A Bertini-type theorem for free arithmetic linear series
aa r X i v : . [ m a t h . AG ] J a n A BERTINI-TYPE THEOREM FOR FREE ARITHMETICLINEAR SERIES
HIDEAKI IKOMA
Abstract.
In this paper, we prove a version of the arithmetic Bertini theoremasserting that there exists a strictly small and generically smooth section of agiven arithmetically free graded arithmetic linear series.
Contents
0. Introduction 1Notation and conventions 31. Bertini’s theorem with degree estimate 32. Proofs 7References 90.
Introduction
When we generalize results on arithmetic surfaces to those on higher-dimensionalarithmetic varieties, it is sometimes very useful to cut the base scheme by a “good”global section s of a given Hermitian line bundle and proceed to induction ondimension. To do this, we have in the context of Arakelov geometry the followingresult. Fact ([5, Theorems 4.2 and 5.3]) . Let A be a C ∞ -Hermitian line bundle on a gener-ically smooth projective arithmetic variety X , and let x , . . . , x q be points (not nec-essarily closed) on X . Suppose that (i) A is ample, (ii) c ( A ) is positive definite,and (iii) H ( X, mA ) has a Z -basis consisting of sections with supremum norms lessthan for every m ≫ . Then there exist a sufficiently large integer m > and anonzero section s ∈ H ( X, mA ) such that (1) div( s ) Q is smooth over Q , (2) s ( x i ) = 0 for every i , and (3) k s k sup < . For example, this technique plays essential roles in the proofs of the arithmeticBogomolov-Gieseker inequality on high-dimensional arithmetic varieties (see [5]),of the arithmetic Hodge index theorem in codimension (see [6], [10]), of thearithmetic Siu inequality of Yuan (see [9]), and so on. A purpose of this paper is Date : November 23, 2013. Version 2.0.1991
Mathematics Subject Classification.
Primary 14G40; Secondary 11G50, 37P30.This research is supported by Research Fellow of Japan Society for the Promotion of Science. to give a simple elementary proof of the above fact and to strengthen it to the caseof arithmetically free graded arithmetic linear series.Let K be a number field. Let X be a projective arithmetic variety that isgeometrically irreducible over Spec( O K ) , and let L be an effective line bundle on X . A graded linear series belonging to L is a subgraded O K -algebra R • := M m > R m ⊆ M m > H ( X, mL ) . We consider norms k · k m on R m ⊗ Z R , and assume that the family of norms k · k • := ( k · k m ) m > is multiplicative , that is, k s ⊗ t k m + n k s k m k t k n holds for every s ∈ R m and t ∈ R n . Theorem A.
Let X be a generically smooth projective arithmetic variety, and let A be an effective line bundle on X . We consider a graded linear series R • := M m > R m belonging to A and a multiplicative norm k · k • on R • ⊗ Z R . Suppose the followingconditions. • R is base point free, • R • ⊗ Z Q is generated by R over Q , and • T m > { x ∈ X Q | t ( x ) = 0 for every t ∈ R m with k t k m < } = ∅ .Let Y , . . . , Y p be smooth closed subvarieties of the complex manifold X ( C ) , andlet x , . . . , x q be points (not necessarily closed) on X . Then, for every sufficientlylarge integer m ≫ , there exists a nonzero section s ∈ R m such that (1) div( s | Y ) , . . . , div( s | Y p ) are all smooth, (2) s ( x i ) = 0 for every i , and (3) k s k m < . Let L be a continuous Hermitian line bundle on X , and let k · k ( m )sup be thesupremum norm on H ( X, mL ) ⊗ Z R . We define a Z -submodule of H ( X, mL ) by F ( X, mL ) := D s ∈ H ( X, mL ) (cid:12)(cid:12)(cid:12) k s k ( m )sup < E Z . Then L m > F ( X, mL ) is a graded linear series belonging to L . We denote thestable base locus of L m > F ( X, mL ) by SBs ( L ) . Corollary B.
Let X be a generically smooth projective arithmetic variety, and let A be a continuous Hermitian line bundle on X . Suppose that SBs( A ) = ∅ and SBs ( A ) ∩ X Q = ∅ . Let Y , . . . , Y p be smooth closed subvarieties of the complexmanifold X ( C ) , and let x , . . . , x q be points (not necessarily closed) on X . Thenthere exist a sufficiently large integer m > and a nonzero section s ∈ H ( X, mA ) such that (1) div( s | Y ) , . . . , div( s | Y p ) are all smooth, (2) s ( x i ) = 0 for every i , and (3) k s k ( m )sup < . BERTINI-TYPE THEOREM FOR FREE ARITHMETIC LINEAR SERIES 3
Corollary C.
Let X be a generically smooth normal projective arithmetic variety,let L := ( L, | · | L ) be a continuous Hermitian line bundle on X , and let x , . . . , x q be points (not necessarily closed) on X \ SBs ( L ) . If SBs ( L ) ( X , then thereexist a sufficiently large integer m > and a nonzero section s ∈ H ( X, mL ) suchthat (1) div( s ) Q is smooth off SBs ( L ) , (2) s ( x i ) = 0 for every i , and (3) k s k ( m )sup < . Notation and conventions.
Let k denote a field, and let P n := P ( k n +1 ) denotethe projective space of one-dimensional quotients of k n +1 . Let pr : P n × k P m → P m denote the second projection. We denote the natural coordinate variables of P n (resp., of P m ) by X , . . . , X n (resp., by Y , . . . , Y m ) or simply by X • (resp., by Y • ).Let Y be a smooth variety over k . The singular locus of a morphism ϕ : X → Y over k is a Zariski-closed subset of X defined as Sing( ϕ ) := { x ∈ X | ϕ is not smooth at x } . A projective arithmetic variety X is a reduced irreducible scheme that is pro-jective and flat over Spec( Z ) . We say that X is generically smooth if X Q := X × Spec( Z ) Spec( Q ) → Spec( Q ) is smooth.1. Bertini’s theorem with degree estimate
In this section, we consider the geometric case. Let X ⊆ P n be a projectivevariety over an algebraically closed field k that is defined by a homogeneous primeideal I X ⊆ k [ X , . . . , X n ] , let O X (1) be the hyperplane line bundle on X , and let deg X := deg( c ( O X (1)) · dim X ) be the degree of X in P n . Let k [ X ] := k [ X , . . . , X n ] /I X be the homogeneouscoordinate ring of X , and let k [ X ] l be the homogeneous part of k [ X ] of degree l .There exists a polynomial function ϕ X ( l ) such that deg ϕ X = dim X , all coefficientsare nonnegative, and(1.1) dim k k [ X ] l ϕ X ( l ) for all l > . Let Z ⊆ X × k P m be a Zariski-closed subset defined by a system ofpolynomial equations: u ( X • ; Y • ) = 0 (mod I X ) , . . . , u h ( X • ; Y • ) = 0 (mod I X ) , where u i ∈ k [ X , . . . , X n ; Y , . . . , Y m ] has homogeneous degree deg X • u i (resp., deg Y • u i )in the set of variables X • (resp., Y • ). We recall the following fact from the elimina-tion theory. Lemma 1.1.
Let p := max i { deg X • u i } and let q := max i { deg Y • u i } . If the set-theoretic image pr ( Z ) does not coincide with P m , then pr ( Z ) is contained in ahypersurface of P m defined by a single homogeneous polynomial of degree less thanor equal to ϕ X (deg X · p dim X +1 ) · q. HIDEAKI IKOMA
Proof.
First, we can take a geometric point y , • = ( y , : · · · : y ,m ) ∈ P m \ pr ( Z ) .By an effective Nullstellensatz [3, Corollary 1.4], there exists a positive integer ℓ deg X · p dim X +1 such that ( X , . . . , X n ) ℓ ⊆ ( u ( X • ; y , • ) , . . . , u h ( X • ; y , • )) (mod I X ) . Next, we consider the k -linear maps T ( y • ) : k [ X ] ℓ − deg X • u ⊕ · · · ⊕ k [ X ] ℓ − deg X • u h → k [ X ] ℓ ( f ( X • ) , . . . , f h ( X • )) P i u i ( X • ; y • ) f i ( X • ) defined for y • = ( y : · · · : y m ) ∈ P m . By fixing basis for the above k -vectorspaces, we can represent T ( y • ) by a matrix whose entries are homogeneous poly-nomials of y • of degree less than or equal to q . By the choice of ℓ , we can see thatthere exists a certain dim k k [ X ] ℓ × dim k k [ X ] ℓ -minor of the representation matrixof T ( y • ) whose determinant is nonzero (see [8, Theorem (2.23)]). Then the image pr ( Z ) is contained in the hypersurface defined by the nonzero determinant, whichis homogeneous of degree less than or equal to (dim k k [ X ] ℓ ) · q . Since dim k k [ X ] ℓ ϕ X ( ℓ ) ϕ X (deg X · p dim X +1 ) , we have the result. (cid:3) Remark . For example, we consider the case where X = P n . Then dim k k [ X ] l = (cid:0) l + nn (cid:1) ( l + n ) n /n ! . Thus the bound in the above lemma becomes less than orequal to ( p n +1 + n ) n q/n ! . Moreover, by applying the theory of resultants (see [8,page 35]) to pr : P n × k A m → A m , one can obtain a weaker bound less than orequal to (2 p ) n − q + 1 in the above lemma (where the added is for the hyperplaneat infinity).Let A be an effective line bundle on X , and let R • be a subgraded ring of L m > H ( X, mA ) with Kodaira-Iitaka dimension κ ( R • ) := tr . deg k R • − . Supposethat R is base point free. Let φ m : X → P ( R m ) be a k -morphism associated to R m , and set(1.2) N m := dim k R m − for m > . We recall that the rational function field k ( X ) of X is given by k ( X ) = (cid:26) u (mod I X ) v (mod I X ) (cid:12)(cid:12)(cid:12)(cid:12) u, v ∈ k [ X , . . . , X n ] are homogeneousof the same degree and v / ∈ I X (cid:27) . Given a nonzero section e ∈ R , we define the degree of a nonzero section s ∈ H ( X, mA ) for m > with respect to e by deg X • ,e s := min (cid:26) deg X • u = deg X • v (cid:12)(cid:12)(cid:12)(cid:12) div s = ( u/v (mod I X )) + m div e,u/v (mod I X ) ∈ k ( X ) × (cid:27) . (Compare the definition with Jelonek’s in [3, §2].) Then, for any other nonzerosection s ′ ∈ H ( X, m ′ A ) , we have deg X • ,e ( s ⊗ s ′ ) deg X • ,e s + deg X • ,e s ′ . Theorem 1.3.
Let X ⊆ P n be a smooth projective variety over k , and let A be aline bundle on X . Let R • be a graded linear series belonging to A with Kodaira-Iitaka dimension κ ( R • ) . Suppose that the following three conditions are satisfied. • R is base point free. • R • is generated by R . BERTINI-TYPE THEOREM FOR FREE ARITHMETIC LINEAR SERIES 5 • (i) char( k ) = 0 or (ii) char( k ) = 0 and φ m : X → P ( R m ) is unramified forevery m > .Then one can find a polynomial function P ( m ) and hypersurfaces Z m ⊆ P ( R ∨ m ) for m = 1 , , . . . having the following two properties. (1) deg P dim X (dim X + 1)( κ ( R • ) + 1) . (2) For every m > , the hypersurface Z m ⊆ P ( R ∨ m ) contains the set { H ∈ P ( R ∨ m ) | φ m ( X ) ⊆ H or φ − m ( H ) is not smooth } and the homogeneous degree of Z m in P ( R ∨ m ) is less than or equal to P ( m ) .Remark . Throughout this paper, we assume that the empty set ∅ is smooth, sothat, if H / ∈ Z m , then φ − m ( H ) is empty or smooth of pure dimension dim X − . Proof.
Let I X ⊆ k [ X , . . . , X n ] denote the homogeneous prime ideal defining X .We consider the universal hyperplane section(1.3) W m := { ( x, H ) ∈ X × k P ( R ∨ m ) | φ m ( x ) ∈ H } endowed with the reduced induced scheme structure, and consider the restrictionof the second projection pr : X × k P ( R ∨ m ) → P ( R ∨ m ) to W m , which we denote by(1.4) π m : W m → P ( R ∨ m ) . Note that W m is the inverse image of the canonical bilinear hypersurface in P ( R m ) × k P ( R ∨ m ) via φ m × id : X × k P ( R ∨ m ) → P ( R m ) × k P ( R ∨ m ) . Since the restriction of thefirst projection to W m , W m → X , is surjective with fiber a projective space ofdimension N m − , W m is irreducible. The set-theoretic image of the singular locusof π m is given by π m (Sing( π m )) = { H ∈ P ( R ∨ m ) | φ m ( X ) ⊆ H or φ − m ( H ) is not smooth } . We fix a basis e , . . . , e N for R . From now on, we explain the method toconstruct an equation w that vanishes along W m from the section e . First, we set(1.5) D ,e := max i N { deg X • ,e e i } , and take rational functions u (1)1 /v (1)1 , . . . , u (1) N /v (1) N ∈ k ( X , . . . , X n ) × such that div e i = u (1) i v (1) i (mod I X ) ! + div e and deg X • u (1) i = deg X • v (1) i D ,e for i = 1 , . . . , N . Next, for m > , we can choose sections e ( m )1 , . . . e ( m ) N m ∈ R m suchthat e ( m ) i ∈ n e ⊗ α ⊗ · · · ⊗ e ⊗ α N N (cid:12)(cid:12)(cid:12) α + · · · + α N = m o and e ⊗ m , e ( m )1 , . . . , e ( m ) N m form a basis for R m . By identifying P ( R ∨ m ) with P N m viathe dual basis of e ⊗ m , e ( m )1 , . . . , e ( m ) N m , we can write φ m : X → P ( R ∨ m ) as φ m : X e → P N m , x u ( m )1 ( x ) v ( m )1 ( x ) : · · · : u ( m ) N m ( x ) v ( m ) N m ( x ) ! over X e := { x ∈ X | e ( x ) = 0 } , where u ( m ) i /v ( m ) i ∈ k ( X , . . . , X n ) × satisfies div e ( m ) i = u ( m ) i v ( m ) i (mod I X ) ! + m div e and deg X • u ( m ) i = deg X • v ( m ) i D ,e m. HIDEAKI IKOMA
We set(1.6) w := v ( m )1 · · · v ( m ) N m Y + u ( m )1 v ( m )2 · · · v ( m ) N m Y + · · · + v ( m )1 · · · v ( m ) N m − u ( m ) N m Y N m , which is homogeneous in X • (resp., in Y • ) of degree less than or equal to D ,e mN m (resp., ). Then w (mod I X ) vanishes along W m and defines W m in X e × k P N m .By the same method starting from e j ∈ R , we can construct an equation w j = X (homogeneous in X • of degree at most D ,e j mN m ) × (linear in Y • )that vanishes along W m and defines W m in X e j × k P N m . Let w N +1 , . . . , w h ∈ k [ X , . . . , X n ] be homogeneous polynomials that generate I X . Notice that the bi-homogeneous ideal(1.7) ( w , . . . , w N , w N +1 , . . . , w h ) ⊆ k [ X , . . . , X n ; Y , . . . , Y m ] may not be prime but the closed subscheme defined by ( w , . . . , w h ) in P n × k P N m coincides with W m .Set(1.8) D := max i N { D ,e i } , D := max N +1 j h { deg X • w j } , which does not depend on m . By the Euler rule together with the Jacobian criterionin the affine case, we conclude that the singular locus Sing( π m ) ⊆ X × k P ( R ∨ m ) isdefined by the determinants of certain ( n − dim X + 1) × ( n − dim X + 1) -minorsof the Jacobian matrix (cid:16) ∂w i ∂X j (cid:17) , whose degrees in X • (resp., in Y • ) are all boundedfrom above by ( N + 1)( D mN m −
1) + ( n − dim X )( D − (resp., by N + 1 ).We choose a positive constant D ′ > such that ( N + 1)( D mN m −
1) + ( n − dim X )( D − D ′ m κ ( R • )+1 for all m > . Let ϕ X ( l ) be as in (1.1) and set(1.9) P ( m ) := ϕ X (deg X ( D ′ m κ ( R • )+1 ) dim X +1 ) · ( N + 1) . Then deg P = dim X (dim X + 1)( κ ( R • ) + 1) . Since π m (Sing( π m )) is properly con-tained in P ( R ∨ m ) due to Kleiman [4, Corollaries 5 and 12], we can apply Lemma 1.1to this situation by setting p = D ′ m κ ( R • )+1 and q = N + 1 . Then we conclude that there exists a hypersurface Z m ⊆ P ( R ∨ m ) having degree lessthan or equal to P ( m ) and containing π m (Sing( π m )) . (cid:3) By applying Theorem 1.3 to the image of R m via H ( X, mA ) → H ( Y, mA | Y ) ,we have the following. Corollary 1.5.
Under the same assumptions as in Theorem 1.3, let Y be a smoothclosed subvariety of X , and let y , . . . , y q be closed points on X . Then one canfind a polynomial function P ( m ) and hypersurfaces Z m ⊆ P ( R ∨ m ) for m = 1 , , . . . having the following two properties. (1) deg P dim Y (dim Y + 1)( κ ( R • ) + 1) + q . (2) For every m > , the hypersurface Z m ⊆ P ( R ∨ m ) contains the set (cid:26) H ∈ P ( R ∨ m ) (cid:12)(cid:12)(cid:12)(cid:12) φ m ( Y ) ⊆ H , φ − m ( H ) ∩ Y is not smooth,or H contains one of y , . . . , y q (cid:27) and the homogeneous degree of Z m in P ( R ∨ m ) is less than or equal to P ( m ) . BERTINI-TYPE THEOREM FOR FREE ARITHMETIC LINEAR SERIES 7 Proofs
In this section, we turn to the arithmetic case and give proofs of Theorem A andCorollaries B and C. To prove Theorem A, we use Lemmas 2.1, 2.2, and 2.4.
Lemma 2.1 (Combinatorial Nullstellensatz [5, Lemma 5.2], [1, Theorem 1.2]) . Let V be a finite-dimensional vector space over a field k , and let u : V → k be a nonzero polynomial function with maximal total degree deg u . Let e , . . . , e N begenerators of V over k , and let S , . . . , S N be subsets of k . If Card( S j ) > deg u + 1 for every j , then there exist a ∈ S , . . . , a N ∈ S N such that u ( a e + · · · + a N e N ) = 0 . Lemma 2.2.
Let X be a projective arithmetic variety, let A be a line bundle on X ,and let R • be a graded linear series belonging to A . Suppose that R is base pointfree. Let y , . . . , y l ∈ X be distinct closed points on X such that char( k ( y i )) = 0 for every i , and let e ( m )1 , . . . , e ( m ) N m ∈ R m be generators of the Z -module R m . Set F := Q p : prime ∃ i, p | char( k ( y i )) p . Then, for every sufficiently large m , there exist integers a , . . . , a N m such that a j < F for every j , and ( a + F b ) e ( m )1 ( y i ) + · · · + ( a N m + F b N m ) e ( m ) N m ( y i ) = 0 for every integer b , . . . , b N m and for every i .Proof. First, we need the following claim.
Claim 2.3.
For every sufficiently large m , there exists an s ∈ R m such that s ( y i ) =0 for every i .Proof. Let φ : X → P N Z be the morphism associated to R such that φ ∗ X j = e (1) j for every j , and let O (1) be the hyperplane line bundle on P N Z . Then, for everysufficiently large m , the homomorphism H ( P N Z , O ( m )) → M i O ( m )( φ ( y i )) is surjective. Let t ∈ H ( P N Z , O ( m )) be a section such that t ( φ ( y i )) = 0 for every i .Then s := φ ∗ t has the desired property. (cid:3) Next, let s ∈ R m as above. Since F e ( m ) j ( y i ) = 0 for every i, j , we have that ( s + F t )( y i ) = s ( y i ) = 0 for every t ∈ R m and for every i . Thus we conclude the claim. (cid:3) Lemma 2.4 (Zhang-Moriwaki [7, Theorem A and Corollary B]) . Under the sameassumptions as in Theorem A, take an m ≫ , and fix e , . . . , e N ∈ R m such that { x ∈ X Q | e ( x ) = · · · = e N ( x ) = 0 } = ∅ and such that k e j k m < for every j . Then there exists a positive constant C > such that, for every sufficiently large m , one can find a Z -basis e ( m )1 , . . . , e ( m ) N m for R m such that max i n k e ( m ) i k m o Cm (dim X +2)(dim X − (cid:18) max j {k e j k m } (cid:19) m/m . HIDEAKI IKOMA
Proof of Theorem A.
Let r := [ K : Q ] , and let X ( C ) = X ∪ · · · ∪ X r be the de-composition into connected components. Let R m,α be the image of R m ⊗ Z C via H ( X, A ) ⊗ Z C → H ( X α , A C | X α ) , and let φ m,α : X α → P M m C be a morphism associ-ated to R m,α , where we set M m := rk Z R m /r . By Lemma 2.4, there exist constants C, Q with
C > and < Q < such that there exists a Z -basis e ( m )1 , . . . , e ( m ) rM m for R m consisting of the sections with supremum norms less than or equal to(2.1) Cm (dim X +2)(dim X − Q m . For each Y j , there exists a unique component X α ( j ) that contains Y j . Supposethat char( x i ) = 0 for i = 1 , . . . , q and char( x i ) = 0 for i = q +1 , . . . , q = q + q andlet y i be a closed point in { x i } . By applying Corollary 1.5 to X α ( j ) , Y j , y , . . . , y q ,and R • ,α ( j ) , one can find a polynomial function P j ( m ) of degree less than or equalto dim Y j (dim Y j − κ ( R • ,α ( j ) ) + 1) + q and hypersurfaces Z m,j ⊆ P ( R ∨ m,α ( j ) ) defined by homogeneous polynomials u m,j of degree less than or equal to P j ( m ) ,respectively, such that Z m,j contains all the hyperplanes H in P ( R ∨ m,α ( j ) ) such that φ m,α ( j ) ( Y j ) ⊆ H , φ − m,α ( j ) ( H ) ∩ Y j is not smooth, or φ − m,α ( j ) ( H ) contains one of y , . . . , y q . Set u m,α := Y α ( j )= α u m,j and consider the homogeneous polynomial function u : R m ⊗ Z C ∼ −→ r M α =1 R m,α Q α u m,α −−−−−−→ C of degree less than or equal to(2.2) P ( m ) := P ( m ) + · · · + P p ( m ) . Set F := Q q : prime ∃ i, q | char( y i ) q. Since e ( m )1 , . . . , e ( m ) rM m ∈ R m generate R m ⊗ Z C over C ,one can find integers a , . . . , a rM m and b , . . . , b rM m such that a i < F for every i , b j P ( m ) for every j , and u (( a + F b ) e ( m )1 + · · · + ( a rM m + F b rM m ) e ( m ) rM m ) = 0 by use of Lemmas 2.2 and 2.1. Hence, for each m ≫ , there exists a section t m ∈ R m such that t m | X α is not contained in any of Z m,j and k t m k m CF rm (dim X +2)(dim X − M m (1 + P ( m )) Q m . Since the right-hand side tends to zero as m → ∞ , we conclude the proof. (cid:3) Corollary B is a direct consequence of Theorem A.
Proof of Corollary C.
We can take a ≫ such that Bs F ( X, a L ) = SBs ( L ) .Let b ( a L ) := Image(F ( X, a L ) ⊗ Z ( − a L ) → O X ) , let µ : X ′ → X be ablowup such that X ′ is generically smooth and such that µ − b ( a L ) · O X ′ isCartier, and let E be an effective Cartier divisor on X ′ such that O X ′ ( − E ) = µ − b ( a L ) · O X ′ . We can assume that µ is isomorphic over X \ SBs ( L ) (see[2]). Set x ′ i := µ − ( x i ) ∈ X ′ \ E for i = 1 , . . . , q . Let B := O X ′ ( E ) and let B bethe canonical section. BERTINI-TYPE THEOREM FOR FREE ARITHMETIC LINEAR SERIES 9
Lemma 2.5. (1)
We can endow B with a continuous Hermitian metric | · | B such that | B | B ( x ) = max e ∈ H ( X,a L )0 < k e k ( a < ( | e | a L ( µ ( x )) k e k ( a )sup ) for all x ∈ X ′ ( C ) . (2) We set B := ( B, | · | B ) and A := a µ ∗ L − B . Then A is a continuousHermitian line bundle on X ′ such that Bs F ( X ′ , A ) = ∅ and c ( A ) > as a current.Proof. Set { e ∈ H ( X, a L ) \ { } | k e k ( a )sup < } = { e , . . . , e N } .(1): We choose an open covering { U ν } of X ′ ( C ) such that a µ ∗ L C | U ν is trivialwith local frame η ν , and E C ∩ U ν is defined by a local equation g ν . Then we canwrite µ ∗ e j = f j,ν · g ν · η ν on U ν , where f ,ν , . . . , f N,ν are holomorphic functions on U ν satisfying { x ∈ U ν | f ,ν ( x ) = · · · = f N,ν ( x ) = 0 } = ∅ . Since max j ( | e j | a L ( µ ( x )) k e j k ( a )sup ) = max j ( | f j,ν ( x ) |k e j k ( a )sup ) · | η ν | a µ ∗ L ( x ) · | g ν ( x ) | on x ∈ U ν , we have (1).(2): For each x ∈ X ′ ( C ) , we take indices ν and j such that x ∈ U ν and f j ,ν ( x ) = 0 . Let ε j be the section of A such that µ ∗ e j = ε j ⊗ B , and set h j,ν := f j,ν /f j ,ν . Then − log | ε j | A ( x ) = max j (cid:26) log | h j,ν ( x ) | − log (cid:16) k e j k ( a )sup (cid:17) (cid:27) is plurisubharmonic near x .We claim that k ε j k sup = k e j k ( a )sup , so that ε j ∈ F ( X ′ , A ) . The inequality k ε j k sup > k e j k ( a )sup is clear. Since | ε j | A ( x ) = | e j | a L ( µ ( x )) · min i ( k e i k ( a )sup | e i | a L ( µ ( x )) ) k e j k ( a )sup for all x ∈ ( X ′ \ E )( C ) , we have k ε j k sup = k e j k ( a )sup . This means that Bs F ( X ′ , A ) = ∅ . (cid:3) We apply Corollary B to A and we can find an m ≫ and a σ ∈ H ( X ′ , mA ) such that div( σ ) Q is smooth, σ ( x ′ i ) = 0 for every i , and k σ k sup < . Since X isnormal, there exists an s ∈ H ( X, ma L ) such that µ ∗ s = σ ⊗ ⊗ mB . Since µ isisomorphic over X \ SBs ( L ) , s has the desired properties. (cid:3) References [1] N. Alon. Combinatorial Nullstellensatz, Recent trends in combinatorics (Mátraháza, 1995).
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Graduate School of Mathematical Sciences, The University of Tokyo, Tokyo,153-8914, Japan
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