Normal hyperplane sections of normal schemes in mixed characteristic
aa r X i v : . [ m a t h . A C ] A p r NORMAL HYPERPLANE SECTIONS OF NORMAL SCHEMES INMIXED CHARACTERISTIC
JUN HORIUCHI AND KAZUMA SHIMOMOTO
Abstract.
The aim of this article is to prove that, under certain conditions, an affine flatnormal scheme that is of finite type over a local Dedekind scheme in mixed characteristicadmits infinitely many normal effective Cartier divisors. For the proof of this result, weprove the Bertini theorem for normal schemes of some type. We apply the main resultto prove a result on the restriction map of divisor class groups of Grothendieck-Lefschetztype in mixed characteristic.
Dedicated to Prof. Gennady Lyubeznik on the occasion of his 60th birthday. Introduction
Let X be a connected Noetherian normal scheme. Then does X have sufficiently manynormal Cartier divisors? The existence of such a divisor when X is a normal projectivevariety over an algebraically closed field is already known. This fact follows from theclassical Bertini theorem due to Seidenberg. In birational geometry, it is often necessaryto compare the singularities of X with the singularities of a divisor D ⊂ X , which isknown as adjunction (see [12] for this topic). In this article, we prove some results relatedto this problem in the mixed characteristic case. The main result is formulated as follows(see Corollary 5.4 and Remark 5.5). Theorem 1.1.
Let X be a normal connected affine scheme such that there is a surjectiveflat morphism of finite type X → Spec A , where A is an unramified discrete valuation ringof mixed characteristic p > . Assume that dim X ≥ , the generic fiber of X → Spec A is geometrically connected and the residue field of A is infinite.Then there exists an infinite family of pairwise distinct effective Cartier divisors { D λ } λ ∈ Λ of X such that each D λ is normal and flat over A . The proof of the theorem is obtained as an immediate corollary of the
Bertini typetheorem for normal schemes in mixed characteristic. The hypothesis that A is unramified is forced in the proof of the local Bertini theorem in [15], where we needed to estimate Key words and phrases.
Bertini theorem, graded ring, hyperplane section, normal scheme.2010
Mathematics Subject Classification : 12F10, 13B40, 13J10, 14G40, 14H20. the bound of the number of minimal generators of the (completed) module of K¨ahler dif-ferentials over an unramified complete discrete valuation ring (see Proposition 4.6 below).However, we believe that the theorem holds true for an arbitrary discrete valuation ringwith infinite residue field (see also Remark 5.5). Let X ⊂ P nK be a projective variety overan algebraically closed field K . Let P be one of the following properties: smooth, normalor reduced. Assume that X has P . Then is it true that the scheme theoretic intersection X ∩ H has the same property for a generic hyperplane H ⊂ P nK ? It is a classical theoremof Bertini that the generic hyperplane section of a smooth projective variety X ⊂ P nK is smooth. It is also a classical result of Seidenberg [17] that, if X ⊂ P nK is a normalconnected variety, then X ∩ H is a normal connected variety for a generic hyperplane H ⊂ P nK . Now let X be a scheme such that there is a flat surjective morphism of finitetype X → S and S is a Dedekind scheme of mixed characteristic. The main reason thatwe assume X → S to be surjective is that we want to avoid to include the following case: X = Spec Q p → S = Spec Z p and Q p ≃ Z p [ X ] / ( pX − global Bertinitheorem when X is a normal connected scheme (see Theorem 5.3 and Corollary 5.6 when X is a regular connected scheme). Theorem 1.2.
Let X be a normal connected projective scheme over Spec A such that X → Spec A is flat and surjective, where ( A, π A , k ) is an unramified discrete valuation ringof mixed characteristic p > . Assume that dim X ≥ , the generic fiber of X → Spec A is geometrically connected and k is an infinite field.Then there exists an embedding X ֒ → P nA , together with a Zariski-dense open subset U ⊂ P n ( k ) ∨ , where P n ( k ) ∨ is the dual projective space of P n ( k ) , such that the schemetheoretic intersection X ∩ H is normal and flat over A for any H ∈ Sp − A ( U ) , where Sp A : P n ( A ) ∨ → P n ( k ) ∨ is the specialization map (see Definition 4.4). A crucial ingredient for the proof the above theorem is Lemma 5.2; a constructionof certain Noetherian graded normal domains. As an application, we prove a mixedcharacteristic version of a result of Ravindra and Srinivas on Weil divisor class groupson normal schemes in the final section (see Corollary 6.4). We intend that the presentpaper provides the users with basic tools to study the connection between the Bertini-typeproblem for local (graded) rings and the Bertini-type problem for projective schemes.2.
Notation and conventions
All rings are commutative and unitary. We refer the reader to [3] and [14] as standardreferences for commutative algebra. A local ring is a Noetherian ring with a unique X → S is usually called an arithmetic scheme . However, we will not use this terminology in the article,because it seems that different authors use it in slightly different settings. ORMAL HYPERPLANE SECTIONS OF NORMAL SCHEMES 3 maximal ideal. Let R be a Noetherian ring. Let us denote by Reg( R ) the regular locus ofSpec R and denote by Nor( R ) the normal locus of Spec R . The symbol ( R, m , k ) will denotea local ring such that k = R/ m is the residue field. Let ( A, π A , k ) be a discrete valuationring of mixed characteristic p > π A is a uniformizer of A and p / ∈ π A A .Such a discrete valuation ring A is called unramified . A ring map ( A, π A , k ) → ( R, m , k )of complete local rings is called a coefficient ring map , if it is a local flat map, A isan unramified complete discrete valuation ring of mixed characteristic and k = A/π A A ≃ R/ m . Assume that both ( R, m , k ) and ( A, π A , k ) are complete local rings. Let Ω R/A denotethe module of K¨ahler differentials of R over A . It is known that Ω R/A is not a finitelygenerated R -module when dim R ≥
2, for which we refer the reader to [13, Examples 5.5(a)]. Let b Ω R/A be the m -adic completion of Ω R/A . Then b Ω R/A is a finitely generated R -module (see [9, Proposition 20.7.15] for the proof of this fact). The book [13] is a goodsource for differential modules. In the present article, the completed module b Ω R/A willplay a vital role. Let R be a ring and I be its ideal. Then let V ( I ) denote the set of allprimes of R containing the ideal I . Let us put D ( I ) := Spec R \ V ( I ). Let ( R, m , k ) be alocal ring. Then we say that the elements x , . . . , x n in R are minimal generators of m , ifthey span the k = R/ m -vector space m / m .We follow [7] for the notation in algebraic geometry. When we speak of a divisor, italways means a Cartier divisor . A Weil divisor is used only when we discuss the
Weildivisor class group on normal schemes (see [7, Chapter 11] for divisors and Weil divisorclass groups).3.
Localization of an affine cone attached to a projective scheme
In this section, we recall basic theory on Noetherian graded rings and introduce somegeometric method using the affine cone of projective schemes. The notion of affine coneswill be necessary to relate the global Bertini theorem to the local Bertini theorem inmixed characteristic as proved in [15]. A general reference for graded rings is [3]. Let R = L n ∈ Z R n be a Z -graded Noetherian ring with a prime ideal p . Let p ∗ be the homogeneousideal of R which is generated by homogeneous elements contained in p . It is known that p ∗ is a prime ideal (see [3, Lemma 1.5.6] for the proof of this fact). We denote by R ( p ) the homogeneous localization of R with respect to p . That is, the n -th graded piece of R ( p ) isgiven by ( R ( p ) ) n = n ab ∈ R ( p ) (cid:12)(cid:12)(cid:12) a and b are homogeneous, deg a − deg b = n o . Then R ( p ) is a subring of R p . We consider the following condition on graded rings. The main point is that when R is a domain, then the transcendence degree of the field extensionFrac( A ) → Frac( R ) is infinite and apply [13, Corollary 5.3]. J. HORIUCHI AND K. SHIMOMOTO ( St ) Let R = L n ≥ R n be a positively graded Noetherian ring such that R is generatedby R over R , where ( R , m , k ) is a local ring with the maximal ideal m .A graded ring satisfying the condition ( St ) is a special case of standard graded rings .In practice, such a graded ring arises as the homogeneous coordinate ring of a projectivescheme. Let X ⊂ P nR = Proj (cid:0) R [ X , . . . , X n ] (cid:1) be a closed subscheme over Spec R . Thenthere is a homogeneous ideal I such that X ≃ Proj R , where R = R [ X , . . . , X n ] /I (see [7,Proposition 13.24]). Then R satisfies ( St ). We put m := m ⊕ L n> R n . Since R is a localring, m is the unique homogeneous maximal ideal of R . We also let m + := L n> R n , whichis the irrelevant ideal of R . We define the affine cone by C ( R ) := Spec R . Its pointed affinecone is defined by C ( R ) := C ( R ) \ i ( V ), where i : V = Spec R ֒ → C ( R ) is the closedimmersion induced by R ։ R/ m + = R . Let R m be the localization of R at the maximalideal m . Then we define the localized pointed affine cone by C ( R ) := Spec R m \ j ( V ),where j : V → Spec R m is induced by R m ։ R m / m + R m = R . For a homogeneous ideal I ⊂ R , we set D + ( I ) to be the set of all primes p ∈ Proj R such that p does not contain I . The following proposition is found in [7, Proposition 13.37]. However, we give its prooffor the convenience of the readers. Proposition 3.1.
Let the notation be as above. Then there is a sequence of morphismsof schemes: C ( R ) ψ −→ C ( R ) φ −→ X. The following properties hold:(1) φ ◦ ψ is a flat surjective morphism with smooth fibers of relative dimension one.(2) Assume that P is Serre’s condition ( R n ) or ( S n ) . Then C ( R ) has P if and onlyif so does X .Proof. Since R is generated by R over R by our assumption, there exists a finite set ofelements f , . . . , f s ∈ R such that X = S ni =1 D + ( f i ) and since each R f i has an invertibleelement f i of degree 1, there is an isomorphism:( R f i ) [ T, T − ] ∼ = R f i ( T f i ) . Using this description, since φ is locally induced by the natural inclusion ( R f i ) ֒ → R f i and ψ is induced by the localization map R → R m , it follows that φ ◦ ψ is flat. Moreover,for any prime ideal p ⊂ ( R f i ) , the extension p R f i is also prime. Hence φ is surjective.Every point p ∈ X is contained in m , but not containing m + and so we find that φ ◦ ψ issurjective. On the other hand, we have φ − ( D + ( f i )) = D + ( f i ) × Spec R Spec (cid:0) R [ T, T − ] (cid:1) , where R [ T ] is a polynomial algebra over R with a variable T . That is, the fiber of φ atthe point p ∈ X is the punctured affine line A k ( p ) \ { } = Spec (cid:0) k ( p )[ T ] (cid:1) \ { } , where ORMAL HYPERPLANE SECTIONS OF NORMAL SCHEMES 5 k ( p ) = ( R ( p ) ) / p ( R ( p ) ) . Moreover, we have ( φ ◦ ψ ) − ( p ) ≃ Spec (cid:0) k ( p )[ T ] ( T ) (cid:1) \ { } , whichis a smooth scheme over Spec k ( p ). We have thus proved the assertion (1). For theassertion (2), it suffices to apply [14, Theorem 23.9]. (cid:3) Example . Let k be a field and consider X = P nk = Proj (cid:0) k [ X , . . . , X n ] (cid:1) . Then wehave C ( R ) = A n +1 k \ { (0 , . . . , } , where (0 , . . . ,
0) is the origin of A n +1 k . Then A n +1 k \{ (0 , . . . , } → P nk gives a construction of the projective space as a set of lines through theorigin of A n +1 k . Moreover, we have C ( R ) = Spec (cid:0) k [ X , . . . , X n ] ( X ,...,X n ) (cid:1) \ { , . . . , } .4. Basic elements and symbolic power ideals
For the proof of the main theorem of the article, let us prepare notation from [15].Especially, the notion of basic elements plays an important role.
Definition 4.1.
Let R be a Noetherian ring and let M be a (not necessarily finitelygenerated) R -module. Then we say that an element m ∈ M is basic at a prime ideal p of R , if the image of m under the map M → M p / p M p is not zero (in particular, M p = 0).We will also need the derivation for modules. Definition 4.2.
Let R be a ring and let M be an R -module. A set-theoretic map d : R → M is called a derivation , if the following equalities are satisfied: d ( a + b ) = da + db and d ( ab ) = a ( db ) + b ( da ) for any elements a, b ∈ R .The importance of basic elements is expressed by the following lemma (see [5, Lemma2.2] for the proof). Lemma 4.3.
Let d : R → M be a derivation and let a ∈ R . Assume that da ∈ M is basicat a prime p of R . Then we have a / ∈ p (2) , where p ( n ) := p n R p ∩ R is the n -th symbolicpower ideal of p . We will need the lemma in the case that M is the (completed) module of K¨ahler differen-tials and d is the canonical derivation. Let ( A, π A , k ) be a discrete valuation ring of mixedcharacteristic p >
0. Let P n ( A ) denote the n -dimensional projective space with coordinatesin A such that its A -rational point ( α : · · · : α n ) ∈ P n ( A ) is normalized , which means that π A ∤ α i for some 0 ≤ i ≤ n . By this convention, we have ( α : · · · : α n ) = ( β : · · · : β n ) in P n ( A ) if and only if the equality β i = vα i holds for some v ∈ A × and all 0 ≤ i ≤ n . Wenotice that P n ( A ) may be identified naturally with P n ( K ) as sets, where K is the field offractions of A . Definition 4.4 (Specialization map) . Let us define the set-theoretic map Sp A : P n ( A ) → P n ( k ) in the following way. Let us pick a point α = ( α : · · · : α n ) ∈ P n ( A ) with its lift J. HORIUCHI AND K. SHIMOMOTO e α = ( e α , . . . , e α n ) ∈ A n +1 \ { , . . . , } . Then we defineSp A ( α ) := ( α : · · · : α n ) ∈ P n ( k ) , where we put α i := e α i (mod π A ).Every point of P n ( A ) is normalized and it is easy to check that this map is independentof the lift of α = ( α : · · · : α n ). Thus, the specialization map is well defined. Let ( R, m , k )be a Noetherian local A -algebra and fix a system of elements x , . . . , x n in the maximalideal m and let us choose a point α = ( α : · · · : α n ) ∈ P n ( A ). Let us put x e α := n X i =0 e α i x i , where e α = ( e α , . . . , e α n ) ∈ A n +1 \ { , . . . , } is a lift of α = ( α : · · · : α n ) ∈ P n ( A ) throughthe quotient map A n +1 \ { , . . . , } → P n ( A ). The principal ideal x e α R does not dependon the lift of α ∈ P n ( A ).The following lemma is a modification of [15, Lemma 4.2], which plays a role in theproof of the Bertini theorem. The proof given in [15, Lemma 4.2] applies without anyessential change. Lemma 4.5.
Let ( R, m , k ) be a complete Noetherian local domain of mixed character-istic p > with infinite residue field k and a coefficient ring ( A, π A , k ) . Fix a set ofelements x , . . . , x n in the maximal ideal m , together with a prime ideal p of R such that ( x , . . . , x n ) p . Then there exists a Zariski-dense open subset U ⊂ P n ( k ) such that x e α / ∈ p for every point α = ( α : · · · : α n ) ∈ Sp − A ( U ) . We shall prove the following proposition which slightly generalizes [15, Theorem 4.3] inthe form we need. We explain the meaning of the conclusion after the proof is finished.
Proposition 4.6.
Let ( R, m , k ) be a complete Noetherian local domain of mixed charac-teristic p > and assume the following conditions: (i) Let A → R be a coefficient ring map, where ( A, π A , k ) is an unramified completediscrete valuation ring. (ii) Let π A , x , . . . , x d be a fixed set of minimal generators of m , which in particularimplies that π A / ∈ m . (iii) The residue field k is infinite.Then there exist only finitely many prime ideals { q , . . . , q n } of R such that the ideal ( x , . . . , x d ) is contained in every q i with i = 1 , . . . , n . Let us put x e α = d X i =1 e α i x i for α = ( α : · · · : α d ) ∈ P d − ( A ) . ORMAL HYPERPLANE SECTIONS OF NORMAL SCHEMES 7
Then, there is a Zariski-dense open subset U ⊂ P d − ( k ) such that x e α / ∈ p (2) holds for every p ∈ Spec R \ { q , . . . , q n } and α ∈ Sp − A ( U ) .Proof. For the first assertion, we observe that ht( x , . . . , x d ) ≥ dim R − π A , x , . . . , x d ) = dim R . This implies that the ideal ( x , . . . , x d ) is containedin only finitely many primes q , . . . , q n and m ∈ { q , . . . , q n } .Let us prove the second assertion. Let b Ω R/A be the m -adic completion of the module ofK¨ahler differentials of the A -algebra R . Since π A ∈ A , we have dπ A = 0 in b Ω R/A , where d : R → b Ω R/A is the universal derivation. So it follows that dx , . . . , dx d forms a set ofgenerators of the R -module b Ω R/A . To prove the second assertion, it is necessary to modifythe proof of [15, Theorem 4.3] as follows. Let us put x := π A and let R [ X , . . . , X d ] bea polynomial algebra over R . Then by the proof of [15, Theorem 4.3], there exist finitelymany prime ideals { p , . . . , p s } of R , together with a polynomial G ∈ R [ X , . . . , X d ] \ m R [ X , . . . , X d ]such that for any point e α = ( e α , . . . , e α d ) ∈ A d +1 with G ( e α ) / ∈ m , we have d X i =0 e α i dx i ∈ b Ω R/A is basic at every p ∈ Spec R \ { p , . . . , p s } . Let { q , . . . , q n } be the set of primes of R as in the first assertion. After organizing theset of the primes { p , . . . , p s } in an appropriate order, we may assume that there exists aninteger r such that 0 ≤ r ≤ s and { p , . . . , p r } = { p , . . . , p s } \ { q , . . . , q n } . Then we have( x , . . . , x d ) p when p ∈ { p , . . . , p r } , and ( x , . . . , x d ) ⊂ p when p ∈ { p r +1 , . . . , p s } .By proceeding as in the second part of the proof of [15, Theorem 4.3] and applyingLemma 4.5 to the prime ideals { p , . . . , p r } , we can find a Zariski-dense open subset U ′ ⊂ P d ( k ) ( U ′ is determined by G ∈ R [ X , . . . , X d ]) such that for a given point α = ( α : · · · : α d ) ∈ Sp − A ( U ′ ), the following assertion holds:(4.1) d X i =1 e α i dx i ∈ b Ω R/A is basic at every p ∈ Spec R \ { p , . . . , p s } and x e α = d X i =1 e α i x i / ∈ p for every p ∈ { p , . . . , p r } (notice that dx = dπ A = 0). Then applying Lemma 4.3 to (4 . x e α = d X i =1 e α i x i / ∈ p (2) for every p ∈ Spec R \ { p r +1 , . . . , p s } and every α = ( α : · · · : α d ) ∈ Sp − A ( U ′ ) . J. HORIUCHI AND K. SHIMOMOTO
Let us identify the hyperplane H ⊂ P d ( k ), which is defined by the homogeneous equation X = 0 in k [ X , . . . , X d ], with the projective space P d − ( k ). Then we have an open subset U := U ′ ∩ H of H ∼ = P d − ( k ), which is not empty by the construction. Since we have { p r +1 , . . . , p s } ⊂ { q , . . . , q n } , it follows that Spec R \{ q , . . . , q n } ⊂ Spec R \{ p r +1 , . . . , p s } and the second assertion follows from (4 . (cid:3) Remark . Let us explain the meaning of non-containment in the second symbolic powerideal p (2) . Let R be a Noetherian ring and fix a prime ideal p of R such that R p is regular.Choose an element x ∈ p . Then we can prove that the localization of R/xR at p is regularif x / ∈ p (2) . Indeed, the condition x / ∈ p (2) implies that x is a minimal generator of themaximal ideal of R p . Since R p is regular, R p /xR p is also regular. Thus, Proposition 4.6asserts that the inclusion Reg( R ) ∩ V ( x e α R ) ⊂ Reg( R/ x e α R ) holds outside finitely manypoints q , . . . , q n of Spec R .5. Proof of the main theorem
We need the following lemma. We refer the reader to [8, Corollaire (2.9.4)] for the proofand its generalized version.
Lemma 5.1.
Assume that A is a Noetherian ring and let X ֒ → P NA be a closed subschemewith its ideal sheaf I X , and let Γ ∗ ( I X ) := L n ≥ H ( P NA , I X ( n )) be the homogeneous idealof the polynomial algebra Γ ∗ ( O P NA ) . Then X is an integral scheme if and only if Γ ∗ ( I X ) is a prime ideal of Γ ∗ ( O P NA ) . Lemma 5.2.
Let X be a normal connected projective scheme over Spec A such that X → Spec A is flat and surjective, where A is a Noetherian local normal domain. Suppose thatone of the following conditions holds:(1) The field of fractions of A is of characteristic 0 and the generic fiber of X → Spec A is geometrically connected.(2) The generic fiber of X → Spec A is geometrically integral.Then there exists a Noetherian graded normal domain R = L n ≥ R n such that R satisfiesthe condition ( St ) , R = A and X ≃ Proj R .Proof. Since X is a Noetherian normal connected scheme, it is an integral scheme. Fix anembedding X ֒ → P NA as a closed subscheme with its ideal sheaf I X . By Lemma 5.1, S :=Γ ∗ ( O P NA ) / Γ ∗ ( I X ) is a Noetherian graded domain satisfying ( St ) such that X ≃ Proj S .Let us put R ′ := Γ ∗ ( O X ) = L n ≥ H ( X, O X ( n )). Then by the short exact sequence0 → I X → O P NA → O X →
0, we have an injection
S ֒ → R ′ . By the definition of R ′ , it iscontained in the field of fractions L of S , and R ′ is a finitely generated graded S -module.So R ′ is Noetherian. Consider sf ∈ L , where s ∈ H ( X, O X ( l )) and f ∈ H ( X, O X ( m )) ORMAL HYPERPLANE SECTIONS OF NORMAL SCHEMES 9 with f = 0. Assume that sf is integral over R ′ . Then it can be shown that l ≥ m bylooking at the monic polynomial defining sf . Since O X,x is a normal domain for x ∈ X , itfollows that (cid:16) sf (cid:17) x ∈ O X ( l − m ) x , which implies that sf ∈ H ( X, O X ( l − m )). Hence R ′ is a normal domain.Next, we prove that A = R ′ , where R ′ = H ( X, O X ). Denote by X K the generic fiberof X → Spec A with K := Frac( A ). Then X K is a normal connected projective varietyby assumption. First, assume the condition (1). Since K is a field of characteristic zero, X K := X K × Spec K is a normal projective scheme defined over the algebraic closure K of K . Since X K is geometrically connected, X K is a normal integral projective variety.Second, assume the condition (2). Then X K is an integral projective variety. In any case,we have(5.1) H ( X K , O X K ) = K. Now K ′ := H ( X K , O X K ) is a finite field extension of K . By flat base change, we have(5.2) H ( X K , O X K ) ≃ K ′ ⊗ K K. Combining (5 .
1) and (5 .
2) together, we obtain K ≃ K ′ ⊗ K K . As this isomorphism holdsonly when K ′ = K , we have H ( X K , O X K ) = K. Since there are inclusions
A ֒ → H ( X, O X ) ֒ → H ( X K , O X K ) = K , A ֒ → H ( X, O X ) ismodule-finite and A is integrally closed in K , we have A = R ′ = H ( X, O X ) . Recall that the direct summand of a normal domain is normal. So after replacing R ′ by itsappropriate Veronese subring R , it follows from [1, Proposition 3 at page 159] that thereis a Noetherian graded normal domain R satisfying ( St ), R = A and X ≃ Proj R . (cid:3) Let us give a proof of the main theorem. Let A be a discrete valuation ring and let H bean element of the dual projective space P n ( A ) ∨ . Then we may identify H as a hyperplaneof P nA as a closed subscheme. Theorem 5.3.
Let X be a normal connected projective scheme over Spec A such that X → Spec A is flat and surjective, where ( A, π A , k ) is an unramified discrete valuation ringof mixed characteristic p > . Assume that dim X ≥ , the generic fiber of X → Spec A is geometrically connected and k is an infinite field.Then there exists an embedding X ֒ → P nA , together with a Zariski-dense open subset U ⊂ P n ( k ) ∨ , where P n ( k ) ∨ is the dual projective space of P n ( k ) , such that the scheme theoretic intersection X ∩ H is normal and flat over A for any H ∈ Sp − A ( U ) , where Sp A : P n ( A ) ∨ → P n ( k ) ∨ is the specialization map (see Definition 4.4).Proof. By Lemma 5.2, there is a Noetherian graded normal domain R satisfying ( St ), R = A and X ≃ Proj R . Take the homogeneous maximal ideal m = π A A ⊕ L n> R n of R , together with the localized pointed affine cone C ( R ) = Spec R m \ j (Spec A ) =Spec R m \ { m + R m , m R m } , where j : Spec A ֒ → Spec R m is the closed immersion induced by R m ։ R m / m + R m = A . Then R m is a Noetherian local ring such that π A / ∈ m R m . Thereis a flat surjective smooth morphism (see Proposition 3.1): h : C ( R ) = Spec R m \ j (Spec A ) → X. Let us fix a system of minimal generators x , . . . , x d ∈ R such that m = ( π A , x , . . . , x d )and hence m + = ( x , . . . , x d ). Since A ≃ R/ ( x , . . . , x d ), m + = ( x , . . . , x d ) is a primeideal such that ht( x , . . . , x d ) = dim R −
1. Let us pick a point α = ( α : · · · : α d ) ∈ P d − ( A ) and let H α be the hyperplane of P d − A defined by the homogeneous polynomial: e α X + · · · + e α d X d ∈ A [ X , . . . , X d ], to which we give the standard grading: deg( X ) = · · · = deg( X d ) = 1. There is a surjection of graded rings: A [ X , . . . , X d ] ։ R by mapping X i to x i . This defines a closed immersion X ֒ → P d − A . Letting n := d −
1, we shall provethat
X ֒ → P d − A has all the properties as stated in the theorem.Let us put x e α := d X i =1 e α i x i and consider the following fiber product diagram: C ( R/ x e α R ) −−−−→ X ∩ H α y y C ( R ) h −−−−→ X where all the vertical maps are natural closed immersions. Then by Proposition 3.1, itfollows that(5.3) X ∩ H α is normal . ⇐⇒ C ( R/ x e α R ) is normal.Moreover, we have(5.4) C ( R/ x e α R ) is normal . ⇐⇒ R m / x e α R m is normal after localization at p ∈ ∆ , where we put ∆ := V ( x e α R m ) ∩ C ( R ) ⊂ Spec R m . We find that ∆ = V ( x e α R m ) ∩ C ( R ) = V ( x e α R m ) \ { m + R m , m R m } . Now consider thenatural map R m → R ∧ m , where R ∧ m is the m R m -adic completion of R m . Then this is a flatlocal map between Noetherian local rings. Let A ∧ be the ( π A )-adic completion of A . Since ORMAL HYPERPLANE SECTIONS OF NORMAL SCHEMES 11 R m is essentially of finite type over the discrete valuation ring A whose field of fractionsis of characteristic zero, it is an excellent local normal domain. Hence R ∧ m is a completelocal normal domain with A ∧ its coefficient ring (by our assumption, A is an unramifieddiscrete valuation ring). It defines an affine scheme map: g : Spec R ∧ m → Spec R m . Moreover, we have a commutative diagram of specialization maps: P d − ( A ) −−−−→ P d − ( A ∧ ) Sp A y Sp A ∧ y P d − ( k ) = −−−−→ P d − ( k )Let us pick a prime q ∈ Spec R ∧ m and put p := g ( q ) ∈ Spec R m . Then since g is a regularlocal map, we have R m / x e α R m is normal after localization at p . ⇐⇒ R ∧ m / x e α R ∧ m is normal after localization at q . Hence in view of (5 .
3) and (5 . U ⊂ P d − ( k ) such that the following holds:(5.5) R ∧ m / x e α R ∧ m is normal after localization at q ∈ g − (∆) for α ∈ Sp − A ∧ ( U ) . So let us prove (5 .
5) below. By elementary set theory, we have g − (∆) = V ( x e α R ∧ m ) ∩ g − (cid:0) C ( R ) (cid:1) ⊂ Spec R ∧ m and g − (∆) is a locally closed subset of Spec R ∧ m . The set of minimal primes in Sing( R ∧ m )is finite, where Sing( R ∧ m ) denotes the singular locus of Spec R ∧ m . By recalling that m =( π A , x , . . . .x d ) and m + = ( x , . . . , x d ), it follows that { m + R ∧ m , m R ∧ m } is the set of all primeideals containing m + R ∧ m . Then we have { m + R ∧ m , m R ∧ m } ∩ g − (∆) = ∅ , because we know { m + R ∧ m , m R ∧ m } ∩ g − (cid:0) C ( R ) (cid:1) = ∅ .To prove (5 . R ) and ( S ). Taking the open subset U ′ ⊂ P d − ( k ) as in Proposition 4.6, it follows from Remark 4.7 that(5.6) Reg( R ∧ m ) ∩ V ( x e α R ∧ m ) ⊂ Reg( R ∧ m / x e α R ∧ m ) holds at every point q ∈ g − (∆)for every α ∈ Sp − A ∧ ( U ′ ) ⊂ P d − ( A ∧ ) . In other words, if we have q ∈ Reg( R ∧ m ) ∩ V ( x e α R ∧ m ) ∩ g − (∆), then q ∈ Reg( R ∧ m / x e α R ∧ m ).Let us put Q = { p ∈ g − (∆) | p is a minimal prime in Sing( R ∧ m ) } . This is a finite set. Notice that every prime contained in Q has height at least 2 due tothe ( R ) condition on R ∧ m . Let us put Q = { p ∈ g − (∆) | depth( R ∧ m ) p = 2 and dim( R ∧ m ) p > } . Since R ∧ m is a complete local domain, we find that the set of primes p such that ( R ∧ m ) p satisfies Serre’s ( S ) forms an open subset of Spec R ∧ m by [10, Proposition (6.11.2)]. Thenthe ( S ) condition on R ∧ m allows us to apply [5, Lemma 3.2] to conclude that Q is a finiteset. Set Q ∪ Q = { p , . . . , p m } . By the same reasoning as in the Step 1 of the proofof [15, Theorem 4.4] and proceeding as in the Step 2 of the proof of [15, Theorem 4.4]together with (5 . p i a Zariski-dense open subset U i ⊂ P d − ( k )with the properties stated below: Let us put U ′′ := m \ i =1 U i ⊂ P d − ( k ) . Then we obtain the following assertion: R ∧ m / x e α R ∧ m is normal after localization at q ∈ g − (∆) for α ∈ Sp − A ∧ ( U ′ ∩ U ′′ ) . This shows that the desired conclusion (5 .
5) is fulfilled by letting U := U ′ ∩ U ′′ . Thisfinishes the proof of the theorem. (cid:3) We have the following corollary.
Corollary 5.4.
Let X be a normal connected affine scheme such that there is a surjectiveflat morphism of finite type X → Spec A , where A is an unramified discrete valuation ringof mixed characteristic p > . Assume that dim X ≥ , the generic fiber of X → Spec A is geometrically connected and the residue field of A is infinite.Then there exists an infinite family of pairwise distinct effective Cartier divisors { D λ } λ ∈ Λ of X such that each D λ is normal and flat over A .Proof. We can embed X into P nA as a dense open subset of an integral projective scheme Y . Let us consider the normalization h : Y → Y . Then Y is a normal connected projectivescheme over Spec A and h − ( X ) ≃ X because X ⊂ Nor( Y ). Hence, we may assume that X is embedded into a normal connected projective scheme Y as a dense open subset.Since the generic fiber of X → Spec A is geometrically connected, so is the generic fiberof Y → Spec A .Since the residue field of k is infinite, every non-empty Zariski open subset of P n ( k ) isinfinite. Now we can apply Theorem 5.3 to the normal connected projective scheme Y and find a family of pairwise distinct effective Cartier divisors { D ′ λ } λ ∈ Λ of Y such that D ′ λ is normal and flat over A and X ∩ D ′ λ = ∅ , because X ֒ → Y is an open immersion. Then,the set { D λ := X ∩ D ′ λ } λ ∈ Λ is the one as desired. (cid:3) Remark . (1) We have stated and proved Theorem 5.3 only when A is an unramifieddiscrete valuation ring. Suppose that we are given a flat and finite type morphism X → Spec B such that B is a ramified discrete valuation ring. If there is a ORMAL HYPERPLANE SECTIONS OF NORMAL SCHEMES 13 finite morphism Spec B → Spec A such that A is an unramified discrete valuationring and the generic fiber of X → Spec A is geometrically connected, Corollary5.4 still applies to the composite morphism X → Spec B → Spec A . However,since Spec B → Spec A is ramified, it does not happen that the generic fiber of X → Spec A is geometrically connected. Also the authors do not know if anyramified discrete valuation ring admits such an unramified discrete valuation ring.(2) The assumption on the geometric connectedness of the generic fiber of X → Spec A seems to be unavoidable in the proof of Theorem 5.3. Fix a prime p = 2 andconsider the following example:Proj (cid:0) Z p [ x, y ] / ( x − py ) (cid:1) → Spec Z p with deg( x ) = deg( y ) = 1. Then Z p [ x, y ] / ( x − py ) is not integrally closed andwe get the normalization map Z p [ x, y ] / ( x − py ) ֒ → Z p [ x, y, xy ] / ( x − py ) ≃ Z p [ √ p ][ t ] =: R by letting xy
7→ √ p , x
7→ √ pt and y t . The graded normal domain R is as shownin Lemma 5.2. However, we have R = Z p [ √ p ].We also obtain the Bertini theorem for regular projective schemes. Corollary 5.6.
Let X be a regular connected projective scheme over Spec A such that X → Spec A is flat and surjective, where ( A, π A , k ) is an unramified discrete valuation ringof mixed characteristic p > . Assume that dim X ≥ , the generic fiber of X → Spec A is geometrically connected and k is an infinite field.Then there exists an embedding X ֒ → P nA , together with a Zariski-dense open subset U ⊂ P n ( k ) ∨ , where P n ( k ) ∨ is the dual projective space of P n ( k ) , such that the schemetheoretic intersection X ∩ H is regular and flat over A for any H ∈ Sp − A ( U ) .Proof. The proof is modeled after that of Theorem 5.3 and let us only sketch the idea. As X is a regular scheme, it is normal. Keeping the notation as in the proof of Theorem 5.3, wecan find a Noetherian graded normal domain R such that R satisfies ( St ) and X ≃ Proj R .We have a flat surjective morphism C ( R ) = Spec R m \ { m + R m , m R m } → X = Proj R with smooth fibers by Proposition 3.1. Then X ∩ H α is regular . ⇐⇒ R m / x e α R m is regular after localization at p ∈ ∆ . So after completing R m , an application of Proposition 4.6 together with Remark 4.7 givesinfinitely many regular hyperplane sections of X . (cid:3) Applications to Weil divisor class groups of normal schemes
Let us recall the following theorem.
Theorem 6.1.
Let X ⊂ P nk be an irreducible and reduced projective variety over analgebraically closed field k such that dim X ≥ . Then there exists a Zariski-dense opensubset U ⊂ P n ( k ) ∨ such that X ∩ H is irreducible and reduced for every H ∈ U .Proof. Since the ground field k is assumed to be algebraically closed, if K ( X ) denotesthe function field of X , the extension of fields K ( X ) /k satisfies the assumption of [17,Theorem 11]. Hence X ∩ H is reduced and irreducible for all hyperplanes H belonging toa Zariski-dense open subset of P n ( k ) ∨ . The reducedness property of X ∩ H also followsfrom [4, Corollary 2]. (cid:3) Remark . If one takes a “hypersurface” instead of a “hyperplane” in Theorem 6.1 inthe positive characteristic case, one cannot get even reduced Cartier divisors as shownin the following example. Let k be a perfect field of characteristic p > P nk =Proj (cid:0) k [ X , . . . , X n ] (cid:1) . For α = ( α : · · · : α n ) ∈ P n ( k ). Let H α ⊂ P nk be the hypersurfacedefined by the homogeneous polynomial α X p + · · · + α n X pn . Then the Cartier divisor H α cannot be reduced for any choice of α .Let X be a Noetherian integral scheme with a Cartier divisor D . Since D ֒ → X is aregular immersion of codimension one, we can define the Gysin map CH i ( X ) → CH i − ( D ),where CH i ( X ) is the Chow group of algebraic cycles of dimension i on X (see [6, (2.6) or(6.2)]). When X is a normal scheme and dim X = d , then we have Cl( X ) = CH d − ( X ),where Cl( X ) denotes the Weil divisor class group of X . So we get the restriction map ofWeil divisor class groups Cl( X ) → Cl( D ). Let us prove the mixed characteristic versionof Grothendieck-Lefschetz theorem by Ravindra and Srinivas which is stated below [16,Theorem 1]. Theorem 6.3 (Ravindra-Srinivas) . Let X be a normal connected projective variety definedover an algebraically closed field of characteristic zero. Let | V | be a base-point free linearsystem associated to a linear subspace V ⊂ H ( V, L ) for an ample invertible sheaf L .Then there exists a Zariski-dense open subset U ⊂ | V | such that for Y ∈ U , Y is a normalconnected variety and the restriction map of the Weil divisor class groups Cl( X ) → Cl( Y ) is an isomorphism if dim X ≥ , and injective with finitely generated cokernel if dim X =3 . ORMAL HYPERPLANE SECTIONS OF NORMAL SCHEMES 15
The book [11] is a good introduction to the study of divisor class groups of projectivevarieties. [2] is an excellent survey which discusses relevant topics from the viewpoint ofHodge theory begun by Griffiths and his collaborators.
Corollary 6.4.
Let X → Spec A be a surjective projective flat morphism, where ( A, π A , k ) is an unramified discrete valuation ring with mixed characteristic and algebraically closedresidue field. Assume that X is a normal scheme, the generic fiber is geometrically con-nected and if X k = P ti =1 m i Γ i denotes the closed fiber of X → Spec A , then each Γ i is aprincipal, irreducible and reduced divisor.Then there exists an infinite family of pairwise distinct effective Cartier divisors { D λ } λ ∈ Λ of X such that each D λ is normal, connected, and the restriction map ofthe Weil divisor class groups Cl( X ) → Cl( D λ ) is an isomorphism if dim X ≥ , and injective with finitely generated cokernel if dim X =4 . We should note that the closed fiber X k is not necessarily a normal scheme. Proof.
Let K be the field of fractions of A . It is a field of characteristic zero. Let X K bethe generic fiber of X → Spec A . Then X K is a geometrically connected normal projectivevariety over K and dim X K = dim X −
1. Let us take
X ֒ → P nA as in Theorem 5.3. Thenthere is a commutative diagram: X k ֒ → −−−−→ P nk y y X ֒ → −−−−→ P nA x x X K ֒ → −−−−→ P nK Applying [7, Proposition 11.40] to the pair (
X, X K ), there is an exact sequence of groups t X i =1 Z [Γ i ] → Cl( X ) → Cl( X K ) → . Since Γ i is a principal divisor for i = 1 , . . . , t by assumption, the image of [Γ i ] in Cl( X ) istrivial. Hence we have an isomorphism:(6.1) Cl( X ) ≃ Cl( X K ) . By Theorem 5.3, there exists a Zariski-dense open subset U ⊂ P n ( k ) ∨ such that X ∩ H isa normal Cartier divisor for every H ∈ U := Sp − A ( U ) ⊂ P n ( A ) ∨ . On the other hand, applying Theorem 6.3 together with remarks at page 584 of [16],we have the following assertion:- There exists a Zariski-dense open subset
V ⊂ P n ( K ) ∨ = P n ( A ) ∨ such that X K ∩ H on X K is connected and normal for every H ∈ V and the restriction map Cl( X K ) → Cl( X K ∩ H ) is an isomorphism if dim X K ≥
4, and injective with finitely generatedcokernel if dim X K = 3.It is clear that Λ := U ∩ V ⊂ P n ( A ) ∨ is an infinite set.Denote by H λ the hyperplane of P nA corresponding to λ ∈ Λ. Then we have an infinitefamily of pairwise distinct effective Cartier divisors { D λ := X ∩ H λ } λ ∈ Λ on X . Since X K ∩ D λ = X K ∩ H λ is connected and D λ is flat over Spec A , the divisor D λ is the Zariskiclosure of X K ∩ D λ in X and thus, D λ is connected. It follows by the normality of D λ that each D λ is irreducible and reduced. After shrinking Λ ⊂ P n ( A ) ∨ , we can assume thatCl( D λ ) → Cl( X K ∩ D λ ) is an isomorphism. Indeed, it suffices to choose D λ so that thefollowing assertion holds by applying Theorem 6.1 to each Γ i (notice that Γ i ∩ D λ is ahyperplane section of Γ i ⊂ P nk ).- The closed fiber of the map D λ → Spec A is given as X k ∩ D λ = P ti =1 m i (Γ i ∩ D λ )and each Γ i ∩ D λ is irreducible and reduced.Then, since the restriction of a principal Cartier divisor is principal, it follows that Γ i ∩ D λ is a principal, irreducible and reduced divisor of D λ . Since the generic fiber of D λ → Spec A is X K ∩ D λ , applying [7, Proposition 11.40], we have t X i =1 Z [Γ i ∩ D λ ] → Cl( D λ ) → Cl( X K ∩ D λ ) → D λ ) ≃ Cl( X K ∩ D λ ) . Combining (6 .
1) and (6 .
2) together, we have the commutative diagram of the Weildivisor class groups: Cl( X K ) −−−−→ Cl( X K ∩ D λ ) (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) Cl( X ) −−−−→ Cl( D λ )Since the upper horizontal map has all the required properties as in Theorem 6.3, thelower horizontal map fulfills the similar properties. (cid:3) Acknowledgement .
The authors are grateful to the referee for pointing out errors andproviding remarks.
ORMAL HYPERPLANE SECTIONS OF NORMAL SCHEMES 17
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