹ of those reducible surfaces by line bundlesof degree 2g−2 (as described in Proposition3.7) are smoothabletoEnriquessurfacesofdegree2g−2;moreprecisely,theyrepresentsmoothpointsintheHilbert scheme of suchsurfaces.This resultis theEnriques version of[7,Thm. 1] forK3 surfaces and webelieve thatitisofindependentinterestandthatitwillhavefurtherapplications;indeed,althoughdegenerationsof Enriques surfaceshavebeen widelystudied(cf.,e.g.,[19,22,20,25]),aconcrete resultsuchasTheorem3.10 hasnotbeenavailableyet,cf.Remark5.12.Asecond keyresultisProposition3.12statingthatundercer- tainconditions(whichwillturn outtobeequivalent tonumericalnon-2-divisibility)theprojective models byboth aline bundleanditsadjointlieinthesameirreduciblecomponentoftheHilbertscheme.

In §4weproveTheorem 1.1bydegeneration,using theresultsfrom §3.Finally, in§5weintroducethe notionsoffundamentalpresentationandφ-vectormentionedaboveandproveTheorems1.3and1.4,aswell as Proposition1.5.

2. Isotropic10-sequencesandsimpleisotropicdecompositions

Let us explain some notionsfrom [4]. Any eﬀective line bundle L with L² 0 on an Enriques surface maybewrittenas(cf. [4,Cor. 4.6])

L∼a₁E₁+· · ·+a_nE_n+εK_S, (4) such thatall Ei are eﬀective, non–zero,isotropic (i.e., E_i² = 0) and primitive (i.e., indivisible inNumS), alla_i arepositive integers,ε∈ {0,1},n10 and

⎧⎪

⎪⎨

⎪⎪

⎩

eithern= 9,Ei·Ej = 1 for alli=j,

or n= 10,E1·E2= 2 andEi·Ej= 1 for all other indicesi=j, or E1·E2=E1·E3= 2 andEi·Ej = 1 for all other indicesi=j,

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upto reorderingindices. Wecallthisasimple isotropicdecomposition,cf.[4,Def.4.1].

WesaythattwopolarizedEnriquessurfaces(S,L) and(S,L) inEgadmitthesamesimpledecomposition type (cf.[4,Def.4.13])ifonehassimpleisotropicdecompositions

L∼a₁E₁+· · ·+a_nE_n+εK_S and L ∼a₁E₁ +· · ·+a_nE_n +εK_S, with ε∈ {0,1} (6) suchthatEi·Ej=E_i·E_j foralli=j.Similarly,wesaythattwo numericallypolarizedEnriques surfaces (S,[L]) and(S,[L]) inEg admitthesamesimpledecompositiontypeif(6) holdsmoduloKS andKS.

We note thatε = 1 is only needed in(6) when alla_is are even, otherwiseone maysubstitute any E_i having odd coeﬃcient with Ei +KS. Also note that a given line bundle may admit decompositions of diﬀerent types,cf. [4, Rmk.4.14],butneverthelessthepropertyofadmittingthesamedecompositiontype is anequivalence relationonEg and Eg, cf.[4,Prop.4.15].

Werecallthefollowingfrom [9,p. 122]:

Deﬁnition2.1.Anisotropic10-sequenceonanEnriquessurfaceSisasequenceofisotropiceﬀectivedivisors {E₁,. . . ,E₁₀}suchthatE_i·E_j= 1 for i=j.

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Itiswell-knownthatanyEnriquessurfacecontainssuchsequences.Notethatwe,contraryto[9],require thedivisorsto beeﬀective, whichcanalways bearrangedbychanging signs.Wewill alsomakeuse ofthe followingresult,cf. [4, Lemma3.4(a)],[8, Lemma1.6.2(i)] or[9,Cor. 2.5.5]:

Lemma 2.2. Let {E1,. . . ,E10} be an isotropic 10-sequence. Then there exists a divisor D on S such that D²= 10,φ(D)= 3 and3D ∼E1+· · ·+E10.Furthermore,forany i=j,wehave

D∼E_i+E_j+E_i,j, withE_i,j eﬀective isotropic, E_i·E_i,j=E_j·E_i,j= 2, (7) andE_k·E_i,j= 1 for k=i,j.Moreover,E_i,j·E_k,l=

1, if{i, j} ∩ {k, l} =∅, 2, if{i, j} ∩ {k, l}=∅. Inparticular,fori,j,kdistinct,we haveEi+Ej+Ei,j ∼Ei+Ek+Ei,k,sothat

E_j+E_i,j ∼E_k+E_i,k. (8)

Thenextresultyieldsa“canonical”wayof writingsimpleisotropicdecompositions:

Proposition2.3.LetLbeanyeﬀectivelinebundleonanEnriquessurfaceS suchthatL²>0.Thenthereis anisotropic10-sequence{E1,. . . ,E10}(dependingonL)suchthatthereisasimpleisotropicdecomposition L∼a₁E₁+· · ·+a₇E₇+a₉E₉+a₁₀E₁₀+a₀E_9,10+ε_LK_S, (9) whereE9,10∼ ¹₃(E1+· · ·+E10)−E9−E10 anda0,a1,. . . ,a10 are nonnegativeintegers satisfying

a₁≥ · · · ≥a₇, and (10)

a₉+a₁₀≥a₀≥a₉≥a₁₀. (11)

Proof. By[4,Cor.4.7] combinedwith[4,Rem.4.11],afterrenamingindices,thereisanisotropic10-sequence {E1,. . . ,E10}andnonnegative integersa0,a1,. . . ,a10 suchthat

L∼a1E1+· · ·+a10E10+a0E9,10+εLKS

witha0= 0 ora8= 0.Wehavelefttoprovethatwecanmakesurethecoeﬃcientssatisfy(10) and(11).

Assumea0= 0.Byrenamingindices wemayassumea1≥ · · · ≥a10,so that(10) issatisﬁed.Ifa8 = 0, thena9=a10= 0 and(11) issatisﬁed.Ifa8>0,then,using (7):

L∼a₁E₁+· · ·+a₁₀E₁₀+ε_LK_S

∼a8(E1+· · ·+E10) + (a1−a8)E1+· · ·+ (a10−a8)E10+εLKS

∼3a8(E9+E10+E9,10) + (a1−a8)E1+· · ·+ (a10−a8)E10+εLKS

∼

7

i=1

(a_i−a₈)E_i+ (2a₈+a₉)E₉+ (2a₈+a₁₀)E₁₀+ 3a₈E_9,10+ε_LK_S.

Settinga_i:=ai−a8fori∈ {1,. . . ,7},a_i := 2a8+aifori∈ {9,10}anda₀:= 3a8,weseethata₁≥ · · · ≥a₇ and

a₉+a₁₀= 4a8+a9+a10≥3a8=a₀≥2a8+a9=a₉≥2a8+a10=a₁₀, sothecoeﬃcientsa_i satisfy(10) and(11).

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Assumehenceforth thata₀>0,so thata₈ = 0. Byrenamingindices wemayassumea₁ ≥ · · · ≥a₇,so that(10) issatisﬁed,and a₉≥a₁₀.Wesee that(11) issatisﬁedunless a₀< a₉ or a₉+a₁₀< a₀.Wetreat these twocasesseparately.

Casea0< a9.Wesetb:= min{a9−a0,a7}.Recalling(7) and(8),wehave L∼a₁E₁+· · ·+a₇E₇+a₉E₉+a₁₀E₁₀+a₀E_9,10+ε_LK_S

∼b(E1+· · ·+E10) +

7

i=1

(ai−b)Ei−bE8+a0(E9+E9,10) +(a9−a0−b)E9+ (a10−b)E10+εLKS

∼3b(E8+E10+E8,10) +

7

i=1

(ai−b)Ei−bE8+a0(E8+E8,10) +(a9−a0−b)E9+ (a10−b)E10+εLKS

∼

7

i=1

(ai−b)Ei+ (a9−a0−b)E9

+(a₀+ 2b)E₈+ (a₁₀+ 2b)E₁₀+ (a₀+ 3b)E_8,10+ε_LK_S.

By deﬁnitionof b, wesee thatat least oneamong E7 andE9 appearswith coeﬃcient0. Hence,inthe expression7

i=1(a_i−b)E_i+ (a₉−a₀−b)E₉thereareatmost7 nonzeroterms,andwemayrearrangethem so thatthe coefficientsappearindecreasingorder, thatis, so that(10) issatisfied. Moreover,we seethat (a₀+ 2b)+ (a₁₀+ 2b)=a₀+a₁₀+ 4b≥a₀+ 3b.Ifa₀≥a₁₀,weseethatalsoa₀+ 3b≥a₀+ 2b≥a₁₀+ 2b, whence(11) is satisfied.If insteada₀ < a₁₀, weset E₉ :=E₁₀,a₉ :=a₁₀+ 2b, E₁₀ :=E₈, a₁₀:=a₀+ 2b, E_9,10:=E8,10anda₀:=a0+ 3b;then werewrite

(a₀+ 2b)E₈+ (a₁₀+ 2b)E₁₀+ (a₀+ 3b)E_8,10=a₉E₉ +a₁₀E₁₀ +a₀E_9,10 ,

with a₉+a₁₀ ≥a₀,a₉> a₁₀ anda₀≥a₁₀. Ifa₀≥a₉ (whichhappensifandonly ifa0+b≥a10),we are done.Ifa₀< a₉,werepeat theprocess fromthestart,whichthistimewill givethedesireddecomposition, sincea₀≥a₁₀.

Casea₉+a₁₀< a₀.Recalling(8),we have

L∼a1E1+· · ·+a7E7+a9E9+a10E10+a0E9,10+εLKS

∼

7

i=1

a_iE_i+a₉(E₉+E_9,10) +a₁₀(E₁₀+E_9,10) + (a₀−a₉−a₁₀)E_9,10+ε_LK_S

∼

7

i=1

a_iE_i+a₉(E₈+E_8,10) +a₁₀(E₈+E_8,9) + (a₀−a₉−a₁₀)E_9,10+ε_LK_S

∼

7

i=1

aiEi+ (a9+a10)E8+a9E8,10+a10E8,9+ (a0−a9−a10)E9,10+εLKS.

Wenotethat{E1,. . . ,E7,E8,10,E8,9,E9,10}isanisotropic10-sequence,andE8∼¹₃(E1+· · ·+E7+E8,10+ E8,9+E9,10)−E8,10−E8,9 (cf. Lemma 2.2). Thus, setting E₈ := E9,10, E₉ := E8,10, E₁₀ := E8,9 and E_9,10:=E8,andb:= min{a7,a0−a9−a10},wemayrewriteas

L∼

7

i=1

a_iE_i+ (a₀−a₉−a₁₀)E₈ +a₉E₉ +a₁₀E₁₀ + (a₉+a₁₀)E_9,10+ε_LK_S

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∼b(E₁+· · ·+E₇+E₈ +E₉ +E₁₀ ) +

7

i=1

(a_i−b)E_i+ (a₀−a₉−a₁₀−b)E₈ +(a₉−b)E₉ + (a₁₀−b)E₁₀ + (a₉+a₁₀)E_9,10 +ε_LK_S

∼3b(E₉ +E₁₀ +E_9,10 ) +

7

i=1

(ai−b)Ei+ (a0−a9−a10−b)E₈ +(a₉−b)E₉ + (a₁₀−b)E₁₀ + (a₉+a₁₀)E_9,10 +ε_LK_S

∼

7

i=1

(ai−b)Ei+ (a0−a9−a10−b)E₈

+(a₉+ 2b)E₉ + (a₁₀+ 2b)E₁₀ + (3b+a₉+a₁₀)E_9,10 +ε_LK_S.

Bydeﬁnitionofb,weseethatatleastoneofE₇andE₈ appearswithcoeﬃcient0.Hence,intheexpression 7

i=1(a_i−b)E_i+ (a₀−a₉−a₁₀−b)E₈ there are at most 7 nonzero terms, and we mayrearrange them so that the coefficients appearin decreasing order, thatis, so that (10) is satisfied. We also see that the coefficientsinfrontofE₉,E₁₀ andE_9,10 satisfytheconditions(11).

Finally,thefactthat(9) isasimpleisotropicdecompositioniseasyto check.

Remark2.4.Recalling(2),wehaveby[4,Lemma4.8] that εL =

0, if some ai is odd 0 or 1, if allai are even, withtheaisasinProposition2.3.

3. FlatlimitsofEnriquessurfaces

LetE beasmoothellipticcurve.Denote by⊕(and)thegroupoperationonE andbye0 theneutral element.LetR:= Sym²(E) and π:R→E be the(Albanese)projectionmap sendingx+y to x⊕y. We denotetheﬁberofπoverapointe∈E by

fe:=π⁻¹(e) ={x+y∈Sym²(E)|x⊕y=e(equivalently,x+y∼e+e0)},

whichistheP¹deﬁnedbythelinearsystem|e+e0|.Wedenotethealgebraicequivalenceclassoftheﬁbers byf.

For eache ∈E, we deﬁne the curves_e (called D_e in[3]) as theimage of the section E →R mapping xtoe+ (xe).Weletsdenote thealgebraicequivalence classofthese sections,whichare theoneswith minimalself-intersection,namely 1,cf. [3]. Wenote forlaterusethatforx=y wehave

s_x∩s_y={x+y}. (12)

Wealsonotethatwehave

KR∼ −2se0+fe0. (13)

Foranyofthethreenonzero2-torsionpointsηofEthemapE→Rdeﬁnedbymappingetoe+ (e⊕η) realizesE asanunramiﬁeddoublecoverofitsimagecurve

T_η:={e+ (e⊕η)|e∈E}.

(9)

Wehave

Tη ∼ −KR+fη−fe0∼2se0−2fe0+fη, (14) by[3,(2.10)].Inparticular,

T_η−K_R and 2T_η ∼ −2K_R. (15)

Wehenceforthﬁxη andset T :=Tη.Forlaterusewegatherafewlemmas here:

Lemma 3.1.We havehⁱ(TR(−T))= 0foralli.

Proof. Weﬁrstnotethat(14),Serre dualityandRiemann-Rochimplythat

hⁱ(OR(−KR−T)) =hⁱ(OR(−T)) = 0 for all i. (16) Then thelemmafollowsfrom thesequence

0−→ OR(−K_R−T)−→ TR(−T)−→ OR(−T)−→0, whichisthedual ofthesequenceofrelativediﬀerentialsofπtensoredbyOR(−T).

Lemma 3.2.We have

sx+fy∼sy+fx for all x, y∈E. (17)

In particular,

sx+fη ∼sx⊕η+fe0 for all x∈E. (18) Proof. Restrictingtose0 andusing theisomorphismπ|se0 :se0 →E,wehave

(sx−sy)|se0 ∼π|^∗se0(x−y)∼(fx−fy)|se0,

and (17) followsfromthespecialcaseof[3,Prop.(2.11)] statingthattwo linebundlesonR withthesame restrictiontoasectionareisomorphic.Settingy:=x⊕ηin(17) andusingthegrouplaw(x⊕y∼x+y−e0) onE,we obtain(18).

Lemma 3.3.If B ∈Pic⁰E such thatπ^∗B|T is trivial,then B OE orB OE(η−e₀).

Proof. Any B ∈ Pic⁰E can be written as B OE(x−e0) for some x∈ E. Assume thatx = e0. Since π^∗B|T OT(f_x−f_e₀) istrivial,wehave,using(14),ashortexactsequence

0−→ OR(fx−fe0−T) OR(KR+fx−fη)−→ OR(fx−fe0)−→ OT −→0.

As hⁱ(OR(f_x−f_e₀))= 0 for i= 0,1,2,we musthaveh²(OR(K_R+f_x−f_η))=h¹(OT)= 1, whencex=η bySerre duality,ﬁnishingtheproof.

(10)

EmbedT asacubicinP². Considerdistinctpointsx₁,. . . ,x₉∈T suchthat

x1+· · ·+x9∈ |NT /R⊗ NT /P²|. (19) Let σR :R →R be theblow upat x1,x9 and σP :P →P := P² be the blow upat x2,. . . ,x8. We will alwaysassumethepointsxi to besuﬃcientlygeneral,sothat

P is Del Pezzo (whence contains no (−2)-curves), and (20) x1 andx9 lie on distinct ﬁbers ofπ:R→E. (21) We denote by on P the pullback of a general line on P² and by ei the exceptional divisor over x_i, i∈ {2,. . . ,8}.ByabuseofnotationwedenotebysandfthepullbacksofsectionsandﬁbersonRandbyei

theexceptionaldivisoroverxi,i∈ {1,9}.Westill denotebyπthecomposedmap R→R→E.Byabuse ofnotationwedenote byT thestricttransformofT inbothR andP. Wehave(cf.(14)-(15))

T ∼2s_e₀−2f_e₀+fη−e1−e9−K_R, 2T ∼ −2K_R on R, (22) T ∼3 −e2− · · · −e8∼ −K_P on P . (23) Deﬁne X :=R∪T P as thesurfaceobtainedbygluing R and P alongT.Theﬁrst cotangent sheaf T_X¹ :=

ext¹_O_X(Ω_X,OX) ofX (cf. [24,Cor.1.1.11] or[11, §2]),satisﬁes

T_X¹ N_{T /}R⊗ N_{T /}P OT, (24) by[11,Prop.2.3],becauseof(19).Thus,X issemi-stable,cf.[11,Def.(1.13)] and[12,(0.4)].Wewilldenote byDthefamilyofsurfacesX obtainedinthisway.It iseasytoseethatDisirreducibleofdimension9.

Lemma3.4. Wehave h⁰(OX)= 1,h¹(OX)=h²(OX)= 0.Inparticular,PicX H²(X,Z).

Proof. Considerthedecompositionsequence

0 OR(−T) OX OP 0.

Since h⁰(OR(−T))= 0 and h²(OR(−T))=h⁰(OR(K_R+T))= 0 (because T is notanticanonical on R), alsoh¹(OR(−T))= 0 byRiemann-Roch.Wethusgeth^j(OX)=h^j(OP) forj= 0,1,2,whichhasthestated values,asP isrational.Thelaststatementfollowsfrom thecohomologyoftheexponentialsequence.

Werecall thataCartierdivisor, or aline bundle, L∈PicX, isapair (L_R,L_P) such thatL_R ∈PicR, L_P∈PicPandL_R|T L_P|T.WeremarkthatsinceT isnumericallyequivalenttotheanticanonicaldivisor onbothR andP,wehave

L²=L²

R+L²

P = 2pa(L_R)−2 + 2pa(L_P)−2 + 2d, d:=L_R·T =L_P·T, (25) sois even.AsOT(K_R+T)ωT OT,thecanonicaldivisorKX isrepresentedby

K_X = (K_R+T,0) = (f_η−f_e₀,0) in PicR×PicP . (26) Inparticular,by(22)-(23) wehave

K_X= 0 and 2K_X= 0. (27)

(11)

By[12,(3.3)] thesurfaceX alsocarriesaCartierdivisorξrepresentedbythepair

ξ= (T,−T)∼(2se0−2fe0+fη−e1−e9,−3 +e2+· · ·+e8) in PicR×PicP . (28) Lemma 3.5.The CartierdivisorKX istheonlynonzero torsionelementof PicX.

Proof. Assume that L, represented by (L_R,L_P) as above, is a torsion element of PicX. Since PicP is torsion-free, wemusthaveL_P=OP.Moreover,sinceL_R istorsioninPicRσ_R^∗PicR⊕Z[e₁]⊕Z[e₉],we must haveL_Rπ^∗BforaB ∈Pic⁰(E) suchthatπ^∗B|T L_P|T OT.ByLemma3.3wehaveπ^∗B OR

or π^∗B OR(fη−fe0).Thus,by(26),wehaveL OX or L OX(KX),as desired.

Wenow ﬁndspecialeﬀectiveprimitiveisotropicdivisorsonX thatwillbe usedlater.

For j∈ {1,9},the linearsystem| ⊗ Jxj|onP isapencilinducingag¹₂ onT,whichhas,byRiemann- Hurwitz, two membersthatalsobelong toaﬁberofπ_|T :T →E.In otherwords, thereare twoﬁbersfαj

and f_α_j ofπ:R→E suchthat

(fαj∪ej)∩T ∈ | ||T and (fα_j ∪ej)∈ | ||T, j∈ {1,9}.

Oneeasily veriﬁesthatα_j =α_j⊕η.Inparticular, therearetwo uniquelydeﬁnedpointsα_j and α_j⊕η on E suchthatthepairs

(fαj +ej, ) and (fαj⊕η+ej, ), j∈ {1,9},

deﬁne Cartier divisorsonX.It is easyto check,using thegrouplawon E and (26), thatoneisobtained from theotherbytensoringwithKX.Wedeﬁne

E9:= (fα9+e9, ) and E9+KX = (fα9⊕η+e9, ).

Similarly,foreachi∈ {2,. . . ,8}therearetwo uniquelydeﬁnedpointsα_i andα_i⊕ηonE suchthatthe pairs

Ei:= (fαi, −ei) and Ei+KX = (fαi⊕η, −ei), for i∈ {2, . . . ,8}, deﬁne CartierdivisorsonX.

Consideringeachpointxi∈T,fori∈ {1,9}asapointinR= Sym²(E) wemaywritexi=pi+ (pi⊕η).

There aretwosectionsinRpassingthroughxi,namelyspi andspi⊕η,cf. (12).Thus,onR thepairs (spi−ei,0) and (spi⊕η−ei,0), i∈ {1,9} (29) deﬁneCartierdivisorsonX.Using(18) and(26) onechecksthatoneisobtainedfromtheotherbytensoring with K_X.Wedeﬁne

E₁:= (s_p₁−e₁,0) and E₁+K_X = (s_p₁_⊕η−e₁,0). (30) We haveπ(xi)=pi⊕pi⊕η andf_π(x_i₎ istheunique fiberofπ:R→E passingthroughxi. Itssecond intersectionpointwithT isx_i:= (pi⊕η1)+ (pi⊕η2),whereη1andη2arethetwononzero2-torsionpoints ofEinadditiontoη.Theg₂¹cutoutonT bythepenciloflinesthroughx_ihas,againbyRiemann-Hurwitz as above,two elementsthatarefibersofπ_|_T :T →E,saythefibersoverβ_i∈E andβ_i⊕η∈E.It follows thatfori∈ {1,9}there aretwouniquelydefinedpoints β_i andβ_i⊕η onE suchthatthepairs