Moduli of curves on Enriques surfaces Advances in Mathematics

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Advances in Mathematics

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Moduli of curves on Enriques surfaces

Ciro Ciliberto^a, Thomas Dedieu^b,∗, Concettina Galati^c, Andreas Leopold Knutsen^d

aDipartimentodiMatematica,UniversitàdiRomaTorVergata,ViadellaRicerca Scientifica,00173Roma,Italy

bInstitutdeMathématiquesdeToulouse–UMR5219,UniversitédeToulouse– CNRS,UPSIMT,F-31062ToulouseCedex9,France

cDipartimentodiMatematicaeInformatica,UniversitàdellaCalabria, via P. Bucci,cubo 31B,87036ArcavacatadiRende(CS),Italy

dDepartmentofMathematics,UniversityofBergen,Postboks7800,5020Bergen, Norway

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

Articlehistory:

Received27June2019 Receivedinrevisedform10 December2019

Accepted17January2020 Availableonlinexxxx

CommunicatedbytheManaging Editors

Keywords:

Enriquessurfaces Moduli

CurvesonEnriquessurfaces Extension

Enriques–Fanothreefolds

We compute the number of moduli of all irreducible com- ponents of themoduli spaceof smooth curves on Enriques surfaces.Inmostcases,themodulimapstothemodulispace of Prym curves are generically injective or dominant. Ex- ceptional behavior is related to existence of Enriques–Fano threefoldsandtocurveswithnodalPrym-canonicalmodel.

©2020ElsevierInc.Allrightsreserved.

* Correspondingauthor.

E-mailaddresses:[email protected](C. Ciliberto),[email protected] (T. Dedieu),[email protected](C. Galati),[email protected](A.L. Knutsen).

https://doi.org/10.1016/j.aim.2020.107010 0001-8708/©2020ElsevierInc. Allrightsreserved.

1. Introduction

Moduliof curvesonprojectivesurfaces havebeenthe objectofintensivestudy fora longtime.Inmorerecenttimestheso-calledMukaimapcgfromthe(19+g)–dimensional modulispaceofsmoothK3 sectionsofgenusg(thatis,pairs(S,C),whereSisasmooth K3 surfaceandC⊂Sisasmoothgenusgcurve)toMg hasbeengivenmuchattention inrelationto thebirationalgeometry ofMg andofthemoduli spaceofK3 surfacesof genus g. Inparticular cg isdominantforg11 andg= 10,isbirational ontoitsimage forg11 andg= 12,and itsimageisadivisoringenus10and ithasgenericallyone- dimensional fibers ingenus 12 [35,33,34,31,9]. Notable are the relations of pathologies of cg with the existence of Fano and Mukai manifolds [10,8]. Also recall that Mukai’s programtowardsreconstructingafiberofc_gisnowproven[35,1,19],andthattheimage of cg hasbeenrecentlycharacterized,viatheGauss–Wahl map,forBrill-Noether-Petri generalcurves[2,45].

In this paper we consider smooth curves on Enriques surfaces. The moduli of such curveshavenotbeensystematicallyinvestigatedso far.Probablythisisduetothefact that the Enriques case is much more complicated and rich compared to the K3 case due to the presence of many irreducible components of the moduli space of polarized such surfaces,whencealso of themoduli space of smooth curveson Enriques surfaces, even whenfixingthegenusofthepolarization.Remarkably enoughourresultsgivethe numberof moduliof allsuch components,equivalently,thedimensionof theimage(or of a generalfiber) of the moduli map.It should be notedthatthere aresome striking analogies with the K3 case, including behavior induced by the existenceof Enriques–

Fanothreefolds,as wellasmoreexceptionalbehavior,e.g.,relatedtocurveswithnodal Prym–canonicalmodels.

Wenowpresentourresults.LetEdenotethesmoothirreducible10-dimensionalmod- uli space parametrizing smooth, complex Enriques surfaces and Eg,φ the (in general reducible)modulispaceofpairs(S,H) suchthatS isamemberofE andH isanample line bundleonS satisfyingH²= 2g−2 andφ(H)=φ,where

φ(H) := min

E·H |E∈NS(S), E²= 0, E >0

. (1)

Recall thatφ²2g−2 by [15,Cor.2.7.1].

Denote by ECg,φ the moduli space of triples (S,H,C) where (S,H) isa memberof Eg,φ andC∈ |H|isasmoothirreduciblecurve.NotethatECg,φ hasasmanyirreducible componentsas Eg,φ.There arenaturalmorphisms

ECg,φ pg,φ χg,φ cg,φ

Eg,φ Rg Mg,

(2)

whereRg isthemoduli spaceofPrym curves,thatis,ofpairs(C,η),withC asmooth, irreducible,genus gcurveandη anon-zero2-torsionelementofPic⁰(C).Themap χg,φ

sends(S,H,C) tothePrymcurve(C,ωS⊗ OC).Themorphismcg,φisthecomposition of the latter with the forgetful covering map Rg → Mg, which has degree 2^2g −1.

Byadimension count, onecould expect χg,φ and cg,φ to be dominant (onsome or all components)forg6 andgenericallyfinite (onsomeorallcomponents)forg6.

Asfarasweknow,theonlyknownresultsofarconcerningthemapsχ_g,φ andc_g,φ is theoneofVerra [43] statingthatχ_6,3 is dominant,equivalently genericallyfinite (note thatE6,3 isirreducible).

Our main resultsare the following.We present the cases φ 3,φ = 2 and φ = 1 separately.Werefertothetablesin§2andNotation3.4forthedefinitionofthevarious componentsof Eg,φ andECg,φ showingupintheresultsbelow.

Theorem1.Assume that φ3(whence g6).The map χg,φ:ECg,φ → Rg isgeneri- callyinjectiveonany irreduciblecomponentof ECg,φnot appearinginthelistbelow,for whichthedimensionof ageneralfiberis indicated:

Comp. EC7,3 EC^(II)9,3 EC⁺9,4 EC⁻9,4 EC^(II)10,3 EC^(II)13,3 EC^(II)13,4⁺ EC^(II)13,4⁻ EC^(IV)17,4⁺ EC^(IV)17,4⁻

Fib. dim. 1 1 3 0 2 1 1 0 1 0

Inparticular, we obtain thatχ6,3 :EC6,3 −→ R6 isbirational, improvingthe result of [43]. Moreover, inanalogy with theK3 case, forany g 8 thereis acomponent of Eg,φ on which χ_g,φ is generically injective, whereas on E7,3 (which is irreducible), the map χg,φ hasgenericallyone–dimensional fibers.However,in contrastto the K3 case, there are more componentsof Eg,φ forg 8 where χg,φ is notgenerically finite.This phenomenoncanbeexplainedbytheexistenceofEnriques–Fanothreefolds,see§4.

Forφ= 2 weobtain:

Theorem2. Themapχg,2:ECg,2→ Rgisgenericallyfiniteonallirreduciblecomponents of ECg,2 wheng10.Forg9thedimensionof ageneral fiberof χg,2 onthevarious irreduciblecomponentsofECg,2 isasfollows:

Comp. EC^(I)9,2 EC^(II)9,2⁺ EC^(II)9,2⁻ EC8,2 EC^(I)7,2 EC^(II)7,2 EC6,2 EC^(I)5,2 EC^(II)5,2⁺ EC^(II)5,2⁻ EC4,2 EC3,2

Fib. dim. 0 2 1 0 1 3 2 3 6 4 4 6

Inparticular, χg,2 is dominantprecisely on EC3,2 and EC4,2 andis genericallyfinite onat leastonecomponentof ECg,2 precisely for g8.The positive-dimensionalfibers ofχ_9,2onEC^(II)9,2⁺andEC^(II)9,2⁻ canagainbeexplainedbytheexistenceofEnriques–Fano threefolds, see Corollary 4.3. The other positive-dimensional fibers are due to thefact thattheimage ofχg,2liesinquitespecialloci,aswenowexplain. Define:

• R⁰g — the locally closed locus in Rg of pairs (C,η) for which the complete linear system |ωC(η)| is base point free and the map C → P^g−2 it defines (the so-called Prym–canonicalmap)isnotanembedding.Thislocusisirreducible(andunirational) of dimension 2g+ 1 for g 5 by [7, Thm. 1]. (Obviously, R⁰g is dense in Rg for g 4.) Moreover, for the general element, the Prym–canonical map is birational ontoitsimage,whichhaspreciselytwo nodes,cf.[7,Prop. 1.2].

• R^0,nbg —theclosedlocusinR⁰gofpairs(C,η) forwhichthePrym–canonicalmapis notbirational ontoitsimage.This locusisirreducible ofdimension2g−2 forg4 and dominates the bielliptic locus in Mg via the forgetful map Rg → Mg by [7, Cor. 2.2].

• D⁰5 — the locally closed locus in R⁰5 of pairs (C,η) with 4-nodal Prym-canonical model.By[7,Prop. 5.3] thislocusisanirreducible(unirational)divisorinR⁰5whose closure in R5 coincides with closure of the locus of pairs (C,η) carrying a theta- characteristicθ suchthath⁰(θ)=h⁰(θ+η)= 2.

The image of χ_g,2 (on any component of ECg,2) always lies in R⁰g, cf. Lemma 3.5(ii)- (v), andconsequently, bycounting dimensionsoneapriori seesthatχ_g,2 hasexpected fiberdimension max{0,8−g}. Furthermore,as aconsequenceofTheorem 2, themaps χg,2 dominate someof the peculiar lociabove invarious cases. Indeed,it follows from Proposition9.1(i)-(ii)andCorollary 8.7that:

• χ_5,2 onEC^(I)5,2 (respectively,χ_6,2, χ_7,2 onEC^(I)7,2, χ_8,2)dominatesR⁰5 (resp., R⁰6, R⁰7, R⁰8). In particular, the image of χ5,2 on EC^(I5,2⁾ is adivisor inR5; this parallels the situationofimc10 intheK3 case.

• χ5,2 onEC^(II)5,2⁺ dominatesR^0,nb5 .

• χ5,2 onEC^(II)5,2⁻ dominatesD5⁰.

Forφ= 1 themodulispacesEg,1areirreducibleforallgandtheimageofχ_g,1(andof c_g,1)liesinthehyperellipticlocus,cf.Lemma3.5(i),hencetheexpectedfiberdimension is max{10−g,0}.Weprovethatthisisindeedthedimensionofageneralfiber:

Theorem3.Thedimensionofageneralfiberofχg,1andofcg,1ismax{10−g,0}.Hence, cg,1 dominates thehyperelliptic locusifg10and isgenericallyfinite ifg10.

Animmediateconsequenceoftheaboveresultsis:

Corollary1.1. Ageneralcurveofgenus2,3,4and6liesonanEnriquessurface,whereas ageneral curveof genus5or7doesnot.Ageneralhyperelliptic curveofgenus g lies on anEnriquessurfaceif andonly ifg10.

The proof of Theorem 1 also has an application to the classification of projective varieties havingEnriques surfaces aslinearsections. Werecallthataprojective variety

V ⊂P^Nissaidtobek-extendableifthereexistsaprojectivevarietyW ⊂P^N+k,different from a cone, such that V = W ∩P^N (transversely). The question of k-extendability of Enriques surfaces is still open, although it is proved in [38,27] that N 17 is a necessaryconditionfor1–extendability,andterminalthreefoldshavingEnriquessurfaces ashyperplanesectionshavebeenclassifiedin[4,41,30].

Corollary 1.2. LetS ⊂P^N be an Enriquessurface not containing any smoothrational curve.If S is1–extendable,then (S,OS(1)) belongstothefollowinglist:

E17,4^(IV)+, E13,4^(II)+, E13,3^(II), E10,3^(II), E9,4⁺, E9,3^(II), E7,3.

Furthermore,the membersof this list areall atmost 1–extendable, except formembers ofE10,3^(II),whichareatmost 2–extendable, andof E9,4⁺ ,whichare atmost3–extendable.

Thisresultissharpinthecaseof1-extendability:Thegeneralmembersofthemoduli spacesof Corollary 1.2indeedoccur as hyperplanesections ofthreefolds differentfrom cones, cf. Remark 4.9. Furthermore, one cannot remove the assumption about S not containingsmoothrationalcurves, asthere arethreefoldsdifferent fromconesenjoying the peculiar property that their Enriques hyperplane sections belong to E8,3 and E6,3

and contain a smooth rational curve, cf. Corollary 6.5. We remark that the proof of Corollary1.2 isindependentfrom,andmuch simplerthan, theresultsin[38,27], butit needsthetechnicalassumptionaboutrationalcurves,whichcanprobablybeavoided,at theexpenseofaddingmorecases,cf.Remark6.1.WerefertoCorollary4.10foranother variantofCorollary1.2.

Ourgeneralstrategyistocomputethekernelofthedifferentialofthemapc_g,φ,see§3;

tothisendwedevelopin§5toolstocomputethecohomologyoftwistedtangentbundles on Enriques surfaces. In some cases additional arguments are required, involving for instanceextensionstoEnriques–Fanothreefolds(see§4),andspecializationstoEnriques surfaces containing smooth rational curves (see §§8–9). Theorem 1 and Corollary 1.2 are proved in §6; Theorem 2 is obtained by combining Propositions 6.6, 8.1 and 9.1;

Theorem3isprovedin§9.

Inconclusionwe remarkthatourworkleavesseveral interestingopenquestions.For example:isitpossibletocharacterizecurvesonEnriquessurfacesintermsofthesuitable Gauss–Prymmap?Inthecasesofgenericinjectivityofχg,φ,isitpossibletodevelopan analogueofMukai’sprogramme ofexplicitreconstructionof theEnriques surfacefrom its Prym curve section? The latter question was proposed to us by Enrico Arbarello.

Finally, in view of Corollary 1.2, are the general members of E10,3^(II) (respectively, E9,4⁺ ) 2–extendable(resp.,3–extendable)?

Acknowledgments

The authors thank Alessandro Verra and Enrico Arbarello for useful conversations on the subject and acknowledge funding from MIUR Excellence Department Project

CUP E83C18000100006(CC), projectFOSICAVwithintheEUHorizon 2020research andinnovationprogrammeundertheMarieSkłodowska-Curiegrantagreementn.652782 (CC,ThD),GNSAGAofINdAM (CC,CG),theTrondMohnFoundation(ThD,ALK) and grant 261756of the ResearchCouncil of Norway (ALK).Finally the authorswish tothanktherefereefortheextremelycarefulreadingofthepaperandforher/hisuseful comments.

2. ModulispacesofEnriquessurfaces

Wefirstbriefly recallsomewell-knownpropertiesofdivisorsonEnriques surfaces.

Any irreducible curve C on an Enriques surface satisfies C² −2, with equality occurringifandonlyifCP¹.Thelattercurvesarecallednodal,andEnriquessurfaces containing(respectively,notcontaining)themarecallednodal(resp.,unnodal).Itiswell- knownthatthegeneralEnriques surfaceisunnodal,cf. referencesin[14,p. 577].

A divisorE issaidtobeisotropicifE²= 0 and E≡0 (where‘≡’denotesnumerical equivalence)andprimitiveifitisnon-divisibleinNumS.IfE isprimitive,isotropicand nef,then|2E|isabasepointfreepencilwithgeneralmemberasmoothellipticcurve,cf.

[15,Prop. 3.1.2].Inthis case,dim(|E|)= 0 andE is called ahalf-fiber,cf. [15, p. 172].

Conversely,anyellipticpencil|P|containspreciselytwodoublefibers2Eand2E,where E istheonlymemberof|E+KS|.Itisclearthat,whenH isbigandnef,theinvariant φ(H) in(1) iscomputedbyahalf-fiber.

Let E (resp. K^ι) denote the 10-dimensional smooth moduli space parametrizing smooth Enriques surfaces (respectively,smooth K3 surfaces with afixed point freein- volution), andletK denotethe20-dimensionalmoduli spaceparametrizingsmoothK3 surfaces,cf.[3,VIII.§12,§§19-21].WehaveanaturalbijectivemapsendingaK3 surface with fixedpointfreeinvolutionto thequotientsurfacebytheinvolution

δ:K^ι−→ E. (3)

Let Eg,φ (respectively,Eg,φ)denote themoduli space ofpolarized (resp.,numerically polarized)Enriquessurfaces,thatis,pairs(S,H) (resp.,(S,[H]))suchthat[S]∈ E and H ∈ Pic(S) (resp., [H] ∈ Num(S)) is ample with H² = 2g−2 2 and φ = φ(H).

ThereisanétalecoverEg,φ→ E,andEg,φissmooth.Thereisalsoanétaledoublecover ρ:Eg,φ→Eg,φmapping(S,H) and(S,H+K_S) to(S,[H]).Wereferto[6,§2] fordetails and referencesandalsorecallthatφ(H)²H² by[15,Cor.2.7.1].

The spaces Eg,φ neednot be irreducible. In[6] various irreducible components were determined and theirunirationality or uniruledness wasproved.In particular, allcom- ponentsaredeterminedanddescribedforφ4 andg20 respectively.Thedescription is intermsofisotropicdecompositions,aswenow explain.

By[6,Cor. 4.6, Cor. 4.7,Rem. 4.11] anyeffectiveline bundleH with H²0 on an Enriques surfaceS canbewrittenas(denotinglinearequivalenceby‘∼’):

H ∼a1E1+· · ·+anEn+εKS (4)

whereallEi areeffective,primitiveand isotropic,allai are positiveintegers,n10, ε=

0, ifH+K_S is not 2-divisible in Pic(S), 1, ifH+KS is 2-divisible in Pic(S), andmoreover

⎧⎪

⎪⎨

⎪⎪

⎩

eithern= 9,Ei·Ej= 1 for alli=j,

orn= 10,E1·E2= 2 and otherwiseEi·Ej = 1 for alli=j, orE₁·E₂=E₁·E₃= 2 and otherwiseE_i·E_j = 1 for alli=j.

(5)

Wecallthisasimple isotropicdecomposition(uptoreorderingindices),cf. [6].

We say that two polarized Enriques surfaces (S,H) and (S,H) in Eg,φ admit the samesimple decompositiontype ifonehastwosimpleisotropicdecompositions

H ∼a1E1+· · ·+anEn+εKS and H∼a1E₁ +· · ·+anE_n +εKS

and Ei·Ej =E_i·E_j forall i =j. This defines an equivalence relation on Eg,φ by [6, Prop.4.15].

By[6, Cor. 1.3 and 1.4] the irreducible components of Eg,φ when φ 4 or g 20 correspondprecisely tothelociconsisting ofpairs(S,H) admittingthesamedecompo- sitiontype. Moreover, by[6, Cor. 1.5],inthe samerange, forC ⊂ Eg,φ any irreducible component,ρ⁻¹(ρ(C)) is reducible if and onlyif C parametrizespairs (S,H) such that H is 2-divisibleinNum(S).Thevarious irreduciblecomponentsofEg,φ werelabeledby romannumbers inthe appendix of [6]. We will use the samelabels for theirreducible componentsof Eg,φ, adding asuperscript “+” and“−” inthecases there aretwo irre- duciblecomponents lying aboveone irreducible componentof Eg,φ. We alsoadopt the followingfrom [6]:

Notation 2.1.When writing a simple isotropic decomposition (4) verifying (5) (up to permutingindices),wewilladopttheconventionthatEi,Ej,Ei,jareprimitiveisotropic satisfyingEi·Ej= 1 fori=j,Ei,j·Ei=Ei,j·Ej= 2 andEi,j·Ek = 1 fork=i,j.

Inparticular,werecallthefollowing(cf. [6,Cor. 1.3andLemma4.18]):

• Eg,1 isirreducibleandunirational,andH ∼(g−1)E1+E2.

• Ifgiseven(resp.,g= 3),thenEg,2isirreducibleandunirational,andH ∼ ^g−₂²E1+ E2+E3 (resp.,H ∼E1+E1,2).

• Ifg7 andg≡3mod 4,thenEg,2hastwoirreducible,unirationalcomponentsEg,2^(I)

andEg,2^(II)corresponding,respectively,tosimpledecompositiontypes:

(I) H ∼^g−1₂ E₁+E_1,2, (II) H ∼^g−₂¹E1+ 2E2.

• If g 5 and g ≡1mod 4, then Eg,2 hasthree irreducible, unirational components Eg,2^(I), Eg,2^(II)+ andEg,2^(II)−,corresponding,respectively,tosimpledecompositiontypes (I) H ∼ ^g−₂¹E1+E1,2,

(II)⁺ H ∼^g−1₂ E₁+ 2E₂, (II)⁻ H ∼^g−1₂ E₁+ 2E₂+K_S.

Forlaterreferencewelistallirreducible componentsofEg,φforφ2 andg10,cf.

[6,Appendix]:

g φ Comp. Dec. type 3 2 E3,2 H∼E1+E1,2

4 2 E4,2 H∼E1+E2+E3

5 2 E5,2^(I) H∼2E1+E1,2

5 2 E5,2^(II)+ H∼2E1+ 2E2

5 2 E5,2^(II)− H∼2E1+ 2E2+KS

6 2 E6,2 H∼2E1+E2+E3

6 3 E6,3 H∼E1+E2+E1,2

7 2 E7,2^(I) H∼3E1+E1,2

7 2 E7,2^(II) H∼3E1+ 2E2

7 3 E7,3 H∼E1+E2+E3+E4

8 2 E8,2 H∼3E1+E2+E3

8 3 E8,3 H∼2E1+E3+E1,2

g φ Comp. Dec. type 9 2 E9,2^(I) H∼4E1+E1,2

9 2 E9,2^(II)+ H∼4E1+ 2E2

9 2 E9,2^(II)− H∼4E1+ 2E2+KS

9 3 E9,3^(I) H∼2E1+E2+E1,2

9 3 E9,3^(II) H∼2E1+ 2E2+E3

9 4 E9,4⁺ H∼2(E1+E1,2) 9 4 E9,4⁻ H∼2(E1+E1,2) +KS

10 2 E10,2 H∼4E1+E2+E3

10 3 E10,3^(I) H∼2E1+E2+E3+E4

10 3 E10,3^(II) H∼3(E1+E2) 10 4 E10,4 H∼2E1,2+E1+E2

Wealsolistallirreducible componentsofE13,3, E13,4 andE17,4:

g φ Comp. Dec. type

13 3 E13,3^(I) H∼3E1+E2+E3+E4

13 3 E13,3^(II) H∼4E1+ 3E2

13 4 E13,4^(I) H∼2E1+ 2E2+E1,2

13 4 E13,4^(II)+ H∼2(E1+E2+E3) 13 4 E13,4^(II)− H∼2(E1+E2+E3) +KS

13 4 E13,4^(III) H∼3E1+ 2E1,2

g φ Comp. Dec. type

17 4 E^(I)17,4 H∼3E1+ 2E2+ 2E3

17 4 E^(II)17,4 H∼3E1+ 2E2+E1,2

17 4 E^(III)17,4⁺ H∼4E1+ 2E1,2

17 4 E^(III)17,4⁻ H∼4E1+ 2E1,2+KS

17 4 E^(IV)17,4⁺ H∼4E1+ 4E2

17 4 E^(IV)17,4⁻ H∼4E1+ 4E2+KS

3. Generalitiesonmodulimaps

RecallthatifadivisorHonanEnriquessurfaceSisbigandnefsuchthatH²= 2g−2, then dim|H|=g−1 and a general memberC of |H| is asmooth irreducible curveof genus g if either g >2, or g = 2 and S is unnodal or H is ample, by [14, Prop. 2.4]

and[12,Thm. 4.1andProp. 8.2].Asweexplainedintheintroduction,onecouldexpect χ_g,φ andc_g,φ fromdiagram(2) tobedominant(onsomeorallirreduciblecomponents) for g6 and genericallyfinite(onsomeor allirreducible components)for g6.This

expectationfailsinthecasesφ= 1,2 forlowgeneraas thecurvesin|H| areallspecial fromaBrill-Noethertheoreticalpointofview,cf.Lemma3.5below.Italsofailsincase ofexistenceofEnriques–Fanothreefolds,aswewill seein§4below.

Recallingthemap(3),setK^ιg,φ=δ⁻¹(Eg,φ);thus K^ιg,φ isacomponentofthemoduli space of polarized K3 surfaces (S, H, ι) of genus 2g−1 with a fixed point free invo- lution ι, and we havea genericallyinjectivemorphism α_g,φ :K^ι_g,φ → K2g−1 forgetting theinvolution,whereK2g−1denotesthemodulispaceofpolarizedK3 surfacesofgenus 2g−1 (asthe generalK3 surface with a fixed point freeinvolution contains onlyone such).Wehavethecommutativediagram

M2g−1

KC^ιg,φ c^ι_g,φ

p^ι_g,φ

αg,φ

KC2g−1 c_2g−1

q2g−1

K^ι_{g,φ α}

g,φ K2g−1

(6)

where KC^ιg,φ is the moduli space of quadruples (S, H, ι,C), with (S, H, ι) in K^ιg,φ and C∈ |H|isasmoothcurveinvariantundertheinvolutionι, themapαg,φforgetsι, and KC2g−1 is themoduli spaceof triples(X,L,Y), with(X,L) inK2g−1 and Y ∈ |L| isa smoothcurve.

Recallnow,forany smoothC∈ |H|,thesheafTSCdefinedby

0 TSC TS NC/S 0, (7)

andfittingintotheexactsequence

0 TS(−C) TSC TC 0. (8)

Wehavethefollowing,cf.[42,§3.4.4] or[5]:

Lemma 3.1. The differential of cg,φ at (S,H,C) (resp., of c2g−1 at (S, H, C)) is the morphism H¹(TSC) → H¹(TC) (resp., H¹(TSC) → H¹(TC)) induced by (8). Its kernel isH¹(TS(−C))(resp.,H¹(TS(−C))).

ThespacesH¹(TS(−C)) andH¹(TS(−C)) in thelemma arerelated inthe following way. Letπ:S→S be the K3 double coverand set H :=π^∗H.As π isétale, wehave π^∗TS TS.Therefore,

H¹(TS(−H)) =H¹(TS(−H))⊕H¹(TS(−H+KS)). (9)

Lemma 3.2.Assume thatφ3(whence g6).Let(C,KS⊗ OC)be ageneralelement of theimage of χg,φ.Denote by C →C itsinduced double cover.If c⁻_2g−1¹ (C) isfinite, thenitconsistsofonlyonepoint,andalsoχ⁻¹_g,φ((C,KS⊗OC))consistsofonlyonepoint.

Proof. Thanks to the bijective map δ in (3) and α_g,φ being generically injective, the fact thatχ⁻_g,φ¹((C,KS⊗ OC)) is apoint is equivalent to thefact that(c^ι_g,φ)⁻¹(C) is a point, where c^ι_g,φ is as in(6).The latter will follow if c⁻_2g−1¹ (C) is apoint. By[9], this propertyfollowsifChasacorankoneGauss-Wahlmap,cf.[28,SketchofproofofProp.

3.3]. Since 2g−2 = C² φ(C)² 9 (using [15, Cor. 2.7.1]), we have g 6, hence 2g−111.Therefore,ifCliff(C) 3,thefiberc⁻¹_2g−1(C) is positivedimensionalassoon astheGauss-WahlmapofChascorank>1,cf.[8,Thm.2.6].Hence,c⁻_2g−1¹ (C) consists of exactlyonepointifitisfinite.

WethushavelefttoprovethatCliff(C) 3.AstheCliffordindexisconstantamong smoothcurvesinthelinearsystem|H˜|(see[22]),wemayassumethatCisgeneralinits linearsystem.Furthermore,Cliff(C) 3 isequivalenttogon(C) 5,whichissatisfiedif gon(C)5.Thecaseswith gon(C)<2φ(H) areclassifiedin[26, Cor.1.5] andadirect check showsthatgon(C) 5 when φ 3 and g 7. Ifg = 6,we use theassumption thatS isgeneral,sothatgon(C) = 2φ(C)= 6 by[39,Thm.1.1].

Corollary 3.3. Let(S,H)be a general element of an irreduciblecomponent Eg,φ of Eg,φ

and letχ_g,φ denotetherestriction ofχg,φ top⁻_g,φ¹(Eg,φ ).

(i) Ifφ3andh¹(TS(−H))=h¹(TS(−H+KS))= 0,thenχ_g,φisgenericallyinjective.

(ii) Ifh¹(TS(−H))= 0,thenχ_g,φ isgenericallyfinite.

In any case, thedimensionofageneral fiberof χ_g,φ ish¹(TS(−H)).

Proof. Thisfollowsfrom Lemmas3.2and 3.1,aswell as(9).

Intherestofthepaperwe willadoptthefollowing:

Notation3.4.ForanyirreduciblecomponentEg,φ ofEg,φweexpresstheirreduciblecom- ponentp⁻_g,φ¹(Eg,φ ),aswellastherestrictionsofthemapsχ_g,φandc_g,φ tothisirreducible component,bythesamesuperscriptsastheonesusedtolabelEg,φ .Forinstance,weset EC^(II)5,2 :=p⁻¹_5,2(E5,2^(II)),c^(II)_5,2 :=c5,2|_EC^(II)

5,2 andχ^(II)_5,2 :=χ5,2|_EC^(II)

5,2.

We finishthissection withalemma thatwill be neededlater.Wereferto theintro- duction forthedefinitionsofthelociR⁰g,R^0,nb5 andD5⁰.

Lemma3.5. (i)Foranyg2theimageofc_g,1liesinthehyperellipticlocus;inparticular thefiber dimensionismax{0,10−g}.

(ii) The image ofχ^(I_5,2⁾lies in R⁰5;inparticular thefiberdimensionis3.

(iii)The image ofχ^(II)

+

5,2 lies inR^0,nb5 ;in particularthefiberdimension is6.

(iv)The imageof χ^(II)_5,2⁻ lies inD⁰5;inparticular thefiberdimensionis4.

(v)For anyg 6theimageof χg,2 restrictedtoany component of ECg,2 lies inR⁰g; inparticular thefiberdimensionismax{0,8−g}.

Proof. Item (i) follows from [15, Prop. 4.5.1, Cor. 4.5.1], items (ii) and (v) from [7, Ex. 5.1],item(iii)from[7,Rem. 5.5] and(iv)from[7,Ex. 5.2].

4. FibersofthemodulimapsandEnriques–Fanothreefolds

AnEnriques–Fanothreefoldof genusgisapair(X,L) whereX isanormalthreefold andL isanample linebundle onX withL³= 2g−2 suchthat|L|containsasmooth EnriquessurfaceS,andXisnotageneralizedconeoverS,thatis,Xisnotisomorphicto avarietyobtainedbycontractingto apointanegative sectionofsomeP¹-bundle over S. Such threefolds with terminal singularities are classified in [4,41,30], and examples withcanonical, nonterminalsingularitiesare givenin[27,38], butafull classificationof thesethreefoldsisstillmissing,althoughitisprovedin[27,38] thatg17.Wesaythat apolarizedEnriques surface(S,H) is extendable toan Enriques–Fanothreefold(X,L) ifS∈ |L|withH =L|S.

Lemma 4.1.Let (X,L) be an Enriques–Fanothreefold of genus g, π:X →X a desin- gularizationandS∈ |L| asmoothsurface.Then

(i) h⁰(X,L)=g+ 1 andtherestrictionmapH⁰(X,L)→H⁰(S,L|S)isonto;

(ii) h¹(OX)= 0andH⁰(X, π^∗L)H⁰(X,L).

Proof. Since π is an isomorphism outside the singular locus of X, we may identify S withπ⁻¹(S).Bythefactthatπ^∗Lisbigandnefand

0 OX(−π^∗L) OX OS 0, (10)

wegeth¹(OX)= 0.Therestof(ii) followsfrom thenormality ofX.Tensoring(10) by π^∗Landtaking cohomology,weget(i).

Inparticular,part(i)impliesthat|L|isbasepointfreeifandonlyif|L|S|is,forany smoothEnriques surfaceS ∈ |L|, whichholds if andonly ifφ(L|S)2 by [12, Thms.

4.1] or[15,Thm.4.4.1].Similarly,themorphismϕ_Ldefinedby|L|isanisomorphismon Sifandonlyifφ(L|S)3 by[12,Thms.5.1] or[15,Thm.4.6.1] (sinceL|S isample),in whichcasewegetthatϕ_L(X)⊂P^g isa(possiblynon-normal)threefoldwhosegeneral hyperplanesectionisasmoothEnriquessurface.

Theconnectiontothetopicofthis paperisgivenby:

Proposition 4.2. Let (X,L) be an Enriques–Fano threefold of genus g 6. LetS ∈ |L|

be general,and C ∈ |L|S| be general, with φ=φ(L|S)2. Thenthe dimension of the fiber ofcg,φ at(S,L|S,C)isatleast 1.

Proof. Consider the linear pencil l in |L| with base locus C, so that S ∈ l. Consider the open subset U of l whose points correspond to smooth sections of X. We claim thattwo generalpoints ofU correspond to non-isomorphicpolarizedEnriques surfaces (S,L|S),(S,L|S).Theassertionclearlyfollowsfrom thisclaim.

To prove the claim, suppose, to the contrary, that all points of U correspond to isomorphic polarizedEnriques surfaces. This implies that two general members in |L|

areisomorphicaspolarizedEnriquessurfaces.Sinceg6 andφ2,Lemma4.1together with [12,Thms.4.1 and5.1] or[15,Thm.4.4.1and Prop.4.7.1] yieldthatthemap ϕ_L determinedby|L|isamorphismthatmapsX birationallyontoitsimage,whichisnota cone. Hence,two generalhyperplanesectionsofY =ϕ_L(X) areprojectivelyequivalent.

By[37,Prop.1.7] (which appliesinfacttoallvarietiesdifferentfromcones)this would imply thatthegeneralhyperplanesectionofY isruled,acontradiction.

Corollary 4.3.The mapsχ^(IV_17,4⁾⁺,χ^(II)_13,4⁺,χ^(II)_9,2⁺,χ^(II)_9,2⁻,χ7,3 are notgenerically finite.

Proof. Thiswill followfrom Lemmas 4.5,4.6, 4.8and Proposition4.7 below,wherewe provethatthegeneralmembersofE17,4^(IV)+,E13,4^(II)+,E9,2^(II)+,E9,2^(II)−,E7,3 areextendable.

Wewill makeuseofthefollowingauxiliaryresult:

Lemma 4.4. Let (S,L) be a polarized Enriques surface of genus g 6 with φ(L) 2. Assume that (S,L+D) is extendable to an Enriques–Fano threefold (Y,H) for an effectivedivisorD,andthatY isunirational.Then(S,L)isextendabletoanEnriques–

Fano threefold(X,L)andtheelements in|L|areinone-toone correspondencewiththe elements in |H ⊗ JD|.

Proof. Letπ:Y →Y beadesingularizationandidentifySwithπ⁻¹(S).Thenh¹(OY)= 0 by Lemma4.1(ii).Therefore,theexactsequence

0 OY π^∗H ⊗ JD OS(L) 0, (11) shows, as |L| isbase point free andbirational by [12, Thms.4.1 and 5.1] or [15,Thm.

4.4.1 andProp.4.7.1], thattheclosure oftheimage oftherationalmap definedbythe linearsystem|π^∗H ⊗ JD|isathreefoldX inP^g,whereL²= 2g−2,havingthesurfaces in|π^∗H ⊗ JD|,includingS,as hyperplanesections.SinceY isunirational,alsoX is.If X wereacone,thenitwouldbebirationaltoH×P¹,forageneralhyperplanesection H of X. Thus, H wouldbe unirational, acontradiction.Hence, X is notacone. Let ν :X →X beitsnormalization andL:=ν^∗OX(1).Then (X,L) is anEnriques–Fano threefoldextending(S,L).IdentifyingD withπ⁻¹(D),weget,asY isnormal,

H⁰(Y,H ⊗ JD)H⁰(Y , π ^∗H ⊗ JD)H⁰(X,OX(1))C^g+1,

and the latter is contained in H⁰(X,ν_∗L) H⁰(X,L). Since h⁰(L) = g + 1 by Lemma4.1,wemusthaveH⁰(Y,H ⊗ JD)H⁰(X,L),provingthelast assertion.

TheclassicalEnriques–Fanothreefold Y ofgenus13 istheimageofP³viathelinear systemofsexticsurfacesthataredoublealongtheedgesofatetrahedron,cf.[11,18].Its smoothhyperplanesections areEnriquessurfaces withpolarization oftheform 2(E1+ E2+E3) (cf.[27,Pf.ofProp.13.1]),thatis,theybelongtoE13,4^(II)+.

Lemma 4.5. Any (S,H = 2(E1+E2+E3))∈ E13,4^(II)+ such that E1,E2,E3 are nef and

|E1+E2+E3| isbirationalisextendable totheclassicalEnriques–Fanothreefold.

Proof. Byassumption,|E1+E2+E3| maps S birationally ontoasextic surfaceinP³ singular along the edges of a tetrahedron, which are the images of all Ei and E_i, the onlymemberof|Ei+KS|, fori= 1,2,3,cf.,e.g.,[15,Thm.4.9.3]. Allsuchsextics are byconstructionhyperplanesectionsoftheclassicalEnriques–Fanothreefold.

Lemma4.6. Ageneral memberof E7,3 isextendable.

Proof. Let (S,H) ∈ E7,3 be generalwith H ∼E₁+E₂+E₃+E₄. Inparticular, S is unnodal, whence|E₁+E₂+E₃| is birational by[12, Thm.7.2]. Thus(S,L:= 2(E₁+ E₂+E₃)) isextendabletotheclassicalEnriques–FanothreefoldY byLemma4.5.Note that(E₁+E₂+E₃−E₄)²= 0,sothatE₁+E₂+E₃∼E₄+F,foraneffectiveisotropic F.Inparticular, L∼H+F,andtheresultfollowsfrom Lemma4.4.

Nextwe consider the only knownEnriques–Fano threefold of genus 17, namely the one constructed by Prokhorov in [38, §3] with canonical nonterminal singularities in thefollowing way:Letx and y_i,j, 0i,j 2 be homogeneouscoordinates inP⁹ and considertheanticanonicalembeddingofP :=P¹×P¹inP⁸={x= 0}⊂P⁹ givenby

(u₀:u₁)×(v₀:v₁)→(y_0,0:· · ·:y_2,2), y_i,j=uⁱ₀u²₁⁻ⁱv₀^jv²₁^−j.

Let V be the projective cone over P and v = (0 : · · · : 0 : 1) its vertex. Then V is aGorenstein Fanothreefold V with canonical singularities.Let π : V → W be the quotientmapoftheinvolutionτdefinedbyτ(x)=−xandτ(yi,j)= (−1)^i+jyi,j.Letting M:= OV(1), we have−KV ∼ 2M by [38, Lemma 3.1] and everysmooth member of

|−KV| is a K3 surface. The τ-invariant ones are precisely the ones cut out on V by quadricsoftheformq₁(y_0,0,y_0,2,y_2,0,y_2,2,y_1,1)+q₂(y_0,1,y_2,1,y_1,0,y_1,2,x),whereq₁ and q₂ are quadratic homogeneousforms,on which theaction of τ isfree. The quotient of any such τ-invariant S by τ is thus an Enriques surface S. Since π^∗S = S we have 2g−2=S³ = ¹₂S³ = ¹₂(2M)³ = 32,whence g= 17. SetL:=OW(S).Then (W,L) is anEnriques–Fanothreefoldofgenus17.