Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 41 (2000), No. 1, 279-290.

Koszul Cohomology and k-Normality of a Projective Variety

A. Alzati G. M. Besana

Dipartimento di Matematica, Università di Milano via C. Saldini 50, 20133-Milano, Italia e-mail: alzati@mat.unimi.it

Department of Mathematics, Eastern Michigan University Ypsilanti, MI-48197, U.S.A. e-mail: gbesana@emunix.emich.edu

Abstract. Let X be a smooth projective variety and let L be a very ample divisor of X embedding it in ℙ^{N}. In this paper we use the Koszul groups of X to get information about the k-normality of X (i.e. the surjectivity of the map H^{0}(ℙ^{N}, 𝒪_{ℙ^{N}}(k)) → H^{0}(X, kL) via an upper bound for the degree of the generators of ⊕_{t≥0}H^{0}(X, tL). The above idea is applied to some scrolls over curves and surfaces and to some other varieties, by using also results due to Green and Butler. MSC 1991: 14J40 Keywords: Koszul cohomology, k-normality, scrolls

1. Introduction

It is well-known that “there are fewer ways to compute Koszul cohomology groups than reasons to compute them” (see [4]). In this paper we want to give another reason to compute them: almost every work on Koszul cohomology of a smooth projective subvariety X of ℙ^{N} considers only the case in which X is projectively normal, (p.n.) (see [6], [7], [8], [13]) in which case Koszul groups give immediately a free resolution of the ideal sheaf ℐ_{X} of X in ℙ^{N} which is the main interest of the above papers. We know only Birkenhake’s works [10] and [11] treating the case in which X is not linearly normal. When X is not p.n. Koszul groups give only an upper bound for the degree of the generators of the ring R(X) = ⊕_{t≥0}H^{0}(X, tL) where L is the very ample line bundle of X giving the embedding of X in ℙ^{N}. Note that R(X)

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 2000 Heldermann Verlag

280 A. Alzati, G. M. Besana: Koszul Cohomology and k-Normality of a Projective Variety

is the coordinate ring of X if X is projectively normal. In some cases, e.g. for some scrolls, the information given by Koszul groups on R(X) is sufficient to establish the k-normality of X.

More precisely in this paper we prove that for a scroll X over a smooth curve, whose dimension is at least three, the ring R(X) is generated in degree 2 if a condition weaker than Butler’s one (see [9]) is satisfied. The same fact is true for varieties, 4-dimensional at least, which are fibered in hypersurfaces of degree 2 and 3 over a smooth curve. Hence these varieties are projectively normal if and only if they are 2-normal, moreover, this fact is true for scrolls over a genus 2 curve without any other assumptions.

As a consequence of a suitable use of corollary 1.d.4 of [4], we get that for a regular surface (X, L) such that there exists a smooth curve in |L − K_{X}|, R(X) is generated in degree 2 and 3. We also obtain some conditions assuring the projective normality of scrolls on surfaces.

The paper is organized as follows: In Section 2 we fix notation and recall some facts about Koszul cohomology; in Section 3 we use Butler’s work to compute some Koszul vanishings for scrolls and varieties which are fibered in hypersurfaces; in Section 4 we show some other vanishings for Koszul cohomology of scrolls; in Section 5 we consider another method to compute vanishings and we apply it to regular surfaces and to scrolls over surfaces.

2. Notation and background material

ℙ^{N}  N-dimensional projective space over ℂ
S  ℂ[x_{0}, x_{1}, …, x_{N}] the coordinate ring of ℙ^{N}
S(a)  the graded ring S twisted by the integer a
X  smooth n-dimensional projective subvariety of ℙ^{N}
K_{X}  canonical divisor of X
L  very ample line bundle embedding X in ℙ^{N} via H^{0}(X, L)
I_{X}  the homogeneous ideal of X in the ring S
ℐ_{X}  the ideal sheaf of X in ℙ^{N}
𝒪_{X}  the structural sheaf of X
Ω_{X}  cotangent bundle of X
R(X)  the graded ring ⊕_{t≥0}H^{0}(X, tL) which is an S-module
C  smooth algebraic curve of genus g
E  rank r vector bundle over a smooth variety X
E^{∗}  its dual
μ(E)  slope of E
μ^{−}(E) minimal slope of a quotient vector bundle of E over C
μ^{+}(E) maximal slope of a subbundle of E over C
ℙ(E)  projectivized of E
p  natural projection from ℙ(E) to X
T  tautological line bundle of ℙ(E)
F  numerical class of a fibre in ℙ(E) or generic fibre of p
∼  linear equivalence among divisors
≡  numerical equivalence among divisors

Let (X, L) be as above, i.e. a smooth, linearly normal subvariety of ℙ^{N}, embedded by H^{0}(X, L), where N =h^{0}(X, L) − 1. R_{t} =H^{0}(X, tL), R_{0} =H^{0}(X, 𝒪_{X}), then R(X)=⊕_{t≥0}R_{t} is a graded S-module having a minimal free resolution ⋯→E_{p+1} →E_{p} →⋯→E_{1} →E_{0} →R→0 in which E_{0} = ⊕_{q≥0}(B_{0,q} ⊗ S(−q)), E_{1} = ⊕_{q≥0}(B_{1,q} ⊗ S(−q)) and so on, where B_{p,q} are ℂ vector spaces whose dimensions b_{p,q} keep track of how many S(−q) appear in E_{p}; the b_{p,q} do not depend on the choice of the minimal free resolution, (see [6]). We will write b_{p,q} instead of b_{p,q}(X), R instead of R(X), when any confusion is impossible.

Note that S = ⊕_{t≥0}H^{0}(X, 𝒪_{ℙ^{N}}(t)), so we have a natural graded map ρ : S → R and the S-module structure on R is given by sr = ρ(s)r. X is p.n. if every graded piece of ρ is surjective.

E_{0} is the free S-module corresponding to the generators of R, b_{0,q} is the number of generators of R whose degree is q. Let us be careful: every R_{t} is also a ℂ vector space of finite dimension, but we are considering R as an S-module: there is only one generator of degree 0, the multiplicative identity 1 of the ring R, which is also the generator of the ℂ vector space R_{0}. There are no generators in degree 1 because, as X is linearly normal, every element of R_{1} comes from S by ρ, so it is the product of an element of S and the generator 1, hence b_{0,1} = 0.

If X is p.n., for the same reason we have no other generators for R as an S-module, so that E_{0} is isomorphic to S and the kernel of the map E_{0} → R is precisely I_{X}, in this case ⋯ → E_{p} → ⋯ → E_{1} → I_{X} → 0 is a free resolution for I_{X}; this is the point of view of [6], [7], [8], [13], but what can we say when X is not p.n.? Let us examine E_{0} firstly. We have the following

Proposition 2.1. Let X be as above, then:

− b_{0,0} = 1,

− b_{0,1} = 0,

− X is 2-normal if and only if b_{0,2} = 0,

− for q ≥ 3, if X is q-normal then b_{0,q} = 0 (but not vice versa), − X is p.n. if and only if b_{0,q} = 0 for any q ≥ 2.

Proof. The values of b_{0,0} and b_{0,1} were discussed above. As X is linearly normal, the 2normality of X is equivalent to the vanishing of b_{0,2}: in fact if X is 2-normal then b_{0,2}=0 because any element of R_{2} is a multiple of 1 by an element of S. If there are no degree 2 generators in R, as S-module, every element of R_{2} must be an S-linear combination of the generators of R of degree 0 or 1, i.e. it must be a multiple of 1 and then it comes from S by ρ. In any case if we consider the ℂ-linear map between H^{0}(ℙ^{N}, 𝒪_{ℙ^{N}}(2)) and R_{2}, b_{0,2} is the ℂ-dimension of the cokernel of this map.

For q ≥ 3 we have that if X is q-normal then b_{0,q} = 0 for the above reason, but not vice versa because b_{0,q} is always the number of the degree q generators of R, but when we consider the ℂ-linear map between H^{0}(ℙ^{N}, 𝒪_{ℙ^{N}}(q)) and R_{q}, b_{0,q} is only less than or equal the ℂ-dimension of the cokernel of this map because in the cokernel there can be also elements which are S-linear combinations of generators of R whose degree is less than q.

If X is p.n., R is generated over S by 1, hence b_{0,q} = 0 for all q ≥ 2, the vice versa is obvious. □

Let us consider E_{1}, i.e. the free S-module of the primitive syzygies among the generators of R, where primitive means, according to Green [4], that every considered degree q syzygy is not an S-linear combination of syzygies whose degree is less than q. Then b_{1,q} is the number of the degree q generators of this S-module. We have the following

Proposition 2.2. Let X be as above, then:

− b_{1,0} = b_{1,1} = 0,

− b_{1,2} = h^{0}(ℙ^{N}, ℐ_{X}(2)),

− if b_{1,q} = 0 for q ≥ k + 1 then b_{0,q} = 0 for q ≥ k.

Proof. The first vanishings are obvious. b_{1,2} is the number of the generators of the S-module of degree 2 syzygies (in this degree every syzygy is primitive). As a degree q syzygy can involve only generators of R whose degree is less than or equal to q − 1, a degree 2 syzygy is always of the following type: s1 = 0 with s ∈ S, because there are no degree 1 generators in R. Moreover, the number of the generators of the S-module of the degree 2 syzygies coincides with the dimension of this S-module viewed as a ℂ vector space. Note that if q ≥ 3 this is not longer true: the submodule of the primitive degree q syzygies of type s1 = 0 always corresponds to degree q ℂ-independent hypersurfaces of ℙ^{N} containing X, but we have only that b_{1,q} is greater than or equal to the ℂ-dimension of the ℂ-vector space of irreducible hypersurfaces of degree q containing X (which is less than or equal to h^{0}(ℙ^{N}, ℐ_{X}(q)): there can be primitive degree q syzygies which have no links with the hypersurfaces containing X.

Now let us assume that k is the maximal degree for the primitive syzygies among the generators of R: 1, x_{1}, x_{2}, …, x_{h}, and, by contradiction, let us assume that one of these generators, say x_{h}, belongs to R_{q} with q ≥ k, and thus there are no primitive syzygies involving x_{h}. However, it is easy to see that any generator is involved by a syzygy, so we have α_{0}1 + α_{1}x_{1} + ⋯ + α_{h}x_{h} = 0 where α_{i} ∈ S, deg(α_{i}) ≥ 1, and this is not a primitive syzygy. Hence it must be an S-linear combination of primitive syzygies, but this is not possible because no primitive syzygy involves x_{h}. □

Let us consider the exact sequence 0 → M → V ⊗ 𝒪_{X} → L → 0 of vector bundles over X, where M is the kernel of the evaluation map V ⊗ 𝒪_{X} → L and V = H^{0}(X, L). Let q ≥ 2; to estimate b_{1,q} we have

Proposition 2.3. Let X, L, M be as above, then b_{1,q} = 0 if H^{1}(X, Λ^{2}M ⊗ L^{q−2}) = 0.

Proof. By [7], we have that b_{1,q} is the dimension of the ℂ-vector space which is the homology at the middle level in the following piece of the Koszul complex:

⋯ → Λ^{2}(V) ⊗ R_{q−2} → V ⊗ R_{q−1} → R_{q} · · · .

Let us call α_{q} : Λ^{2}(V) ⊗ R_{q−2} → V ⊗ R_{q−1} and β_{q} : V ⊗ R_{q−1} → R_{q}. From it we get: 0 → Λ^{2}M → Λ^{2}(V) ⊗ 𝒪_{X} → V ⊗ L → S^{2}(L) = L ⊗ L → 0 which splits as

0 → M ⊗ L → V ⊗ L → L ⊗ L → 0 and 0 → Λ^{2}M → Λ^{2}(V) ⊗ 𝒪_{X} → M ⊗ L → 0.

Hence we have, for any m ∈ ℤ: 0 → Λ^{2}M ⊗ L^{m} → Λ^{2}(V) ⊗ L^{m} → M ⊗ L^{m+1} → 0 and 0 → M ⊗ L^{m} → V ⊗ L^{m} → L^{m+1} → 0. By choosing m = q − 2 in the first case we get the following exact sequence:

0→H^{0}(X, Λ^{2}M ⊗ L^{q−2})→Λ^{2}(V) ⊗ R_{q−2} →H^{0}(X, M ⊗ L^{q−1})→H^{1}(X, Λ^{2}M ⊗ L^{q−2})→ · · · .

By choosing m = q − 1 in the second case we get this exact sequence:

0 → H^{0}(X, M ⊗ L^{q−1}) → V ⊗ R_{q−1} → R_{q} → · · · .

If we call γ_{q} : Λ^{2}(V) ⊗ R_{q−2} → H^{0}(X, M ⊗ L^{q−1}), δ_{q} : H^{0}(X, M ⊗ L^{q−1}) → V ⊗ R_{q−1} and ε_{q} : V ⊗ R_{q−1} → R_{q}, we have that β_{q} = ε_{q} and α_{q} = δ_{q} ∘ γ_{q}.

Hence if H^{1}(X, Λ^{2}M ⊗ L^{q−2}) = 0 we get that γ_{q} is surjective, kern(β_{q}) = Im(α_{q}) and therefore b_{1,q} = 0. □ Remark 2.4. Note that, in the same way, it is possible to get that b_{0,q} = 0 if H^{1}(X, M ⊗ L^{q−1}) = 0, i.e. the condition H^{1}(X, M ⊗ L^{q−1}) = 0 for q ≥ 2 implies that X is p.n.; in this form this condition is used by many authors (see [9], [14], [15] for instance). When X is p.n. and H^{1}(X, Λ^{2}M ⊗ L^{q−2}) = 0 for q ≥ 3 then I_{X} is generated in degree 2, when X is not p.n. the condition yields only some information about the generators of R.

3. Syzygies of scrolls

In this section we consider the vanishing of H^{1}(X, Λ^{2}M ⊗ L^{q−2}), q ≥ 2, for r-dimensional scrolls X = ℙ(E) over smooth curves C, r ≥ 2, where E is a very ample rank r vector bundle over C. In this case L is the tautological bundle T, p_{∗}T = E and we have the exact sequence 0 → M_{E} → H^{0}(C, E) ⊗ 𝒪_{C} → E → 0, where H^{0}(C, E) ⊗ 𝒪_{C} → E is the natural evaluation map. Our strategy will be to calculate h^{1}(X, Λ^{2}M ⊗ T^{q−2}) by using h^{1}(C, p_{∗}(Λ^{2}M ⊗ T^{q−2})). It is well-known that the two numbers are equal if R^{i}p_{∗}(Λ^{2}M ⊗ T^{q−2}) = 0, ∀i ≥ 1 (see [12], p. 253) and this is true if h^{j}(F, (Λ^{2}M ⊗ T^{q−2})_{|F}) = 0, ∀j ≥ 1. We have the following Lemma 3.1. With the above notations h^{j}(F, (Λ^{2}M ⊗ T^{q−2})_{|F}) = 0, ∀j ≥ 1. Proof. Recall that F ≅ ℙ^{r−1}, (T^{q−2})_{|F} = 𝒪_{F} (q − 2), so that h^{j}(F, (Λ^{2}M ⊗ T^{q−2})_{|F}) = h^{j}(F, (Λ^{2}M_{|F} ⊗ 𝒪_{F} (q − 2)).

Now let us consider 0 → 𝒪_{X}(T − F) → 𝒪_{X}(T) → 𝒪_{F} (T_{|F}) → 0 and the long exact sequence 0 → H^{0}(X, T −F) → H^{0}(X, T) → H^{0}(F, 𝒪_{F} (1)) → H^{1}(X, T −F) → · · ·. As p_{∗}T = E is generated by global sections we have that H^{0}(X, T) = H^{0}(X, T − F) ⊕ H^{0}(X, 𝒪_{F} (1)).

By considering the restriction to F of 0 → M → H^{0}(X, T) ⊗ 𝒪_{X} → T → 0 we get 0 → M_{|F} → H^{0}(X, T) ⊗ 𝒪_{F} → 𝒪_{F} (1) → 0.

By using the Euler sequence for F we get the following commutative diagram:

0  0
↓  ↓
H^{0}(X, T − F) ⊗ 𝒪_{F}  H^{0}(X, T − F) ⊗ 𝒪_{F}
↓  ↓
0 ⟶ M_{|F} ⟶ H^{0}(X, T − F) ⊕ H^{0}(X, 𝒪_{F} (1)) ⊗ 𝒪_{F} ⟶ 𝒪_{F} (1) ⟶ 0
↓  ↓
0 ⟶ Ω_{F} (1) ⟶  H^{0}(X, 𝒪_{F} (1)) ⊗ 𝒪_{F}  ⟶ 𝒪_{F} (1) ⟶ 0
↓  ↓
0  0

The left column splits so that M_{|F} ≅ H^{0}(X, T − F) ⊗ 𝒪_{F} ⊕ Ω_{F} (1), Λ^{2}(M_{|F}) = Ω_{F}^{2} (2) ⊕ Ω_{F} (1) ⊗ H^{0}(X, T − F) ⊗ 𝒪_{F} ⊕ Λ^{2}(H^{0}(X, T − F)) ⊗ 𝒪_{F}, and Λ^{2}(M_{|F})⊗𝒪_{F} (q−2) = Ω_{F}^{2} (q)⊕Ω_{F} (q−1)⊗H^{0}(X, T −F)⊗𝒪_{F} ⊕Λ^{2}(H^{0}(X, T −F))⊗𝒪_{F} (q−2), now it is very easy to see that h^{j}(F, (Λ^{2}M ⊗ T^{q−2})_{|F}) = 0, ∀j ≥ 1. □

Now we can prove

Theorem 3.2. Let (X, T) be a scroll as above over a genus g curve, r ≥ 3, q ≥ 4, then H^{1}(X, Λ^{2}M ⊗ L^{q−2}) = 0 if μ^{−}(M_{E}) + μ^{−}(E) > g − 1.

Proof. By Lemma 3.1 we have to show that h^{1}(C, p_{∗}(Λ^{2}M ⊗ T^{q−2})) = 0. As p_{∗}(M ⊗ M ⊗ T^{q−2}) = p_{∗}[Λ^{2}M⊗T^{q−2}]⊕p_{∗}[S^{2}M⊗T^{q−2}] it suffices to show that h^{1}(C, p_{∗}(M⊗M⊗T^{q−2})) = 0. This is true if μ^{−}[p_{∗}(M ⊗ M ⊗ T^{q−2})] > 2g − 2, (see [9]).

We can use Prop. 4.2 of [9], in fact p_{∗}T = E is generated by global sections, T is 0 p-regular and M⊗T^{q−2} is -1 p-regular, hence we have that μ^{−}[p_{∗}(M⊗M⊗T^{q−2})] ≥ μ^{−}(M_{E})+μ^{−}[p_{∗}(M⊗ T^{q−2})]; (note that, in this case, the inequality given by Prop. 4.2 of [9] is very simple because for i = r − 2 we have that R^{i}p_{∗}(T^{−i}) = 0, so that min{μ^{−}(M_{E}) + μ^{−}[p_{∗}(M ⊗ T^{q−2})], +∞} = μ^{−}(M_{E}) + μ^{−}[p_{∗}(M ⊗ T^{q−2})]; see [9] for the notion of k p-regular vector bundles).

Now we can use the same proposition for μ^{−}[p_{∗}(M ⊗ T^{q−2})], in fact T is 0 p-regular and T^{q−2} is -1 p-regular. By the same previous reason we get: μ^{−}[p_{∗}(M ⊗ T^{q−2})] ≥ μ^{−}(M_{E}) + μ^{−}[S^{q−2}(E)] and μ^{−}[S^{q−2}(E)] = (q − 2)μ^{−}(E).

Hence we have μ^{−}[p_{∗}(M ⊗ M ⊗ T^{q−2})] ≥ 2μ^{−}(M_{E}) + (q − 2)μ^{−}(E), but as T is very ample, this inequality is satisfied for q ≥ 4 if it is true for q = 4, so that the condition is simply μ^{−}(M_{E}) + μ^{−}(E) > g − 1. □

Remark 3.3. Assume that E is semistable, g ≥ 2 and μ(E) < 2g, then we have μ^{−}(M_{E}) > r(μ^{−}(E) − 2g) − 2 + 2h^{1}(E) (see [9], Prop. 1.5), so the condition in Theorem 3.2 becomes: (2 + r)μ(E) + 2h^{1}(E) > (2r + 1)g + 1.

Remark 3.4. Although Theorem 3.2 is stated for E very ample, exactly the same proof works also when E is ample and generated by global sections.

For the rest of this section we are concerned with the vanishing of H^{1}(X, Λ^{2}M ⊗L^{q−2}), q ≥ 2, when X is a divisor of W = ℙ(E) where E is an ample, globally generated vector bundle over a smooth, genus g, curve C. We assume that L, the restriction to X of the tautological divisor T of W, is very ample. X is fibered over C and the generic fibre is a smooth hypersurface of ℙ^{r−1} whose degree is fixed. We can prove the following

Proposition 3.5. Let (X, L) be as above, assume that X ≡ aT + bF with a ≥ 2, then H^{1}(X, Λ^{2}M ⊗ L^{q−2}) = 0 if r ≥ 4, q ≥ 4, μ^{−}(M_{E}) + μ^{−}(E) > g − 1 except, possibly, for q = a and q = a + 1.

Proof. As E is generated by global sections we can consider the usual exact sequence

0 → M_{T} → H^{0}(W, T) ⊗ 𝒪_{W} → T → 0.

By restricting it to X we get: 0 → (M_{T})_{|X} → H^{0}(W, T) ⊗ 𝒪_{X} → L → 0 as L = T_{|X}. On the other hand we have 0 → M → H^{0}(X, L) ⊗ 𝒪_{X} → L → 0, but it is easy to see that H^{0}(W, T) ⊗ 𝒪_{X} = H^{0}(X, L) ⊗ 𝒪_{X} so that (M_{T})_{|X} = M. Hence we can tensorize the exact sequence 0 → 𝒪_{W} (−X) → 𝒪_{W} → 𝒪_{X} → 0 with Λ^{2}M_{T} ⊗ T^{q−2} and in cohomology we have

⋯ →H^{1}(W, Λ^{2}M_{T} ⊗T^{q−2})→H^{1}(X, Λ^{2}M ⊗L^{q−2})→H^{2}(W, Λ^{2}M_{T} ⊗𝒪_{W} (−X+(q−2)T)→ ⋯

as H^{1}(X, Λ^{2}M ⊗ L^{q−2}) = H^{1}(X, Λ^{2}(M_{T})_{|X} ⊗ (T_{|X})^{q−2}). By arguing as in 3.2 and by recalling that E ample implies μ^{−}(E) > 0, we have H^{1}(W, Λ^{2}M_{T} ⊗ T^{q−2}) = 0. To deal with H^{2}(W, Λ^{2}M_{T} ⊗ 𝒪_{W} (−X + (q − 2)T)) recalling Remark 3.4, we proceed as in the proof of 3.2. Thus that group vanishes if h^{j}(F, (Λ^{2}M_{T} ⊗ 𝒪_{W} (−X + (q − 2)T))_{|F}) = 0, ∀j ≥ 1.

As in the proof of 3.1 we have that (Λ^{2}M_{T} ⊗ 𝒪_{W} (−X + (q − 2)T))_{|F} is the direct sum of some copies of Ω_{F}^{2} (q − a), Ω_{F}^{1} (q − a − 1) and 𝒪_{F} (q − a − 2) so that we get the vanishing for q ≥ a + 2. If q ≤ a − 1 (if necessary, recall that a ≥ 1 in any case) we can consider H^{r−2}(W, [Λ^{2}M_{T} ⊗ 𝒪_{W} (−X + (q − 2)T)]^{∗} ⊗ K_{W}) and we can proceed analogously as r − 2 ≥ 2.

□

Remark 3.6. If a = 2 or a = 3 (i.e. the fibres are hypersurfaces of degree 2 or 3) Proposition 3.5 shows that under the same assumptions of 3.2 for ℙ(E), with r ≥ 4, b_{0,q}(X) = 0 if q ≥ 3 and therefore X is p.n. if and only if it is 2-normal. If r = 3 the previous proof works only for q ≥ a + 2.

Proposition 3.7. Let (X, L) as above, with the same assumptions of 3.5, then b_{1,a}(X) = 0 if b ≥ 1.

Proof. We have only to show that H^{2}(W, Λ^{2}M_{T} ⊗ 𝒪_{W} (−X + (a − 2)T)) = 0. By using 0 → Λ^{2}M_{T} → Λ^{2}(H^{0}(W, T)) ⊗ 𝒪_{W} → H^{0}(W, T) ⊗ T → S^{2}(T) = T ⊗ T → 0 tensorized by 𝒪_{W} (−X + (a − 2)T) we see that

H^{2}(W, Λ^{2}M_{T} ⊗ 𝒪_{W} (−X + (a − 2)T)) = H^{1}(W, M_{T} ⊗ 𝒪_{W} (−X + (a − 1)T))

as M_{T} ⊗ T is the kernel of H^{0}(W, T) ⊗ T → T ⊗ T.

Let B be a degree b divisor on C such that X = aT + p^{∗}B, then −X + (a − 1)T = −T + p^{∗}(−B) and H^{1}(W, M_{T} ⊗ 𝒪_{W} (−T + p^{∗}(−B)) = H^{0}(C, −B) by using Leray’s spectral sequence as usual. If b ≥ 1 we have the required vanishing. □

Remark 3.8. By arguing as in the previous proof we can show that

H^{2}(W, Λ^{2}M_{T} ⊗ 𝒪_{W} (−X + (a − 1)T)) = H^{1}(C, M_{E} ⊗ B)

which does not vanish for b ≥ 1, so that it is not possible to get conditions under which b_{1,a+1}(X) = b_{1,a}(X) = 0 by using this method.

4. Koszul groups for X = ℙ ℙ(E)

For any smooth n-dimensional X, embedded in ℙ^{N} by a very ample line bundle L and for any vector bundle ℰ over X we can consider the Koszul ℂ-vector spaces K_{p,q}(X, ℰ, L) (see [4]). Let k_{p,q}(X, ℰ, L) be the dimension of K_{p,q}(X, ℰ, L). It is k_{p,q}(X, 𝒪_{X}, L) = b_{p,p+q} for any p, q, so that the computation of these Koszul groups is related to the minimal resolutions of R.

For the convenience of the reader we recall the following basic results, due to M. Green, which will be used in the sequel:

Theorem 4.1. ([4], Th. 3.a.1) K_{p,q}(X, ℰ, L) = 0 if h^{0}(X, ℰ ⊗ L^{q}) ≤ p.

Theorem 4.2. ([4], Th. 2.c.6) K_{p,q}(X, ℰ, L)^{∗} ≅ K_{N−n−p,n+1−q}(X, ℰ^{∗}⊗K_{X}, L) if h^{i}(X, ℰ⊗L^{q−i})=0 and h^{i}(X, ℰ⊗L^{q−i−1})=0 for i = 1, 2, …, n − 1.

In this section E is a rank r vector bundle over a smooth genus g curve C, r ≥ 2, X = ℙ(E) is embedded in ℙ^{N} by a very ample line bundle L ∼ aT + p^{∗}B, where B is a divisor of C, deg(B) = b. X is linearly normal, N = h^{0}(L) − 1, n = r, L ≡ aT + bF, δ = c_{1}(E). We want to compute k_{p,q} by using 4.1 and 4.2 when ℰ = 𝒪_{X}.

First of all we consider h^{i}(X, L^{q−i}) for i = 1, 2, …, r − 1 and q ≥ 2. Recall that L^{q−i} ∼ (q − i)aT + p^{∗}[(q − i)B] and that a ≥ 1 and aμ^{−}(E) + b > 0 as L is very ample; moreover, by using Leray’s spectral sequence and Kodaira’s vanishing we have that all cohomology groups vanish but for i = 1, in this case we have h^{1}(X, L^{q−1}) = h^{1}(C, S^{(q−1)a}(E) ⊗ (q − 1)B) = 0 if (q − 1)(aμ^{−}(E) + b) > 2g − 2.

Now we consider h^{i}(X, L^{q−i−1}) for i = 1, 2, …, r − 1 and q ≥ 3. Reasoning as in the previous case we get that all groups vanish if (q − 2)(aμ^{−}(E) + b) > 2g − 2. Note that if q = 2, i = 1 the corresponding group does not vanish unless g = 0.

We have proved the following

Lemma 4.3. With the notation as in this section let q ≥ 3, g ≥ 1, then K_{p,q}(X, 𝒪_{X}, L)^{∗} ≅ K_{N−r−p,r+1−q}(X, K_{X}, L) if (q − 2)(aμ^{−}(E) + b) > 2g − 2.

Lemma 4.3 and Theorem 4.2 tell us that, under some conditions, for our varieties k_{p,q}(X, 𝒪_{X}, L) = 0 if N − r − p < 0 and k_{p,q}(X, 𝒪_{X}, L) = 0 if N − r − p ≥ 0 and h^{0}(X, K_{X} + (r + 1 − q)L) ≤ N − r − p. If g ≥ 1 it is well-known that N ≥ 2r, hence N − r − p ≥ r − p, so that for p = 0, 1, …, r to get k_{p,q}(X, 𝒪_{X}, L) = 0 it suffices that h^{0}(X, K_{X} + (r + 1 − q)L) = 0.

We have K_{X} + (r + 1 − q)L ≡ [(r + 1 − q)a − r]T + [d + 2g − 2 + (r + 1 − q)b]F, and such a line bundle has no sections if (r + 1 − q)a − r < 0 or (see [9], Lemma 1.12) if [(r + 1 − q)a − r]μ^{+}(E) + δ + 2g − 2 + (r + 1 − q)b < 0 and (r + 1 − q)a − r ≥ 0. Then we have proved the following:

Lemma 4.4. With the notation as in this section let q ≥ 3, r ≥ p ≥ 0, g ≥ 1, then K_{p,q}(X, 𝒪_{X}, L)=0 if (q−2)(aμ^{−}(E)+b)>2g−2 and (r+1−q)a−r <0 or if (r+1−q)a−r ≥0 and [(r + 1 − q)a − r]μ^{+}(E) + δ + 2g − 2 + (r + 1 − q)b < 0.

Corollary 4.5. Let X = ℙ(E) as above with r = 2, g ≥ 1, then b_{0,q}(X) = 0 for q ≥ 3 if aμ^{−}(E) + b > 2g − 2.

Corollary 4.6. Let X = ℙ(E) be a scroll over a curve of genus g ≥ 1 (hence a = 1, b = 0) then b_{0,q}(X) = 0 for q ≥ 3 if μ^{−}(E) > 2g −2, therefore X is p.n. if and only if it is 2-normal.

Corollary 4.7. Let X = ℙ(E) be a scroll over a curve of genus g = 2 then X is p.n. if and only if it is 2-normal; in fact in this case μ^{−}(E) > 3, (see [2]); moreover, b_{p,p+q}(X) = 0 for q ≥ 3, r ≥ p ≥ 0.

Corollary 4.8. Let X = ℙ(E) be a scroll over a curve of genus g ≥ 1 with μ^{−}(E) > 2g, then X is projectively normal (see [9]) and I_{X} is generated in degree two.

Remark 4.9. The previous results hold also if X is embedded by a linear subspace W of H^{0}(X, L), i.e. if X is not linearly normal. To see this it is enough to use Green’s Theorems 4.1 and 4.2, being careful to use N′ = dim W instead of N in the previous formulas.

5. Long exact sequences for Koszul groups and applications

Let X be a smooth variety in ℙ^{N} as usual and let Y be a smooth, one codimensional subvariety of X. Let L be a very ample divisor of X. Then from the natural exact sequence given by Y: 0 → 𝒪_{X}(−Y) → 𝒪_{X} → 𝒪_{Y} → 0 we get 0 → H^{0}(X, qL − Y) → H^{0}(X, qL) → H^{0}(Y, qL_{|Y}) ⋯ for any q ≥ 0.

Assume that the previous sequence is exact for any q ≥ 0, then if we put

𝒜 = ⊕_{q≥0}H^{0}(X, qL − Y), ℬ = ⊕_{q≥0}H^{0}(X, qL) and 𝒞 = ⊕_{q≥0}H^{0}(Y, qL_{|Y})

we get an exact sequence 0 → 𝒜 → ℬ → 𝒞 → 0 of S-modules from which we deduce some long exact sequences for Koszul groups (see [4], Cor. 1.d.4):

⋯ → K_{p,q}(ℬ) → K_{p,q}(𝒞) → K_{p−1,q+1}(𝒜) → K_{p−1,q+1}(ℬ) → ⋯,

where K_{p,q}(𝒜)=K_{p,q}(X, 𝒪_{X}(−Y), L), K_{p,q}(ℬ)=K_{p,q}(X, 𝒪_{X}, L), K_{p,q}(𝒞)=K_{p,q}(Y, 𝒪_{Y} , L_{|Y}).

In [3] the authors considered the projective normality of (X, L) in the case in which L is interesting from the point of view of adjunction theory, i.e. when L = aK_{X} + bA where A is a suitable divisor of X and a, b are integers. Notice that it is the same point of view of [8] and [14], [15], [16]. Here, by using the previous ideas we can prove the following proposition:

Proposition 5.1. Let X be a regular surface and let L be a very ample line bundle on X. Assume that there exists a smooth curve in |L − K_{X}|. Then b_{0,q}(X) = 0 for q ≥ 4.

Proof. Firstly notice that the proposition is true for (X, L) = (ℙ^{2}, 𝒪_{ℙ^{2}}(1)). Let now (X, L) ≠ (ℙ^{2}, 𝒪_{ℙ^{2}}(1)) and let Y be a smooth curve in |L − K_{X}|. From the exact sequence 0 → K_{X} → L → L_{|Y} → 0, as X is regular it is easy to see that h^{1}(X, L) = h^{2}(X, L) = 0 and that Y is linearly normal in the embedding given by L. Now let

𝒜 = ⊕_{q≥0}H^{0}(X, qL − Y), ℬ = ⊕_{q≥0}H^{0}(X, qL), 𝒞 = ⊕_{q≥0}H^{0}(Y, qL_{|Y}).

The regularity of X and Kodaira’s vanishing theorem give an exact sequence of S-modules 0 → 𝒜 → ℬ → 𝒞 → 0, which in turn gives the exact sequences

⋯ → K_{0,q}(𝒜) → K_{0,q}(ℬ) → K_{0,q}(𝒞) → 0

for any q ≥ 0.

K_{0,q}(𝒞) = 0 for q ≥ 2 because Y is p.n. in ℙ^{N−1} as it is canonically embedded by L_{|Y} = K_{Y}, hence it suffices to show that b_{0,q}(𝒜) = K_{0,q}(X, 𝒪_{X}(−Y), L) = 0 for q ≥ 4. Note that here 𝒞 is considered as an S-module, not a ℂ[x_{0}, x_{1}, …, x_{N−1}]-module, however, Y is p.n. in ℙ^{N} too, so that b_{0,q}(𝒞) = k_{0,q}(𝒞) = 0, ∀q ≥ 2.

We can use Theorem 4.2 as h^{1}(X, −Y + (q − 1)L) = h^{1}(X, −Y + (q − 2)L) = 0 for q ≥ 4, so we have to consider K_{N−2,3−q}(X, 𝒪_{X}(Y + K_{X}), L). Now we can use Theorem 4.1 because h^{0}(X, (4 − q)L) ≤ h^{0}(X, L) − 3 for q ≥ 4. □

The previous ideas can be applied in other cases, for instance when X = ℙ(E) is the projectivized of a rank r vector bundle E over a surface Σ. In this case let T be the tautological bundle and p : X → Σ the natural projection as usual. Let C be a smooth curve on Σ, C′ = π^{−1}(C) and let us consider, for any j ≥ 0, the exact sequences:

0 → 𝒪_{X}(jT − p^{∗}C) → 𝒪_{X}(jT) → 𝒪_{C′}(jT_{|C′}) → 0.

If we assume that H^{1}(X, jT − p^{∗}C) = 0, ∀j ≥ 0, K_{0,q}(X, 𝒪_{X}(−p^{∗}C), T) = 0, ∀q ≥ 2, and T_{|C′} embeds C′ p.n. in ℙ^{N−1}, then we have that X is p.n. In fact by these assumptions there is an exact sequence 0 → 𝒜 → ℬ → 𝒞 → 0 of S-modules, and from the sequences ⋯ → K_{0,q}(𝒜) → K_{0,q}(ℬ) → K_{0,q}(𝒞) → … we have that b_{0,q}(ℬ) = 0, ∀q ≥ 2, (i.e. X is p.n.) as b_{0,q}(𝒜) = b_{0,q}(𝒞) = 0, ∀q ≥ 2, by assumptions. Note that here 𝒞 is considered as an S-module, not a ℂ[x_{0}, x_{1}, …, x_{N−1}]-module, however, C′ is p.n. in ℙ^{N} too, so that b_{0,q}(𝒞) = 0, ∀q ≥ 2.

Now we translate our assumptions into conditions on Σ. The first one is simply H^{1}(Σ, S^{j}(E) ⊗ 𝒪_{S}(−C)) = 0, ∀j ≥ 1, the third one is satisfied if we assume that μ^{−}(E_{|C}) > 2g(C) by Butler’s results [9]; for the second one we use Green’s Theorems 4.1 and 4.2. Let us consider h^{i}(X, (q − i)T + p^{∗}(−C)) for i = 1, …, n − 1 = r, by standard calculations they vanish if h^{i}(Σ, S^{q−i}(E) ⊗ 𝒪_{Σ}(−C)) = 0 for i = 1, …, r and q ≥ i, so we have only to assume further that h^{2}(Σ, S^{q−2}(E) ⊗ 𝒪_{Σ}(−C)) = 0 for q ≥ 2. In order to have h^{i}(X, (q − i − 1)T + p^{∗}(−C)) = 0 for i = 1, …, r, it suffices to ask that h^{1}(Σ, 𝒪_{Σ}(−C)) = 0 by similar arguments. Hence we can apply Theorem 4.2 and we consider, for q ≥ 2, K_{N−n,n+1−q}(X, 𝒪_{X}(p^{∗}C + K_{X}), T). This group vanishes if h^{0}(X, p^{∗}C + K_{X} + (n + 1 − q)T) ≤ N − n, i.e. h^{0}(Σ, C + det(E) + K_{Σ}) ≤ h^{0}(E) − r − 2, by Theorem 4.1.

Thus we have proved the following

Theorem 5.2. Let E be a very ample, rank r, vector bundle over a smooth surface Σ, let X = ℙ(E) and let T be the tautological bundle. Moreover, let C be a smooth genus g curve on S. Then (X, T) is p.n. if

1) h^{1}(Σ, S^{j}(E) ⊗ 𝒪_{Σ}(−C)) = 0 for j ≥ 0,

2) h^{2}(Σ, S^{j}(E) ⊗ 𝒪_{Σ}(−C)) = 0 for j ≥ 0,

3) h^{0}(Σ, C + det(E) + K_{Σ}) ≤ h^{0}(E) − r − 2,

4) μ^{−}(E_{|C}) > 2g.

Remark 5.3. If C is a rational curve 4) is satisfied; if C is an ample divisor 1) and 2) are satisfied for j = 0; if −K_{Σ} is effective 3) is more easily satisfied.

Now we want to give some examples in which Theorem 5.2 can be applied.

Example 5.4. Let π : Σ → ℙ^{2} be the blowing up of ℙ^{2} of k points in general position with 1 ≤ k ≤ 5, let L be the generator of Pic(ℙ^{2}), let E_{1}, …, E_{k} be the exceptional divisors. Σ is a well-known Del Pezzo surface and it is known that, in this range, −K_{Σ} is very ample. Let E be −K_{Σ} ⊕ −K_{Σ} and let C be E_{1}. Then Theorem 5.2 proves that (X, T) is p.n.

In fact by looking at the exact sequence 0 → 𝒪_{Σ}(−E_{1}) → 𝒪_{Σ} → 𝒪_{ℙ^{1}} → 0 we get that 1) and 2) are true for j = 0. 4) is true as E_{1} is a rational curve. By recalling that h^{0}(Σ, −K_{Σ}) = 10−k we have that h^{0}(E)−r−2 = 16−2k. Moreover, h^{0}(Σ, C+det(E)+K_{Σ}) = h^{0}(Σ, −K_{Σ}+E_{1}) = h^{0}(Σ, 3π^{∗}L−E_{2}, …, −E_{k}) = 11−k as the k points are in general position, hence 3) is satisfied. Now let us consider 1) and 2) for j ≥ 1. It suffices to show that h^{i}(Σ, −tK_{Σ} − E_{1}) = 0 for t ≥ 1, i = 1, 2. For i = 2 we can use Serre duality. For i = 1 we can use Kodaira vanishing because −tKΣ − E_{1} = K_{Σ} − (t + 1)K_{Σ} − E_{1} and −(t + 1)K_{Σ} − E_{1} is ample by Nakai-Moishezon criterion: (−(t + 1)K_{Σ} − E_{1})^{2} > 0 and for any curve Γ on Σ we have:

(−(t + 1)K_{Σ} − E_{1}) Γ = −(t + 1)K_{Σ}Γ − E_{1}Γ = −tK_{Σ}Γ − K_{Σ}Γ − E_{1}Γ ≥ −tK_{Σ}Γ > 0

because Φ_{|−K_{Σ}|} embeds E_{1} as a line so that

−K_{Σ}Γ = deg [Φ_{|−K_{Σ}|}(Γ)] ≥ Φ_{|−K_{Σ}|}(E_{1})Φ_{|−K_{Σ}|}(Γ) = E_{1}Γ.

Remark 5.5. Obviously, in the previous example, when k = 1 Butler’s criterion can be used (see [9], Theorem 5.1A), to get the projective normality of (X, T).

Example 5.6. Let Σ be ℙ^{2}, let C be a line, let E be a rank 2 very ample vector bundle on ℙ^{2}, let δL and c be, respectively, the first and second Chern classes of E (L is the generator of Pic(ℙ^{2}) as above), let p : ℙ(E) → ℙ^{2} be the natural projection. Under which assumptions can we apply Theorem 5.2 ?

First of all 4) is true as C is a rational curve. 1) and 2) are true for j = 0 as C is ample. h^{0}(Σ, C + det(E) + K_{Σ}) = h^{0}(ℙ^{2}, 𝒪_{ℙ^{2}}(δ − 2)) = δ(δ − 1)/2, so condition 3) becomes: h^{0}(E) ≥ 4+δ(δ−1)/2. Now let us consider 1) and 2) for j ≥ 1. Let Y be a smooth element of |T |, so that Y is isomorphic to the blowing up of ℙ^{2} at c points. Let π be the blowing up and let E_{1}, …, Ec be the exceptional divisors as before. We consider 0 → 𝒪_{X}(−Y) → 𝒪_{X} → 𝒪_{Y} → 0 tensorized with jT − p^{∗}L and we have:

0 → 𝒪_{X}((j − 1)T − p^{∗}L) → 𝒪_{X}(jT − p^{∗}L) → 𝒪_{Y} (jT − p^{∗}L)_{|Y} → 0.

We can proceed by induction on j ≥ 1 as h^{i}(X, −p^{∗}L) = 0 for i = 1, 2, so we have only to consider h^{i}(Y, (jT − p^{∗}L)_{|Y}) = 0 for i = 1, 2. Recall that T_{|Y} = δπ^{∗}L − E_{1} ⋯ − Ec. Now if i = 2 we can use Serre duality, if i = 1 we can use Kodaira vanishing as in Example 5.4 when δ ≥ 4 and 0 ≤ c ≤ 6 (or δ ≥ 2 and 0 ≤ c ≤ 2).

Hence, by using Theorem 5.2, with the previously introduced notation, we get the projective normality of (X, T) if

h^{0}(E) ≥ 4 + δ(δ − 1)/2, δ ≥ 4 and 0 ≤ c ≤ 6.

Example 5.7. Let Σ be any surface, let C be any rational curve on Σ, choose E = L, a very ample line bundle L. When r = 1 Theorem 5.2 is true too, moreover, condition 2) is unnecessary. So we get that (X, L) is p.n. if:

h^{1}(Σ, jL − C) = 0 for j ≥ 0,
h^{0}(Σ, C + (3 − q)L + K_{Σ}) ≤ h^{0}(L) − 3.

Such conditions are satisfied, for example, in many cases when Σ is the blowing up of ℙ^{2} in k points in general position and C = E_{1}.

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Received May 20, 1998