Cyclic Covers over Strongly Liftable Schemes?

Qihong XIE

1 Introduction

Throughout this paper,we always work over an algebraically closed fieldkof characteristicp>0 unless otherwise stated.A smooth schemeXis said to be strongly liftable overW2(k),ifXand all prime divisors onXcan be lifted simultaneously overW2(k).This notion was first introduced in[5]to study the Kawamata-Viehweg vanishing theorem in positive characteristic,and furthermore,many examples and properties of strongly liftable schemes were given in[5–7].

Before stating the main theorem,let us fix some notations and assumptions.

LetXbe a smooth projective variety,andLbe an invertible sheaf onX.LetNbe a positive integer prime top,0s∈H0(X,LN),andD=div0(s)be the ef f ective divisor of zeros ofs.LetA=Y=SpecA,andπ:Y?→Xbe the cyclic cover obtained by taking theN-th root out ofs.

Assume thatXis strongly liftable overW2(k),H1=0 and Sing(Dred)=?.By[6,Theorem 4.1 and Corollary 4.3],Xhas a liftingoverW2(k),Lhas a liftingshas a liftingandYis a smooth projective scheme which is liftable overW2(k).

In this paper,we give a criterion for those cyclic covers over strongly liftable schemes that are still strongly liftable(see Sections 3–4 for more details).

Theorem 1.1With the same notations,assumptions and liftingsandas above,assume further that for any prime divisor E on X which is not contained inSupp(D),there exists a liftingof E?X,such that∈H0is a divisible lifting of s|E∈H0(E,LN|E).Then Y is strongly liftable over W2(k).

As a consequence of Theorem 1.1,we have the following corollaries.

Corollary 1.1Let X be a smooth projective variety satisfying the Hi-vanishing condition for i=1,2.Then X is strongly liftable over W2(k).Let L be an invertible sheaf on X,N be a positive integer prime to p,and D be an ef f ective divisor on X with LN=OX(D)andSing(Dred)=?.Let π:Y?→X be the cyclic cover obtained by taking the N-th root out of D.Then Y is a smooth projective scheme which is strongly liftable over W2(k).

Corollary 1.2Letwith n≥3,and L be an invertible sheaf on X.Let N be a positive integer prime to p,and D be an ef f ective divisor on X with LN=OX(D)andSing(Dred)=?.Let π:Y?→X be the cyclic cover obtained by taking the N-th root out of D.Then Y is a smooth projective scheme which is strongly liftable over W2(k).

In Section 2,we recall some definitions and preliminary results of strongly liftable schemes.In Section 3,we give some preliminary results of cyclic covers.The main theorem will be proved in Section 4.For the necessary notions and results on the cyclic cover trick,we refer the reader to[2].

Notation 1.1We useto denote the round-down(resp.round-up,fractional part)of awhere for a real numberb,We use Sing(Dred)(resp.Supp(D))to denote the singular locus of the reduced part(resp.the support)of a divisorD.

2 Preliminaries on Strongly Liftable Schemes

Definition 2.1Let W2(k)be the ring of Witt vectors of length two of k.Then W2(k)is flat overand W2The following definition(see[2,Definition8.11])generalizes the definition in[1,Subsection1.6]of liftings of k-schemes over W2(k).

Let X be a Noetherian scheme over k,andbe a reduced Cartier divisor on X.A lifting of(X,D)over W2(k)consists of a schemeand closed subschemesall de fined and flat over W2(k),such that X=×Speck and×Speck.We writeand say thatis a lifting of(X,D)over W2(k),if no confusion is possible.

Let L be an invertible sheaf on X.A lifting of(X,L)consists of a lifting of X over W2(k)and an invertible sheafonsuch that=L.For simplicity,we say thatis a lifting of L onif no confusion is possible.

Letbe a lifting ofXoverW2(k).Thenis flat overW2(k),and hence flat overNote that there is an exact sequence ofmodules

and amodule isomorphismp:?→Tensoring the above bywe obtain an exact sequence of-modules

and an-module isomorphism

whereris the reduction modulopsatisfyingp(x)==xforx

Definition 2.2Let X be a smooth scheme over k.X is said to be strongly liftable over W2(k),if there is a liftingof X over W2(k),such that for any prime divisor D on X,(X,D)has a liftingover W2(k)as in Definition2.1,whereis fixed for all liftings

LetXbe a smooth scheme overk,be a lifting ofXoverW2(k),Dbe a prime divisor onX,andLD=OX(D)be the associated invertible sheaf onX.Then there is an exact sequence of abelian sheaves

whereq(x)=p(x)+1 forx∈OX,p:OX?→pis the isomorphism(2.2)andris the reduction modulop.The exact sequence(2.3)gives rise to an exact sequence of cohomology groups

Ifr:H1is surjective,thenLDhas a liftingWe combine(2.1)and(2.2)to obtain an exact sequence of-modules

Tensoring(2.5)bywe have an exact sequence of-modules

which gives rise to an exact sequence of cohomology groups

There is a criterion for strong liftability overW2(k)(see[6,Proposition 2.5]).

Proposition 2.1Let X be a smooth scheme over k,andbe a lifting of X over W2(k).If for any prime divisor D on X,there is a liftingof LD=OX(D)onsuch that the natural map r:?→H0(X,LD)is surjective,then X is strongly liftable over W2(k).

3 Preliminaries on Cyclic Covers

For convenience of citation,we recall the following result(see[6,Theorem 4.1 and Corollary 4.3])with a sketch of the proof.

Theorem 3.1Let X be a smooth projective variety,and L be an invertible sheaf on X.Let N be a positive integer prime to p,0s∈H0(X,LN),and D=div0(s)be the divisor ofzeros of s.LetY=SpecA,and π:Y?→X be the cyclic cover obtainedby taking the N-th root out of s.Assume that X is strongly liftable over W2(k),H1(X,LN)=0andSing(Dred)=?.Then X has a liftingover W2(k),L has a liftings has a liftingand Y is a smooth projective scheme,which is liftable over W2(k).

ProofSinceXis strongly liftable overW2(k),there is a liftingofXand a liftingofLonSinceH1(X,LN)=0,the exact sequence(2.6)gives rise to a surjectionH0(X,LN),and henceshas a lifting∈H0Let=div0Thenis a lifting ofD.Letand=SpecThenis a lifting ofY.ThusYis a smooth projective scheme,which is liftable overW2(k).

The above result says that cyclic covers over strongly liftable schemes are liftable overW2(k)under certain conditions,however,in general,they are not strongly liftable overW2(k)(see[6,Remark 4.6]for more details).In order to prove Theorem 1.1,some elementary results on cyclic covers over integral schemes are needed.First of all,we recall an easy lemma in[2,Lemma 3.15(a)].

Lemma 3.1Let X be an integral scheme,and L be an invertible sheaf on X.Let N be a positive integer prime to p,0s∈H0(X,LN),and D=div0(s)be the divisor of zeros of s.LetY=SpecA,and π:Y?→X be the cyclic cover obtained by takingthe N-th root out of s.Then Y is reducible if and only if there is an integer μ>1dividing N and a section t∈H0(X,LN/μ),such that s=

ProofWe can consider the problem over a dense open subset SpecSinceH0we may assume thatcorresponds to an elementu∈B.Since Specis a dense open subset ofY,Yis reducible if and only ifxN?uis reducible inB[x],which is equivalent to the existence of somev∈Bwithu=vμ.

Definition 3.1Let X be a scheme,L be an invertible sheaf on X,N be a positive integer,and0s∈H0(X,LN).The section s is said to be μ-divisible,if μ>0divides N and there exists a section t∈H0(X,LN/μ),such that s=The section s is said to be maximally μ-divisible,if s is μ-divisible,and if s is also ν-divisible,then ν≤μ.

Lemma 3.2With the notation and assumptions as in Lemma3.1,Y then has exactly μ irreducible components if and only if the section s is maximally μ-divisible.

ProofFirst of all,we prove that ifsisμ-divisible,thenYhas at leastμirreducible components.Indeed,assume thatsisμ-divisible,then there is a sectiont∈H0(X,LN/μ),such thats=andD=μD1,whereD1=div0(t).It follows from a direct calculation thatπ:Y?→Xfactorizes into the composition of two cyclic covers:whereπ1:Y1?→Xis the cyclic cover obtained by taking theμ-th root out of 1∈H0H0(X,OX),andπ2:Y?→Y1is the cyclic cover obtained by taking theN/μ-th root out ofSinceπ1is unrami fied,Y1has at leastμirreducible components,and so doesY.

If the sectionsis maximallyμ-divisible,thenYhas at leastμirreducible components by the above argument.IfYhas exactlyνirreducible components withν>μ,then by the proof of Lemma 3.1,sis alsoν-divisible withν>μ,which is absurd.Conversely,ifYhas exactlyμirreducible components,thensisμ-divisible by the proof of Lemma 3.1,and furthermore,sis maximallyμ-divisible by the above argument.

Definition 3.2With the notation and assumptions as in Definition3.1,assume furtherthat X has a liftingover W2(k),and L has a liftingonA sectionH0is called a divisible lifting of s∈H0(X,LN),if the following conditions hold:

(i)is a lifting of s,i.e.,=s.

(ii)If there is an integer μ>0dividing N and a section t∈H0(X,LN/μ),such that s=t?μ,then there exists a sectionlifting t,such that

It is easy to see that if s is maximally μ-divisible andis a divisible lifting of s,thenis also maximally μ-divisible.

Lemma 3.3With the notation and assumptions as in Lemma3.1,assume further that X has a liftingover W2(k),L has a liftingand s has a liftingH0Let=and=SpecIf s is maximally μ-divisible andis adivisible lifting of s,thenhas exactly μ irreducible components.

ProofBy factorizinginto the composition of two cyclic covers,we can prove thathas at leastμirreducible components,whose proof is almost identical to the argument given in the proof of Lemma 3.2 by changing the usual data into the lifted ones.Assume thathas exactlyνirreducible components withν>μ.SinceSpeck=Yand irreducible components ofhave distinct underlying topological spaces,Yhas at leastνirreducible components withν>μ,which contradicts Lemma 3.2.Thushas exactlyμirreducible components.

Lemma 3.4With the notation and assumptions as in Lemma3.1,let E be a primedivisor on X,which is not contained inSupp(D),B=and A|E=be the restriction of A to E.Then there is a natural finite surjective morphismτE:SpecB?→SpecA|E.

ProofIt is easy to see thatholds for anyi≥0 andm≥1.Thus there areinjective homomorphisms for all 0≤i≤N?1,which induce a natural injective homomorphism ofOE-algebras:By[4,Subsection 6.D,Lemma 2],there is a natural dominant morphismτE:SpecB?→SpecA|E,which f its into a commutative diagram

where:SpecA|E=E×XY=π?1(E) ?→Eis the restriction ofπtoπ?1(E)overE,andσ:SpecB?→Eis the cyclic cover obtained by taking theN-th root out ofSinceBis a finiteOE-module,a finiteA|E-module,τEis finite.Since a finite morphism is closed(see[3,Exercise II.3.5]),τEis surjective.

Corollary 3.1With the notation and assumptions as in Lemma3.4,assume further that E is smooth.Then τE:SpecB?→SpecA|Eis the normalization morphism ofSpecA|E.

ProofDenote=Then there are natural injective homomorphisms ofOE-algebras:B,which induce morphisms:SpecB?→Spec?→SpecSinceEis smooth andSpecB?→Eis the cyclic cover obtained by taking theN-th root out ofs|E,by[2,Subsections 3.5 and 3.10],SpecBis the normalization ofSpecand hence ofSpecA|E.

Lemma 3.5With the notation and assumptions as in Lemma3.3,let E be a prime divisor on X,which is not contained inSupp(D),?be a lifting of E?X,=be the restriction oftoThen thereis a natural finite surjective morphism:Spec?→Specwhich is a lifting of τE:SpecB?→SpecA|Econstructed as in Lemma3.4.

ProofIt is similar to that of Lemma 3.4.

We give a simple example to show the dif f erence betweenSpecBandSpecA|Edefined as in Lemma 3.4.

Example 3.1LetX==Projk[x,y,z],L=OX(1),N=2,ands=x2?yz∈H0(X,LN)withD=(x2?yz=0),char(k)=p≥3 andE=(y=0).Consider the cyclic coverπ:Y?→Xobtained by taking the square root out ofs.Look atπover the affine piece=Speck[u,v],whereu=andv=ThenYis defined by the equationt2=u2?v,andEis defined by the equationv=0.It is easy to see thatπ?1(E)consists of two irreducible components,i.e.,E1andE2,which are defined by the equationst+u=0 andt?u=0,respectively.ThusE1andE2are smooth,intersect transversally and map isomorphically ontoE.

Sinces|E,the restriction ofstoE,is defined byx2=0 onE=Projk[x,z],we haveD|E=2Q,whereQis the point[0:1]onE.Therefore,OE-algebrasB=By assumption,SpecA|E=E×XY=π?1(E)=E1+E2,whereas by Corollary 3.1,SpecB=Specis a disjoint union ofF1andF2,such thatτE:F1F2?→E1+E2is the normalization morphism.

4 Proof of the Main Theorem

In this section,we prove the main theorem as follows.

Theorem 4.1With the notation and assumptions as in Theorem3.1, fix such liftingsandas in Theorem3.1.Assume further that for any prime divisor E on X,which is not contained inSupp(D),there exists a liftingof E?X,such that∈H0is a divisible lifting of s|E∈H0(E,LN|E).Then Y is strongly liftable over W2(k).

Before proving Theorem 4.1,we use Example 3.1 to illustrate the meaning of the further assumption made in Theorem 4.1.

Example 4.1With the notation and assumptions as in Example 3.1,take liftings ofX,L,s,DandEas follows:==ProjW2(k)[x,y,z],=(1),=x2?yz∈=(x2?yz=0)and=(y?pz=0).Denote=Specand:?→to be the induced morphism.Look atover the affine piece=SpecW2(k)[u,v],whereu=andv=is defined byt2=u2?v,andis defined byv=p.It is easy to see thatis defined byt2=u2?p,which is irreducible.Hence by[6,Lemma 2.2],is not a lifting ofE1orE2orE1+E2.

The further assumption made in Theorem 4.1 guarantees that the choices of liftingsofEare so adequate that the above situation can be avoided.In our example,s|Eis maximally 2-divisible,if we can choose a liftingofE,such thatis a divisible lifting ofs|E(sois also maximally 2-divisible),and then we have a liftingSpecofSpecB=F1such thatis a lifting ofFifori=1,2.LetThenis a lifting ofEifori=1,2,sinceis a lifting of

Proof of Theorem 4.1Consider the following Cartesian square,whereis the natural projection induced by the definition ofin the proof of Theorem 3.1:

LetEYbe a prime divisor onY,andE=π?(EY)be the induced prime divisor onX.

IfE?Supp(D),then letSupp()be the corresponding lifting ofE.We can take an irreducible componentofsuch that×=EY,i.e.,is a lifting ofEY.

IfESupp(D),thenπ?1(E)may be reducible.Assume thatπ?1(E)=withE1=EY.SinceESupp(D),0s|E∈H0(E,LN|E)determines the ef f ective divisorD|EonE.LetτE:SpecB=?→SpecA|E=π?1(E)=be the natural morphism defined as in Lemma 3.4,whereFjare distinct irreducible components.SinceτEis finite and surjective,we may assume thatτE(F1)=E1.SinceSpecB=?→Eis the cyclic cover obtained by taking theN-th root out ofs|E,by Lemma 3.2,the sectionis maximallyμdivisible.Thus there exists a sectionsuch thatBy assumption,there is a liftingofE?X,such thatis a divisible lifting ofs|E,i.e.,there is a sectionliftingtE,such that

Consider:Spec?→Specdefined as in Lemma 3.5,whereSpec?→is the cyclic cover obtained by taking theN-th root out ofandSpecSinces|Eis maximallyμ-divisible andis a divisible lifting ofs|E,by Lemma 3.3,we may assume thatSpecwhereare distinct irreducible components,and hencehave distinct underlying topological spaces.SinceSpecSpeck=SpecB=Speckare distinct,up to permutation of indices,we can assume thatSpeck=Fjfor any 1≤j≤μ.

By Lemma 3.5,is finite and surjective,so there is an irreducible component ofi.e.,such that:?→is surjective.Sinceis a lifting ofτE,we haveSpeck=E1.Finally,we show thatis flat overW2(k),whenceis a lifting ofE1=EY,and thusYis strongly liftable overW2(k).

SinceW2(k)is an Artin local ring,to prove thatis flat overW2(k),by the local criteria of flatness in[4,Subsection 20.C,Theorem 49],it suffices to show Tor=0.LetZ=π?1(E),Ibe the ideal sheaf ofE1inZ,andbe the ideal sheaf ofinThen the structure sheaf ofiswhich is locally free overandis flat overW2(k),sois flat overW2(k)andSpeck=Z.Locally,E1is defined by one of the factors of the equationxN=s|E,andis defined by one of the factors of the equationSinceis a divisible lifting ofs|E,we have that the reductions of the defining equations ofmodulopare just the defining equations ofE1,soSpeck=Iholds.Considering the following exact sequence:

and taking its long exact sequence for?we obtain an exact sequence

which impliessincek=Iand 0?→I?→OZ?→OE1?→0 is exact.

Definition 4.1A Noetherian scheme X is said to satisfy the Hi-vanishing condition,if Hi(X,L)=0holds for any invertible sheaf L on X.For example,the projective spacesatis fies the Hi-vanishing condition for any1≤i≤n?1.

Corollary 4.1Let X be a smooth projective variety satisfying the Hi-vanishing condition for i=1,2.Then X is strongly liftable over W2(k),and for any cyclic cover π:Y?→X constructed as in Theorem3.1,Y is also strongly liftable over W2(k).

ProofFrom the exact sequences(2.4)and(2.6)and Proposition 2.1,it follows thatXis strongly liftable.By Theorem 4.1,we only have to show that for any prime divisorEonX,there exists a liftingofE?X,such thatis a divisible lifting ofs|E.

Assume thats|E∈H0(E,LN|E)isμ-divisible.Thus there is a sectiontE∈H0(E,),such thats|E=Take an arbitrary liftingofE?Xand consider the following commutative diagram:

where the surjectivity of the upper horizontal maprand the right vertical mapqEfollows from theH1-vanishing condition forXby observing the exact sequence(2.6)and the following exact sequence:

Thus fortE∈H0(E,),there exists a sectionH0(),such thatqE=tE.Let=Then∈H0()is a lifting oftE.

The exact sequence(4.1)gives rise to an exact sequence of cohomology groups

so we haveH1(E,OE)=0.Taking cohomology groups of the following exact sequence,which is the exact sequence(2.3)for

we have an exact sequence of cohomology groups

which implies thatif and only ifL1L2,whereare invertible sheaves onandLi=fori=1,2.

Sincethere exists a unitsuch that=1 andSincepN,we havepμ,and hence there exists a unit∈such that=and=1.Redefinebyso thenis a lifting oftEand=isμ-divisible.

Corollary 4.2Let X=with n≥3,and L be an invertible sheaf on X.Let N be a positive integer prime to p,and D be an ef f ective divisor on X with LN=OX(D)andSing(Dred)=?.Let π:Y?→X be the cyclic cover obtained by taking the N-th root out of D.Then Y is a smooth projective scheme which is strongly liftable over W2(k).

ProofSince the projective space(n≥3)satisfies theHi-vanishing condition fori=1,2,the conclusion follows from Corollary 4.1.

By means of Corollary 4.2,we can construct many strongly liftable varieties of the general type.

Example 4.2LetX=L=OX(1),andNbe a positive integer such thatn≥3,(N,p)=1 andN>n+2.LetHbe a general element in the linear system ofOX(N).ThenHis a smooth irreducible hypersurface of degreeNinXwithLN=OX(H).Letπ:Y?→Xbe the cyclic cover obtained by taking theN-th root out ofH.Then by Corollary 4.2,Yis a strongly

liftable smooth pro jective variety.By Hurwitz’s formula,we haveSince the degree ofHisN?(n+2)>0,KYis an ample divisor onY,and henceYis of the general type.

Obviously,theHi-vanishing condition fori=1,2 is too strong to give more applications.Although there are no further evidences besides Corollary 4.2,we would like to put forward the following conjecture,i.e.,cyclic covers over toric varieties should be strongly liftable overW2(k),whereas the liftability has already been proved in[6,Corollary 4.4].

Conjecture 4.1Let X be a smooth projective toric variety,and L be an invertible sheaf on X.Let N be a positive integer prime to p,and D be an e ff ective divisor on X with LN=OX(D)andSing(Dred)=?.Let π:Y?→X be the cyclic cover obtained by taking the N-th root out of D.Then Y is a smooth projective scheme which is strongly liftable over W2(k).

AcknowledgementsThe author would like to express his gratitude to Professor Luc Illusie,Professor Hél`ene Esnault and the referees for many useful comments,which make this paper more readable.

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