Néron-Severi group of a general hypersurface
aa r X i v : . [ m a t h . AG ] J a n N´ERON-SEVERI GROUP OF A GENERAL HYPERSURFACE
VINCENZO DI GENNARO AND DAVIDE FRANCO
Abstract.
In this paper we extend the well-known theorem of Angelo Lopezconcerning the Picard group of the general space projective surface containinga given smooth projective curve, to the intermediate N´eron-Severi group of ageneral hypersurface in any smooth projective variety.
Keywords : Noether-Lefschetz Theory, N´eron-Severi group, Borel-Moore Ho-mology, Monodromy representation, Isolated singularities, Blowing-up.
MSC2010 : 14B05, 14C20, 14C21, 14C22, 14C25, 14C30, 14F43, 14F45, 14J70. Introduction
A well-known result of Angelo Lopez [13], inspired by a previous work of Grif-fiths and Harris [11], provides a recipe for the computation of the N´eron-Severigroup
N S ( S ; Z ) of a general complex surface S of sufficiently large degree in P ,containing a given smooth curve. For a smooth projective variety X , we definethe i -th N´eron-Severi group N S i ( X ; Z ) as the image of the cycle map A i ( X ) → H i ( X ; Z ) ∼ = H X − i ) ( X ; Z ) ([8], § . N S dim X/ ( X ; Z )of a general hypersurface X , in any smooth projective variety. In the previous pa-per [4] we already obtained a generalization, but only in the case of Q -coefficients,i.e. only for N S dim X/ ( X ; Q ) := N S dim X/ ( X ; Z ) ⊗ Z Q . More precisely, in ([4],Theorem 1.2), we proved the following: Theorem 1.1.
Let Y ⊂ P = P ( C ) be a smooth projective variety of dimension m + 1 = 2 r + 1 and set V d := Im( H ( P , O P ( d )) → H ( Y, O Y ( d ))) . Let Z ⊂ Y be a closed subscheme of dimension r contained in a regular sequence of smoothhypersurfaces X ∈ |V d | , G i ∈ |V d i | , ≤ i ≤ r , such that d > d > · · · > d r .Let X ∈ |V d | be a very general hypersurface containing Z , so that Z is a closedsubscheme of the complete intersection ∆ := X ∩ G ∩ · · · ∩ G r , ∆ = Z ∪ R = ( ρ [ i =1 Z i ) ∪ ( σ [ j =1 R j ) . Assume that the vanishing cohomology of X is not of pure Hodge type ( m , m ) .Denote by H m ( X ; Z ) ∆ the subgroup of H m ( X ; Z ) generated by the componentsof ∆ , and by H m ( X ; Z ) ∆ − the subgroup of H m ( X ; Z ) generated by Z , . . . , Z ρ , R , . . . , R σ − . Then we have: (1) H m ( X ; Z ) ∆ is free of rank ρ + σ ; (2) N S r ( X ; Q ) = N S r +1 ( Y ; Q ) ⊕ H m ( X ; Q ) ∆ − . The aim of this paper is to improve previous Theorem 1.1, showing that:
Theorem 1.2.
N S r ( X ; Z ) = [ N S r ( X ; Z ) ∩ H m ( Y ; Z )] ⊕ H m ( X ; Z ) ∆ − . We would like to stress that even though the main troubles in the proof ofTheorem 1.2 come from the singularities of ∆, such a result is not trivial even forsmooth ∆. Indeed, although in this case e Y := Bl ∆ ( Y ) would be smooth, the stricttransform e X := Bl ∆ ( X ) would vary in a linear system which is not very ample on e Y . In fact, as it is proved in Proposition 2.12, this linear system contracts T ri =1 G i to a point. Therefore, one cannot apply Lefschetz Hyperplane Theorem directly.Actually, it is our opinion that even for smooth ∆ it would be difficult to avoid thearguments used in this note.As explained in the body of the paper, the main technical point in the proof ofTheorem 1.2 refers to the following Lefschetz-type problem: Question 1.
Let G ⊆ P be an irreducible, smooth projective variety of dimension m = 2 r ≥ , and fix a hypersurface W ∈ | H ( G, O G ( d )) | ( d ≥ ). To what extentone can assume the Gysin map: (1) H m +1 ( G ; Z ) ∩ u −→ H m − ( W ; Z ) to be injective (here u ∈ H ( G, G − W ; Z ) denotes the orientation class [8] , § . )? Of course the answer to such a question is trivially affirmative in many cases. IfTor H m +1 ( G ; Z ) = 0 or if we would work with Q -coefficients then the Gysin mapis injective by Hard Lefschetz Theorem. If W is smooth then the Gysin map isinjective by Lefschetz Hyperplane Theorem. However, it is easy to find exampleswhere the above Gysin map is not injective, see Example 2.2. Unfortunately, in ourcase W could be singular. The only way to obtain an interesting result is to vary W . If the linear system | W | was very ample outside its base locus, then we coulddeduce the injectivity of (1) from Lefschetz Theorem with Singularities, see ([10],p. 199), and compare with ([4], Lemma 3.2). Unfortunately, in our case | W | maynot be very ample outside its base locus. This is the ultimate reason for which thefollowing Theorem, which is the main technical result of this paper, has required amajor effort. Theorem 1.3.
Keep notations as in Theorem 1.1, set G := G i , m = 2 r := dim C G ,and define W := G ∩ X ( X ∈ |V d | is Zariski general containing Z ). Then the Gysinmap k ⋆ : H m +1 ( G ; Z ) −→ H m − ( W ; Z ) is injective.Remark . The following example shows that the condition d i = d j in Theorem1.1 is necessary. Consider Y = P . Let G be a smooth quadric hypersurface,and let L be a general hyperplane section of G . Let G be a smooth generalquadric hypersurface containing L , so that G ∩ G is equal to the union of L with another smooth quadric threefold L . Let X be a general hypersurface of ´ERON-SEVERI GROUP OF A GENERAL HYPERSURFACE 3 degree d >
2, and define ∆ := Z := X ∩ G ∩ G . Then ∆ has two irreduciblecomponents ∆ = Z ∪ Z , with Z i = X ∩ L i . Now in H ( X ; Z ) we have Z = 2 H ,where H denotes the hyperplane class. Therefore H ( X ; Z ) ∆ is generated by H ,which contradicts Theorem 1.1, (1).2. Some basic facts
Notations . (i) From now on, unless it is otherwise stated, all cohomology andhomology groups are with Z -coefficients.(ii) Borel-Moore homology . We will denote by H BMi ( M ) the Borel-Moore homol-ogy groups of a variety M . Here we recall some properties of these groups, whichwill be needed throughout the paper.a) Borel-Moore homology is equal to ordinary homology for any compact variety([9], p. 217, line 7 from below).b) If U is open in M , and C is the complement of U in M , then there is a longexact sequence(2) · · · → H BMi +1 ( U ) → H BMi ( C ) → H BMi ( M ) → H BMi ( U ) → H BMi − ( C ) → . . . ([9], Lemma 3, p. 219).c) If M is smooth of complex dimension m , then there is a natural isomorphism(3) H BMi ( M ) ∼ = H m − i ( M )([9], (26), p. 217). Example 2.2.
Denote by T an irreducible, projective, smooth threefold such thatTor H ( T ) = 0. Choose a torsion class 0 = c ∈ Tor H ( T ) and assume that l · c = 0for some l ∈ Z with l >
0. Define S := T × P r − ⊂ G := T × P m − ⊂ P , r = m ≥ , and choose a general W ∈ | H ( G, I S,G ( kl )) | , k ≫
0. From dim S = codim S + 4it follows that the hypersurface W gives rise to a section of the normal bundle N S,G ( kl ) which vanishes in dimension four. Therefore, we have dim Sing W = 4.Consider the cycle γ := c ⊗ [ P r − ] ∈ H m +1 ( S ) , and let γ ′ be the image of γ in H m +1 ( G ), via push-forward. Notice that γ ′ = 0.From the commutative diagram γ ∈ H m +1 ( S ) −→ H m +1 ( G ) ↓ ↓ ↓ γ ∩ kl [ H ] ∈ H m − ( S ) −→ H m − ( W ) , where [ H ] ∈ H ( S ) denotes the hyperplane class, it follows that the image of γ ′ in H m − ( W ) vanishes. Hence the map H m +1 ( G ) → H m − ( W ) provides an exampleof Gysin map, which is not injective. VINCENZO DI GENNARO AND DAVIDE FRANCO
Remark . As we have just observed, in the examples above dim Sing W = 4. Wedo not know examples of not injective Gysin maps for hypersurfaces with isolatedsingularities. Keeping notations as in Theorem 1.1, isolated singularities appear forinstance when we define W = G ∩ X , G = G i ([7], Proposition 4.2.6 and proof, p.133). Nevertheless, even in the case dim Sing W = 0 it seems unlikely that Gysinmap must be always injective. Indeed, assume dim Sing W = 0 and defineΓ := Sing W = { x , . . . , x s } , W ′ := W − Γ . Using (2) and (3) we have an isomorphism for m > H m − ( W ) ∼ = H BMm − ( W ′ ) ∼ = H m − ( W ′ ) . Consider the cohomology long exact sequence . . . −→ H m − ( W, W ′ ) −→ H m − ( W ) −→ H m − ( W ′ ) −→ . . . . Choose a small ball S j ⊂ G around each x j , and set B j := S j ∩ W and B j := B j − { x j } . By excision, we have H m − ( W, W ′ ) ∼ = s M j =1 H m − ( B j , B j ) . By ([5], p. 245), we have H m − ( B j , B j ) ∼ = H m − ( K j ) , where K j denotes the link of the singularity x j . By Milnor’s Theorem ([5], Theorem3.2.1, p.76), the link is ( m − m = 2 r is even, for a node and more generally for anordinary singularity one has H m − ( K j ) = 0. Summing up, we have(4) L sj =1 H m − ( K j ) −→ H m − ( W ) −→ H m − ( W ) ↑ ր H m +1 ( G ) ∼ = H m − ( G ) . Although the vertical arrow is injective by Lefschetz Hyperplane Theorem, it seemsunlikely that the oblique one, i.e. the Gysin map, must be injective for any W .However, we remark that for certain very special isolated singularities one knowsthat H m − ( K j ) = 0 ([5], Proposition 4.7, p. 93, Theorem 4.10, p. 94). Finally, onecan infer the injectivity of the Gysin map also when rk H m − ( W ) = rk H m ( W ).Indeed, in this case the exact sequence0 → H m − ( W ) → H m ( W ) → s M j =1 H m − ( K j ) → H m − ( W ) → H m − ( W )shows that the map L sj =1 H m − ( K j ) → H m − ( W ) is injective, because H m − ( K j )is torsion free ([5], (4.1) and (4.2), p. 91). By (4), this implies that the Gysin map H m +1 ( G ) → H m − ( W ) is injective, because its kernel is a torsion group by HardLefschetz Theorem. ´ERON-SEVERI GROUP OF A GENERAL HYPERSURFACE 5 Notations . Consider a smooth quasi-projective variety Y of dimension n anda locally free sheaf E of rank r on Y . Set V := P ( E ), denote by π : V → Y the natural projection and denote by c := c ( O V (1)) ∈ A ( V ) the first Chernclass. The cycle map ([8], p.370) sends A i ( V ) into the Borel-Moore homologygroup H BM n + r − − i ) ( V ), which can be identified with H i ( V ), see (3). Denote by ξ i ∈ H i ( V ) the cohomology class corresponding to c i ∈ A i ( V ). By the Leray-Hirsch Theorem, we have an isomorphism for any fixed integer m : φ = ⊕ r − i =0 φ i : ⊕ r − i =0 H m − i ( Y ) → H m ( V ) , φ i ( · ) = π ∗ ( · ) ∪ ξ i . Now we are going to prove that the Leray-Hirsch Theorem holds true also forBorel-Moore homology groups. The following Lemma is certainly well-known, butwe briefly prove it for lack of a suitable reference.
Lemma 2.5.
We have an isomorphism of Borel-Moore homology groups: ψ = ⊕ r − i =0 ψ i : H BMm ( V ) → ⊕ r − i =0 H BMm − i ( Y ) , ψ i ( · ) = π ∗ ( · ∩ ξ i ) . Proof.
As explained in ([15], Proof of the Leray-Hirsch Theorem, p. 195), we havean isomorphism in the derived category D ∗ ( A Y ), notations as in [6]: π ∗ Z V ∼ = r − M i =0 Z Y [ − i ] . In order to prove the Lemma it suffices to apply the derived functor R • Γ c to theisomorphism above and then take the dual: R • Γ c ( V , Z ) ∼ = r − M i =0 R • Γ c ( Y, Z )[ − i ] , DR • Γ c ( V , Z ) ∼ = r − M i =0 DR • Γ c ( Y, Z )[2 i ] . Compare with ([12], p.374), and use notations as in ([12], pp. 374-78). (cid:3)
Remark . ( i ) In the statement of the Leray-Hirsch Theorem the cohomologyclasses ξ i are defined up to classes in π ∗ ( H i ( Y )), hence ξ r − could be replaced bythe cycle class of any unisecant in A n ( V ).( ii ) Notice that π is a local complete intersection (l.c.i. for short) morphism [8].Set M m := ker (cid:0) ⊕ r − i =0 ψ i (cid:1) . Then ψ r − : M m → H BMm − r +2 ( Y ) is an isomorphism withinverse the Gysin map(5) π ⋆ : H BMm − r +2 ( Y ) → M m ⊂ H BMm ( V ) , which represents the tensor product with the fundamental class of the fiber of π : V → Y . Compare with ([8], Example 19.2.1, p. 382), and with the proof ofTheorem 8 in ([14], Theorem 8, p. 258). Notations . Choose a section in H ( Y, E ), and assume it vanishes on a subscheme D ⊂ Y having the right codimension. Then we have a surjection E ∨ −→ I D,Y −→ . VINCENZO DI GENNARO AND DAVIDE FRANCO
This surjection induces an imbedding e Y := Bl D ( Y ) ⊂ V . Since the natural pro-jection π Y := π | e Y : e Y −→ Y is a l.c.i. morphism of codimension 0, it follows thatthere exists a Gysin map ([8], Example 19.2.1, p. 382): ⋆ : H BM • ( Y ) → H BM • ( e Y ) . Theorem 2.8.
With notations as above we have: π Y ∗ ◦ ⋆ = id : H BM • ( Y ) → H BM • ( Y ) , in particular ⋆ is injective.Proof. We denote by f : e Y → V the inclusion morphism. Applying (5) to thedefinition of Gysin map ([8], Example 19.2.1, p. 382) we have:(6) ⋆ ( x ) = π ⋆ ( x ) ∩ u e Y , ∀ x ∈ H BMm ( Y ) . Here u e Y denotes the orientation class of e Y in V ([8], p. 372), so that: ∩ u e Y : H BM • ( V ) −→ H BM •− r +2 ( e Y ) . Since e Y is unisecant in V , Remark 2.6, ( i ), implies that we may assume ξ r − to bethe cycle class of e Y . We thus get f ∗ ( · ∩ u e Y ) = · ∩ ξ r − . According to (6), we have(7) f ∗ ( ⋆ ( x )) = f ∗ ( π ⋆ ( x ) ∩ u e Y ) = π ⋆ ( x ) ∩ ξ r − , ∀ x ∈ H BMm ( Y ) . Using (7), Lemma 2.5 and Remark 2.6, ( ii ), we may conclude( π Y ∗ ◦ ⋆ )( x ) = π ∗ ( f ∗ ( ⋆ ( x ))) = π ∗ ( π ⋆ ( x ) ∩ ξ r − ) = ψ r − ◦ π ⋆ ( x ) = x, for any x ∈ H BM • ( Y ). (cid:3) Remark . Consider a quasi-projective smooth variety Y and a complete inter-section ∆ = T ri =1 X i , X i ∈ | H ( Y, O Y ( d i )) | . Fix i , set X := X i , and assume that X is smooth. Applying Theorem 2.8 to Y and E = ⊕ ri =1 O Y ( d i ), and to X and E = ⊕ ri =1 ,i = i O X ( d i ), we see that the Gysin maps are injective: ⋆ : H BM • ( Y ) ֒ → H BM • ( e Y ) , e Y := Bl ∆ ( Y ) ,ı ⋆ : H BM • ( X ) ֒ → H BM • ( e X ) , e X := Bl ∆ ( X ) . Notice that e X ⊂ e Y ([8], B.6.9, p.436), and that e X is a Cartier divisor on e Y , for ∆is regularly imbedded in both X and Y ([8], B.6.10, p.437). Lemma 2.10.
Denote by ι X : X → Y and ι e X : e X → e Y the inclusions. Then thefollowing diagram of Gysin maps is commutative: H BM • ( Y ) ⋆ ֒ → H BM • ( e Y ) ι X⋆ ↓ ↓ ι f X⋆ H BM •− ( X ) ı ⋆ ֒ → H BM •− ( e X ) . ´ERON-SEVERI GROUP OF A GENERAL HYPERSURFACE 7 Proof.
The natural maps e X π X −→ X ι X −→ Y and e X ι f X −→ e Y π Y −→ Y are equal. Fur-thermore, they are l.c.i. maps because they are both composite of l.c.i. maps.Therefore, by functoriality of the Gysin morphism ([8], Example 19.2.1, p. 382),we have: ı ⋆ ◦ ι X⋆ = ι e X⋆ ◦ ⋆ . (cid:3) Notations . Let Y ⊂ P be a possibly singular quasi-projective variety, andset V d := Im( H ( P , O P ( d )) → H ( Y, O Y ( d ))). Consider a complete intersection∆ = T r X i , X i ∈ |V d i | , with d := d ≥ d ≥ d ≥ · · · ≥ d r . Fix a hypersurface X ∈ |V ∆ ,d | , where V ∆ ,d := V d ∩ H ( Y, I ∆ ,Y ( d )). Then we have e X := Bl ∆ ( X ) ⊂ Bl ∆ ( Y ) =: e Y ([8], B.6.9, p.436). Since ∆ is regularly imbedded in both X and Y , it follows that e X is a Cartier divisor on e Y ([8], B.6.10, p.437). More precisely e X ∈ | H ( O e Y ( d e H − e ∆)) | ,where O e Y ( e H ) denotes the pull-back of O Y (1) via the natural projection e Y → Y ,and e ∆ denotes the exceptional divisor in e Y . Since I ∆ ,Y ( d ) is globally generated,by letting X ∈ |V ∆ ,d | vary, we have a base point free linear system | e X | on e Y and amorphism ν : e Y → P ′ = P ( V ∗ ∆ ,d ) , Q := ν ( e Y ) . Proposition 2.12.
Assume moreover that d > d and set T := T ri =2 X i . Then wehave: (1) T ∼ = e T := Bl ∆ ( T ) ⊂ e Y , e T ∩ e X = ∅ , hence the morphism ν sends e T to apoint p ∈ Q ; (2) the morphism ν is an isomorphism outside e T , namely | e X | is very ample on e Y − e T : ν : e Y − e T ∼ = Q − { p } . Proof. (1) Since ∆ is a Cartier divisor cut out on T by X , it follows that the naturalprojection π : e T → T is in fact an isomorphism. So we have
T ∼ = e T = Bl ∆ ( T ) ⊂ e Y . Furthermore, wehave: O e Y ( − e ∆) ⊗ O e T ∼ = π ∗ ( O T ( − ∆)) ∼ = π ∗ ( I X ∩T , T ) ∼ = O e Y ( − e ∆ − e X ) ⊗ O e T . Hence we find O e Y ( e X ) ⊗ O e T ∼ = O e T , and we are done.(2) Consider the point p ∈ P ′ representing the hyperplane L ⊂ |V ∆ ,d | spanned bydivisors of the form X i ∪ M i , with i ≥ M i ∈ |V d − d i | . Such a hyperplane isspanned by the image of ( V d ∩ H ( Y, I T ,Y ( d )) ⊗ V d − d in |V ∆ ,d | . Since its baselocus is T , it follows that ν ( e T ) = p . On the other hand ([8], B.6.10 p.437), we have: N e T , e Y ∼ = ( π ∗ N T ,Y )( − e ∆) ∼ = ⊕ r − i =2 O e Y ( e X i ) . VINCENZO DI GENNARO AND DAVIDE FRANCO
It follows that e T is a complete intersection also in e Y : r − \ i =2 e X i = e T ⊂ e Y .
But the hyperplane L ⊂ P ′ ∼ = | e X | ∗ is spanned by divisors of the form e X i ∪ f M i ,with i ≥ M i ∈ |V d − d i | , and f M i := strict transform of M i in e Y . Since the baselocus of L is e T , it follows that ν − ( p ) = e T scheme theoretically. Consider a point x ∈ e Y − e T and its image ν ( x ) = p . The corresponding hyperplane L x = L ⊂ V ∆ ,d must contain a divisor X ∈ L x such that ∆ = X ∩ T . If ν did not separate x from another point or a tangent vector, then they both would be contained in e X := Bl ∆ ( X ). This is impossible because I ∆ ,X ( d ) is generated by V ∆ ,d , henceour linear system is very ample on e X (recall that d > d ). (cid:3) Proof of Theorem 1.3
Notations . Let Y be a smooth projective variety of dimension m = 2 r + 1, andlet X ∈ |V d | , G i ∈ |V d i | , 1 ≤ i ≤ r , be a regular sequence of smooth hypersurfaces.Assume moreover that d > d > · · · > d r . Define T := T ri =1 G i and ∆ := T ∩ X and fix G = G i . If X ∈ |V ∆ ,d | denotes a general hypersurface containing ∆, definealso W := X ∩ G. Consider the Gysin map k ⋆ : H • ( G ) −→ H •− ( W ) , where k : W → G denotes the imbedding morphism.Theorem 1.3 will follow from a slightly stronger result: Theorem 3.2.
The Gysin map k ⋆ : H m +1 ( G ) −→ H m − ( W ) is injective for a general W ∈ |V d ∩ H ( G, I ∆ ,G ( d )) | . We start with:
Proposition 3.3.
Assume r ≥ and define T := T ri =1 G i . Assume x ∈ H m +1 ( G ) is such that k ⋆ ( x ) = 0 ∈ H m − ( W ) , for a general W ∈ |V d ∩ H ( G, I ∆ ,G ( d )) | . Then x belongs to the image of the push forward from T : x ∈ Im( h ∗ : H m +1 ( T ) → H m +1 ( G )) . Proof.
Denote by S := Sing ∆ the singular locus of ∆, and set∆ := ∆ − S, T := T −
S, G := G − S, W := W − S. Observe that ∆ , G and W are smooth. Since dim S ≤ r − H m +1 ( S ) = H m ( S ) = 0 ([9], Lemma 4, p. 219).Therefore, from the exact sequence for Borel-Moore homology: . . . −→ H m +1 ( S ) −→ H m +1 ( G ) −→ H BMm +1 ( G ) −→ H m ( S ) −→ . . . , ´ERON-SEVERI GROUP OF A GENERAL HYPERSURFACE 9 we get H m +1 ( G ) ∼ = H BMm +1 ( G ) (compare with (2)). We thus find x ∈ H BMm +1 ( G ),and therefore k ⋆ ( x ) = 0 ∈ H BMm − ( W ).Combining Theorem 2.8 and Lemma 2.10 we have moreover a commutative di-agram with injective horizontal maps: H BMm +1 ( G ) ⋆ ֒ → H BMm +1 ( e G ) k ⋆ ↓ ˜ k ⋆ ↓ H BMm − ( W ) ı ⋆ ֒ → H BMm − ( f W ) , with f G o := Bl ∆ ( G ) and f W := Bl ∆ ( W ). We thus find˜ x := ⋆ ( x ) ∈ H BMm +1 ( f G o ) , with ˜ k ⋆ (˜ x ) = 0 ∈ H BMm − ( f W ) . Let us look at the exact sequence:(8) . . . −→ H BMm +1 ( e T ) σ −→ H BMm +1 ( e G ) ρ −→ H BMm +1 ( e G − e T ) −→ . . . ( T ∼ = e T ∼ = Bl ∆ ( T )). Applying Notations 2.11 and Proposition 2.12 to thelinear system | f W | on e G , we find that f W ∩ e T = ∅ . Then the linear system f W is very ample on the smooth variety e G − e T . Since˜ k ⋆ (˜ x ) = 0 ∈ H BMm − ( f W ) ∼ = H m − ( f W ) , it follows by Lefschetz Theorem with Singularities ([10], p.199) that: ρ (˜ x ) = 0 ∈ H BMm +1 ( e G − e T ) ∼ = H m − ( e G − e T ) . Then (8) implies ˜ x = σ ( y ) ∈ Im( H BMm +1 ( e T ) → H BMm +1 ( e G )). We are done because y ∈ H BMm +1 ( e T ) ∼ = H BMm +1 ( T ) ∼ = H m +1 ( T ), and h ∗ ( y ) ∈ H m +1 ( G ) must coincidewith x . In fact they both go to ˜ x ∈ H BMm +1 ( e G ) ([9], p. 219, Exercise 5), and themap H m +1 ( G ) ∼ = H BMm +1 ( G ) → H BMm +1 ( e G )is injective by Theorem 2.8. (cid:3) Proposition 3.4.
Assume r ≥ and define T := T ri =1 G i . If y ∈ H m +1 ( T ) issuch that h ∗ ( y ) ∈ Tor ( H m +1 ( G )) then y = 0 .Proof. First notice that Tor ( H m +1 ( T )) = 0. In fact, since dim Sing T ≤ r − H m +1 ( T ) ∼ = H BMm +1 ( T −
Sing T ) ∼ = H ( T −
Sing T ) . Furthermore, H ( T −
Sing T ) is torsion free by the Universal Coefficient Theorem([14], p. 243). From Tor ( H m +1 ( T )) = 0 it follows H m +1 ( T ; Z ) ⊂ H m +1 ( T ; Q ),and we may assume y ∈ H m +1 ( T ; Q ) is such that 0 = h ∗ ( x ) ∈ H m +1 ( G ; Q ). Fromnow on, in the rest of the proof, all cohomology and homology groups are with Q -coefficients.We are going to argue by induction on r ≥ • r = 2.In this case, by ([7], Proposition 4.2.6, p.133), we know that T = G ∩ G isa threefold with isolated singularities (see also [3], loc. cit.). Set Γ := Sing T = { x , . . . , x s } , T ′ := T −
Γ. Then y ∈ H ( T ) ∼ = H BM ( T ′ ) ∼ = H ( T ′ ). We claimthat:(9) y ∈ Im( H ( T ) → H ( T ′ )) . From the cohomology exact sequence: . . . −→ H ( T ) −→ H ( T ′ ) −→ H ( T , T ′ ) −→ . . . we see that in order to prove the claim it suffices to show that H ( T , T ′ ) = 0.Choose a small ball S j ⊂ G around each x j , and set B j := S j ∩ W and B j := B j − { x j } . Then by excision we have H ( T , T ′ ) ∼ = s M j =1 H ( B j , B j ) ∼ = s M j =1 H ( K j ) , where K j denotes the link of the singularity x j ([5], p. 245). The claim (9) followsby Milnor’s Theorem ([5], Theorem 3.2.1, p.76). To conclude the proof in the case r = 2 it suffices to observe that any y ∈ H ( T ) ∼ = H ( G ) such that 0 = h ∗ ( y ) ∈ H ( G ) ∼ = H ( G ) vanishes by Hard Lefschetz Theorem. Recall that now we areassuming that all cohomology and homology groups are with Q -coefficients. • r ≥ R := G ∩ G j , j = i , and denote by f : T → R the inclusion morphism. Weclaim that:(10) z := f ∗ ( y ) = 0 ∈ H m +1 ( R ) . First we have(11) ψ ∗ ( z ) = ψ ∗ ( f ∗ ( y )) = ( ψ ◦ f ) ∗ ( y ) = h ∗ ( y ) = 0 ∈ H m +1 ( G ) , with ψ : R → G the inclusion morphism. By ([7], Proposition 4.2.6, p.133), R hasat worst finitely many singularities. SetΓ := Sing R = { x , . . . , x s } , R ′ := R − Γ . Then z ∈ H m +1 ( R ) ∼ = H BMm +1 ( R ′ ) ∼ = H m − ( R ′ ). Consider the cohomology longexact sequence:(12) . . . −→ H m − ( R ) −→ H m − ( R ′ ) −→ H m − ( R, R ′ ) −→ . . . , choose a small ball S j ⊂ G around each x j , and set B j := S j ∩ R and B j := B j − { x j } . By excision we have(13) H m − ( R, R ′ ) ∼ = s M j =1 H m − ( B j , B j ) , and by ([5], p. 245) we get:(14) H m − ( B j , B j ) ∼ = s M j =1 H m − ( K j ) = 0 . Here K j denotes the link of the singularity x j . The last vanishing follows by Milnor’sTheorem ([5], Theorem 3.2.1, p.76), because the link of an isolated singularity ofdimension dim R = m − m − ´ERON-SEVERI GROUP OF A GENERAL HYPERSURFACE 11 Combining (11), (12), (13) and (14) we have z ∈ H m − ( R ) ∼ = H m − ( G ) , ψ ∗ ( z ) ∈ H m +1 ( G ) ∼ = H m − ( G ) , and our claim (10) follows by Hard Lefschetz Theorem.Having proved f ∗ ( y ) = 0, we now recall that dim Sing T ≤ r − H and look at thefollowing commutative diagram: H m +1 ( T ) ∼ = H ( T ′ ) f ∗ −→ H m +1 ( R ) ∼ = H m − ( R ′ ) ↓ ↓ H m − ( T ∩ H ) ∼ = H ( T ′ ∩ H ) −→ H m − ( R ∩ H ) ∼ = H m − ( R ′ ∩ H ) , where T ′ := T −
Sing T , and the vertical maps are injective by Lefschetz Theoremwith Singularities ([10], p. 199). The statement follows by induction. (cid:3) Proof of Theorem 3.2.
Choose an element 0 = x ∈ H m +1 ( G ). We have to prove0 = k ⋆ ( x ) ∈ H m − ( W ). We distinguish two cases, according that either r = 1 or r ≥ r = 1 then we may assume x ∈ H ( G ; Q ) because Tor H ( G ) ∼ = Tor H ( G ) = 0by the Universal Coefficient Theorem. And the claim follows because the compositeof k ⋆ with the push-forward (put m = 2): H m − ( G ; Q ) ∼ = H m +1 ( G ; Q ) k ⋆ −→ H m − ( W ; Q ) −→ H m − ( G ; Q ) ∼ = H m +1 ( G ; Q )is injective by Hard Lefschetz Theorem.Next assume r ≥
2. If x / ∈ Tor ( H m +1 ( G )) then again we may assume x ∈ H m +1 ( G ; Q ), and we may conclude as before. If 0 = x ∈ Tor ( H m +1 ( G )) then wehave k ⋆ ( x ) = 0 just combining Propositions 3.3 and 3.4. (cid:3) Proof of Theorem 1.2
Notations . Applying Proposition 2.12, and Notations 2.11, to the completeintersection W = X ∩ G of Theorem 3.2, we get a morphism e Y := Bl W ( Y ) −→ Q ⊂ P ( V ∗ W,d ) V W,d := V d ∩ H ( Y, I W,Y ( d )) . This map contracts G ∼ = e G := Bl W ( G ) ⊂ e Y to a point p ∈ Q , and sends e Y − e G isomorphically to Q − { p } . By ([1], Remark 3.1), both e Y and Q have at worstisolated singularities. Corollary 4.2.
The push-forward map: H m +2 ( e Y ) −→ H m +2 ( Q ) is surjective, thus the cokernel of the map H m +2 ( e Y ) −→ H m ( X ) is torsion free, for a general X ∈ |V W,d | . Proof.
From the commutative diagram H k ( e G ) → H k ( e Y ) → H k ( e Y , e G ) → H k − ( e G ) ↓ ↓ k ↓ H k ( { p } ) → H k ( Q ) → H k ( Q , { p } ) → H k − ( { p } )we see that H m +2 ( e Y ) → H m +2 ( Q ) is surjective if the push-forward H m +1 ( e G ) → H m +1 ( e Y ) is injective, and this follows simply combining Theorem 3.2 with Corollary2.6 of [4]. The last statement is direct consequence of the first. In factcoker( H m +2 ( e Y ) −→ H m ( X )) ∼ = coker( H m +2 ( Q ) −→ H m ( X )) , and the last group is torsion free by Lefschetz Theorem with Singularities ([10],p.199), because H m +2 ( Q ) ∼ = H BMm +2 ( Q −
Sing Q ) ∼ = H m ( Q −
Sing Q ) . (cid:3) Remark . By Corollary 2.6 of [4], H m +2 ( e Y ) ∼ = H m +2 ( Y ) ⊕ H m ( W ), hence Corol-lary 4.2 implies that the groupcoker( H m +2 ( Y ) ⊕ H m ( W ) → H m ( X ))has no torsion. In the morphism above the first component is intended to be theGysin map followed by Poincar´e duality, and the second one is intended to be thepush forward followed by Poincar´e duality. Notations . Let Y ⊂ P be a smooth projective variety of dimension m + 1 =2 r + 1 ≥
3. Let
X, G , . . . , G r be a regular sequence of smooth divisors in Y , with X ∈ |V d | , each G i ∈ |V d i | , and such that d > d > · · · > d r . Set ∆ := X ∩ G ∩· · · ∩ G r , and W := X ∩ G . For any 1 ≤ l ≤ r − H l ∈ |V µ l | ,with 0 ≪ µ ≪ · · · ≪ µ r − , and for any 0 ≤ l ≤ r − Y l , X l , W l , ∆ l )as follows. For l = 0 define ( Y , X , W , ∆ ) := ( Y, X, W, ∆), X ∈ |V W,d | general.For 1 ≤ l ≤ r − Y l := G ∩ · · · ∩ G l ∩ H ∩ · · · ∩ H l , X l := X ∩ Y l , W l := X ∩ Y l ∩ G l +1 , and ∆ l := ∆ ∩ Y l ( m l := dim X l = m − l ). Notice thatdim Y r − = 3 and that ∆ r − = W r − . Remark . (1) As in Theorem 1.2, define: V l,d := Im( H ( P , O P ( d )) → H ( Y l , O Y l ( d ))) . (2) As in Notations 4.1, define: e Y l := Bl W l ( Y l ) −→ Q l ⊂ P ( V ∗ W l ,d ) , V W l ,d := V l,d ∩ H ( Y l , I W l ,Y l ( d )) . By Corollary 4.2 and Remark 4.3 the group V l := coker( H m l +2 ( Q l ) → H m l ( X l )) == coker( H m l +2 ( Y l ) ⊕ H m l ( W l ) → H m l ( X l ))is torsion free. ´ERON-SEVERI GROUP OF A GENERAL HYPERSURFACE 13 (3) By ([2], Theorem 1.1), V l ⊗ Q supports an irreducible action of the mon-odromy group of the linear system |V W l ,d | | Yl . Moreover, by previous remark,we have V l ⊂ V l ⊗ Q .Theorem 1.2 will follow from a slightly stronger result: Theorem 4.6.
Let Y ⊂ P be a smooth projective variety of dimension m + 1 =2 r + 1 ≥ . Let X, G , . . . , G r be a regular sequence of smooth divisors in Y ,with X ∈ |V d | , each G i ∈ |V d i | , and such that d > d > · · · > d r . Set ∆ := X ∩ G ∩ · · · ∩ G r and let X ∈ |V d ∩ H ( Y, I ∆ ,Y ( d )) | be a very general hypersurfacecontaining ∆ . Assume that the vanishing cohomology of X is not of pure Hodgetype ( m , m ) , denote by H m ( X ; Z ) ∆ the subgroup of H m ( X ; Z ) generated by thecomponents of ∆ and by H m ( X ; Z ) ∆ − the subgroup of H m ( X ; Z ) generated by thecomponents of ∆ except one. Then we have: (1) H m ( X ; Z ) ∆ is freely generated by the components of ∆ ; (2) N S r ( X ; Z ) = [ N S r ( X ; Z ) ∩ H m ( Y ; Z )] ⊕ H m ( X ; Z ) ∆ − ; (3) N S r ( X ; Q ) = N S r +1 ( Y ; Q ) ⊕ H m ( X ; Q ) ∆ − .Proof of Theorem 4.6. The argument is very similar to that already used in theproof of Theorem 3.3 of [4], so we are going to be rather sketchy. Thanks toTheorem 1.2 of [4], it suffices to show that the cokernel of the map H m +2 ( Y ) ⊕ H m (∆) −→ H m ( X )is free. In order to prove this, we argue by decreasing induction on l and prove that W l := coker( H m l +2 ( Y l ) ⊕ H m l (∆ l ) −→ H m l ( X l ))coincides with the group V l defined in Remark 4.5, (2). For l = r − r − = W r − , compare with Notations 4.4. Observe that we only need toprove the following inclusion:Im( H m l +2 ( Y l ) ⊕ H m l (∆ l ) −→ H m l ( X l )) ⊇ Im( H m l +2 ( Y l ) ⊕ H m l ( W l ) −→ H m l ( X l ))for the reverse inclusion is obvious. Notice that the composite: H m l ( W l ) ֒ → H m l +1 ( X l +1 ) −→ W l +1 vanishes because W l +1 = V l +1 by induction, the monodromy acts irreducibly on V l +1 and we can assume that rk H m l ( W l ) ≪ rk V l +1 . Again by induction we findthat Im( H m l ( W l ) ֒ → H m l +1 ( X l +1 ))is contained in Im( H m l +1 +2 ( Y l +1 ) ⊕ H m l +1 (∆ l +1 ) → H m l +1 ( X l +1 )) , and we are done by a simple arrow chasing showing that, if α ∈ H m l ( W l ) goes toIm( H m l +1 +2 ( Y l +1 ) → H m l +1 ( X l +1 )), then its push forward in H m l ( X l ) belongs toIm( H m l +2 ( Y l ) → H m l ( X l )). (cid:3) References [1] Di Gennaro, V. - Franco, D.:
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Universit`a di Roma “Tor Vergata”, Dipartimento di Matematica, Via della RicercaScientifica, 00133 Roma, Italy.
E-mail address : [email protected] Universit`a di Napoli “Federico II”, Dipartimento di Matematica e Applicazioni “R.Caccioppoli”, P.le Tecchio 80, 80125 Napoli, Italy.
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