In [7, 8] a good part of the classification, in the smooth case, was recovered, using the point of view of Gaussian maps

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88(2011) 483-518

ON THE EXTENDABILITY OF PROJECTIVE SURFACES AND A GENUS BOUND FOR

ENRIQUES-FANO THREEFOLDS

Andreas Leopold Knutsen, Angelo Felice Lopez & Roberto Mu˜noz

To the memory of Giulia Cerutti, Olindo Ado Lopez and Sa´ul S´anchez

Abstract

We introduce a technique based on Gaussian maps to study whether a surface can lie on a threefold as a very ample divisor with given normal bundle. We give applications, among which one to surfaces of general type and another to Enriques surfaces.

In particular, we prove the genus bound g ≤ 17 for Enriques- Fano threefolds. Moreover we find a new Enriques-Fano threefold of genus 9 whose normalization has canonical but not terminal singularities and does not admitQ-smoothings.

1. Introduction

One of the most important contributions in algebraic geometry is the scheme of classification of higher dimensional varieties proposed by Mori theory. The latter is particularly clear in dimension three: starting with a threefold with terminal singularities and using contractions of extremal rays, the Minimal Model Program predicts to arrive either at a threefold X with K_X nef or at a Mori fiber space. Arguably the simplest case of such spaces is when X is a Fano threefold. As is well known,smooth Fano threefolds have been classified [18,19,27], while, in the singular case, a classification, or at least a search for the numerical invariants, is still underway.

In [7, 8] a good part of the classification, in the smooth case, was recovered, using the point of view of Gaussian maps. The starting step of the latter method is Zak’s theorem [32], [24, Thm. 0.1]: IfY ⊂P^ris a smooth variety of codimension at least two withh⁰(N_{Y /P}^r(−1))≤r+ 1, then the only variety X ⊂P^r+1 that has Y as hyperplane section is a

Research of the first author partially supported by a Marie Curie Intra-European Fellowship. Research of the second author partially supported by the MIUR project

“Geometria delle variet`a algebriche” COFIN 2002-2004. Research of the third author partially supported by the MCYT project BFM2003-03971.

Received 03/21/2007.

483

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cone over Y. When this happens Y ⊂ P^r is said to be nonextendable.

The key point in the application of this theorem is to calculate the cohomology of the normal bundle. It is here that Gaussian maps enter the picture by giving a big help in the case of curves [31, Prop. 1.10]: if Y is a curve then

(1) h⁰(N_{Y /P}^r(−1)) =r+ 1 + cork Φ_H_Y_,ω_Y

where Φ_H_Y_,ω_Y is the Gaussian map associated to the canonical and hyperplane bundleH_Y of Y. For example when X ⊂P^r+1 is a smooth anticanonically embedded Fano threefold andY is a general hyperplane section, h⁰(N_{Y /P}^r(−1)) was computed in [7] by considering the general curve sectionC ofY. That proof was strongly based on the fact thatC is a general curve on a general K3 surface and that the Hilbert scheme of K3 surfaces is essentially irreducible. As the latter fact is peculiar to K3 surfaces, we immediately realized that if one imposes different hyperplane sections to a threefold, for example Enriques surfaces, it gets quite difficult to rely on the curve section. To study this and other cases one therefore needs an analogue of the formula (1) in higher dimension.

We accomplish this in Section 2 by proving the following:

Theorem 1.1. Let Y ⊂ P^r be a smooth irreducible linearly normal surface and letHbe its hyperplane bundle. Assume there is a base-point free and big line bundle D₀ on Y with H¹(H−D₀) = 0 and such that the general element D∈ |D0| is not rational and satisfies

(i) the Gaussian map Φ_H_D_,ω_D_(D₀₎ is surjective;

(ii) the multiplication maps µ_V_D_,ω_D andµ_V_D_,ω_D_(D₀₎ are surjective where V_D := Im{H⁰(Y, H−D₀)→H⁰(D,(H−D₀)_|D)}. Then

h⁰(N_{Y /P}^r(−1))≤r+ 1 + cork Φ_H_D_,ω_D.

The above theorem is a flexible instrument to study threefolds whose hyperplane sections have large Picard group, since, if both D₀ andH− D₀ are base-point free and the degree of D is large with respect to its genus, the hypotheses are fulfilled unless D is hyperelliptic (note that the case whereY is a K3 andH has no moving decomposition has been considered by Mukai [26]).

As we will see in Section 3, Theorem 1.1 has several applications. A sample of this is for pluricanonical embeddings of surfaces of general type:

Corollary 1.2. Let Y ⊂ PV_m be a minimal surface of general type with base-point free and nonhyperelliptic canonical bundle and V_m ⊆ H⁰(OY(mK_Y + ∆)), where ∆≥0 and either ∆ is nef or ∆is reduced

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andK_Y is ample. Suppose that Y is regular or linearly normal and that

m≥











9 if K_Y² = 2;

7 if K_Y² = 3;

6 if K_Y² = 4 and the general curve in |KY| is trigonal or if K_Y² = 5 and the general curve in|KY|is a plane quintic;

5 if either the general curve in|K_Y| has Clifford index 2 or 5≤K_Y² ≤9 and the general curve in |KY|is trigonal;

4 otherwise.

Then Y is nonextendable.

In general, the conditions on K_Y² and m are optimal (see Remark 3.4).

Besides the applications in Section 3, we will concentrate our atten- tion on the following threefolds:

Definition 1.3. An Enriques-Fano threefold is an irreducible three-dimensional variety X ⊂P^N having a hyperplane section S that is a smooth Enriques surface, and such thatXis not a cone overS. We will say thatX has genusgif gis the genus of its general curve section.

Fano [13] claimed a classification of such threefolds, but his proof contains several gaps. Conte and Murre [9] remarked that an Enriques- Fano threefold must have some isolated singularities and, under special assumptions on the singularities, recovered some of the results of Fano, but not enough to give a classification, nor to bound the numerical invariants. Assuming that the Enriques-Fano threefold is a quotient of a smooth Fano threefold by an involution (this corresponds to having only cyclic quotient terminal singularities), a list was given by Bayle [2, Thm. A] and Sano [29, Thm. 1.1]. Moreover, by [25, MainThm. 2], any Enriques-Fano threefold with at most terminal singularities admits a Q-smoothing, that is, it appears as central fiber of a small deformation over the 1-parameter unit disk such that a general fiber has only cyclic quotient terminal singularities. This, together with the results of Bayle and Sano, gives the bound g ≤ 13 for Enriques-Fano threefolds with at most terminal singularities. Bayle and Sano recovered all of the known examples of Enriques-Fano threefolds. Therefore it has been conjectured that this list is complete or, at least, that the genus is bounded, in analogy with smooth Fano threefolds [18, 19]. In Section 13, we will show that the list of known Enriques-Fano threefolds is not complete(not even after specialization), by finding a new Enriques-Fano threefold enjoying several peculiar properties (for a more precise version, see Proposition 13.1):

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Proposition 1.4. There exists an Enriques-Fano threefold X ⊂ P⁹ of genus 9 such that neither X nor its normalization belong to the list of Fano-Conte-Murre-Bayle-Sano.

Moreover, Xdoes not have aQ-smoothing and in particular X is not in the closure of the component of the Hilbert scheme made of Fano- Conte-Murre-Bayle-Sano’s examples. Its normalization Xe has canonical but not terminal singularities and does not admit Q-smoothings.

Observe thatXe is aQ-Fano threefold of Fano index 1 with canonical singularities not having aQ-smoothing, showing that [25, MainThm. 2]

cannot be extended to the canonical case.

In the present article (and [20]) we apply Theorem 1.1 to get a genus bound on Enriques-Fano threefolds, under no assumption on their singularities:

Theorem 1.5. Any Enriques-Fano threefold has genus g≤17.

A more precise result forg= 15 and 17 is proved in Proposition 12.1.

We remark that simultaneously and independently, Prokhorov [28]

proved the same genus bound at the same time constructing an example of a genus 17 Enriques-Fano threefold, thus showing that the bound g ≤ 17 is optimal. His methods, relying on the MMP, are completely different from ours.

Now a few words on our method of proof. In Section 4 we review some basic results. In Section 5 we apply Theorem 1.1 to Enriques surfaces and obtain the main results on nonextendability needed in the rest of the article. In Section 6 we prove Theorem 1.5 for all Enriques- Fano threefolds except for some concrete embedding line bundles on the Enriques surface section. These are handled case by case in Sections 7-11 by finding divisors satisfying the conditions of Theorem 1.1, thus allowing us to prove our theorem and a more precise statement for g = 15 and 17 in Section 12. A part of the proof for a special class of line bundles has been moved to the note [20]. This part involves no new ideas compared to the parts treated in the present article.

To prove our results we use criteria for the surjectivity of Gaussian maps on curves on Enriques surfaces from [22,23] and of multiplication maps of linear systems on such curves, which we obtain in Lemma 5.6 (which holds on any surface) in the present article.

2. Proof of Theorem 1.1

Let L and M be line bundles on a smooth projective variety. Given V ⊆ H⁰(L) we denote by µ_V,M : V ⊗H⁰(M) −→ H⁰(L⊗M) the multiplication map of sections, µ_L,M when V =H⁰(L), and by Φ_L,M : Kerµ_L,M −→H⁰(Ω¹_X ⊗L⊗M) the Gaussian map [31, 1.1].

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Proof of Theorem 1.1. To prove the bound onh⁰(N_{Y /P}^r(−1)), we use the short exact sequence

0 ^/^/N_{Y /P}^r(−D0−H) ^/^/N_{Y /P}^r(−H) ^/^/N_{Y /P}^r(−H)_|D ^/^/0 and prove that

(2) h⁰(N_{Y /P}^r(−D0−H)) = 0, and (3) h⁰(N_{Y /P}^r(−H)_|D)≤r+ 1 + cork Φ_H_D_,ω_D. To prove (2), note that since dim|D0| ≥1, it is enough to have (4) h⁰(N_{Y /P}^r(−D0−H)_|D) = 0 for a generalD∈ |D0|.

Now, setting D₁ :=D₀+H, (4) follows from the exact sequence (5)

0 ^/^/N_D/Y(−D₁) ^α ^/^/N_D/Pr(−D₁) ^/^/N_{Y /P}r(−D₁)_|D ^/^/0 and the facts, proved below, that h⁰(N_D/P^r(−D1)) = 0 and H¹(α) is injective.

To see that h⁰(N_D/P^r(−D1)) = 0, we note that µ_H_D_,ω_D_(D₀₎ is surjective by the H⁰-lemma [15, Thm. 4.e.1], since |D_0|D| is base-point free, whence D²₀ ≥ 2, therefore h¹(ω_D(D₀−H)) = h⁰((H −D₀)_|D) ≤ h⁰(H_D)−2, as H_D is very ample. Now let P^k ⊆P^r be the linear span of D. Then

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0 ^/^/N_D/Pk(−D1) ^/^/N_D/P^r(−D1) ^/^/(−D0)^⊕(r−k)_|D ^/^/0 implies that h⁰(N_D/P^r(−D1)) =h⁰(N_D/Pk(−D1)), asD²₀ >0. Since Y is linearly normal and H¹(H−D₀) = 0, alsoD is linearly normal. As µ_H_D_,ω_D_(D₀₎ is surjective, h⁰(N_D/P^k(−D₁)) = cork Φ_H_D_,ω_D_(D₀₎ = 0 by [31, Prop. 1.10] because of (i). Henceh⁰(N_D/P^r(−D₁)) = 0. As for the injectivity ofH¹(α), we prove the surjectivity of H¹(α)^∗ with the help of the commutative diagram

(7) H⁰(^I_D/P^r(H))⊗H⁰(ω_D(D₀)) ^/^/

f

H⁰(N_D/P^∗ r⊗ω_D(D₁))

H¹(α)^∗

H⁰(^I_D/Y(H))⊗H⁰(ω_D(D₀)) ^h ^/^/H⁰(N_D/Y^∗ ⊗ω_D(D₁)).

Here f is surjective by linear normality of Y, while h is surjective by (ii) since it factors as the composition of the (surjective) restriction map H⁰(J_D/Y(H))⊗H⁰(ω_D(D₀))→V_D⊗H⁰(ω_D(D₀)) and the multiplication mapµ_V_D_,ω_D_(D₀₎ :V_D ⊗H⁰(ω_D(D₀))→H⁰(N_D/Y^∗ ⊗ω_D(D₁)).

Finally, to prove (3), recall thatµ_H_D_,ω_D is surjective by [1, Thm. 1.6]

sinceDis not rational, whenceh⁰(N_D/Pk(−H)) =k+ 1 + cork Φ_H_D_,ω_D

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by [31, Prop. 1.10]. Therefore, twisting (6) by (D₀)_|D, we find that h⁰(N_D/P^r(−H))≤r+1+cork Φ_H_D_,ω_D and (3) will follow by the sequence (5) tensored by (D₀)_|Dand injectivity ofH¹(α⊗(D0)_|D), which is proved exactly as the injectivity ofH¹(α) above, using now the surjectivity of

µ_V_D_,ω_D. q.e.d.

Remark 2.1. In the above proposition and also in Corollary 2.2 below, the surjectivity of µ_V_D_,ω_D_(D₀₎ can be replaced by either one of the following conditions: (i)µ_ω_D_(H_−D₀_),D_0|D is surjective; (ii)h⁰((2D₀− H)_|D)≤h⁰(D_0|D)−2; (iii)H.D₀ >2D²₀. Indeed, condition (iii) implies h⁰((2D₀−H)_|D) = 0, whence (ii), while (ii) implies (i) by theH⁰-lemma [15, Thm. 4.e.1]. Finally, (i) is enough by surjectivity ofµ_V_D_,ω_D and the commutative diagram

V_D ⊗H⁰(ω_D)⊗H⁰(D_0|D) ^µ^VD,ωD^/^/^⊗Id

H⁰(ω_D(H−D₀))⊗H⁰(D₀_|D)

µ_ωD(H−D0),D0|D

V_D ⊗H⁰(ω_D(D₀)) ^µ^VD,ωD^(D⁰⁾ ^/^/H⁰(ω_D(H)).

Whereas the upper bound in Theorem 1.1 can be applied to control how many times Y can be extended to higher dimensional varieties, we will concentrate on the case of one simple extension.

Corollary 2.2. Let Y ⊂P^r be a smooth irreducible surface which is linearly normal or regular and let H be its hyperplane bundle. Assume there is a base-point free and big line bundleD₀onY withH¹(H−D0) = 0and such that the general elementD∈ |D0|is not rational and satisfies

(i) the Gaussian map ΦHD,ωD is surjective;

(ii) the multiplication maps µ_V_D_,ω_D andµ_V_D_,ω_D_(D₀₎ are surjective, where V_D := Im{H⁰(Y, H−D₀)→H⁰(D,(H−D₀)_|D)}.

Proof. Note thatg(D)≥2, as Φ_H_D_,ω_D is surjective. Sinceµ_V_D_,ω_D_(D₀₎ is surjective, we have thatV_D (whence also|(H−D₀)_|D|) is base-point free, as|ωD(H)|is such. Therefore 2g(D)−2 + (H−D0).D >0, whence h¹(ω_D²(H−D₀)) = 0, and the H⁰-lemma [15, Thm. 4.e.1] implies that µ_ω2

D(H),D0|D is surjective. Now Φ_H_D_,ω_D_(D₀₎ is surjective by (i) and the commutative diagram

Kerµ_H_D_,ω_D⊗H⁰(D0|D)^Φ^HD,ωD^/^/^⊗Id^/^/

H⁰(ω²_D(H))⊗H⁰(D0|D)

µ_ω2

D(H),D0|D

Kerµ_H_D_,ω_D_(D₀₎ ^Φ^HD,ωD^(D⁰⁾ ^/^/H⁰(ω_D²(H+D₀)).

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IfY is linearly normal, we are done by Zak’s theorem [32] and Theorem 1.1.

Assume now that h¹(OY) = 0 and that Y ⊂ P^r is extendable, that is, thatY is a hyperplane section of some nondegenerate threefoldX⊂ P^r+1 which is not a cone over Y. Let π : Xe → X be a resolution of singularities and let L = π^∗OX(1) and Ye = π⁻¹(Y) ∼= Y, as Y is smooth, so that Y ∩SingX = ∅. Using H¹(O_Y_e) = H¹(O_Y) = 0 and Kawamata-Viehweg vanishing, one easily deduces the surjectivity of the restriction map H⁰(X, L)e → H⁰(Y , Le _|_Y_e). Consider the birational map ϕ_L:Xe →P^N whereN ≥r+ 1. ThenY :=ϕ_L(Ye)∼=Y is a hyperplane section ofϕ_L(X) that is linearly normal and extendable and we reducee

to the linearly normal case above. q.e.d.

3. Absence of Veronese embeddings on threefolds It was known to Scorza [30] that the Veronese varieties v_m(Pⁿ) are nonextendable for m, n > 1. For an arbitrary Veronese embedding we can use Zak’s theorem [32], [24, Thm. 0.1] as follows:

Remark 3.1. Let X ⊂ P^r be a smooth irreducible nondegenerate n-dimensional variety, n≥2,L=OX(1) and letϕ_mL(X) ⊂P^N be the m-th Veronese embedding of X.

If H¹(T_X(−mL)) = 0 then ϕ_mL(X) is nonextendable. In particular the latter holds if m >maxn

2, n+ 2 +^K^X^.Lⁿ⁻¹_L^−2r+2n+2n

o .

Proof. Set Y = ϕ_mL(X). From standard sequences and Kodaira vanishing one gets h⁰(N_{Y /P}N(−1)) ≤ h⁰(T_PN(−1)_|Y) +h¹(T_Y(−1)) = N+ 1 +h¹(T_X(−mL)) =N+ 1, and we just apply Zak’s theorem [32].

To see the last assertion, observe that since n ≥ 2 and m ≥ 3 we have, as is well-known,h¹(T_X(−mL)) =h⁰(N_X/P^r(−mL)). If the latter were not zero, the same would hold for a general curve section C ⊂ P^r−n+1. Takingr−n−1 general pointsx_j ∈C, we get from the exact sequence [4, (2.7)] that h⁰(N_C/Pr−n+1(−m)) = 0 for reasons of degree,

a contradiction. q.e.d.

In the case of surfaces, Corollary 2.2 yields an extension of this remark:

Definition 3.2. LetY be a smooth surface and letLbe an effective line bundle on Y such that the general divisor D ∈ |L| is smooth and irreducible. We say that L is hyperelliptic, trigonal, etc., if D is such. We denote by Cliff(L) the Clifford index of D. Moreover, when L² >0, we set

ε(L) = 3 if Lis trigonal; ε(L) = 5 if Cliff(L)≥3;

ε(L) = 0 if Cliff(L) = 2;

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m(L) =











16

L² if L.(L+K_Y) = 4;

25

L² if L.(L+K_Y) = 10 and the general divisor in |L| is a plane quintic;

3L.KY+18

2L² +³₂ if 6≤L.(L+K_Y)≤22 and Lis trigonal;

2L.KY−ε(L)

L² + 2 otherwise.

Corollary 3.3. LetY ⊂PV be a smooth surface withV ⊆H⁰(L^⊗m⊗ OY(∆)), where L is a base-point free, big, nonhyperelliptic line bundle on Y with L.(L+K_Y) ≥ 4 and ∆ ≥ 0 is a divisor. Suppose that Y is regular or linearly normal and that m is such that H¹(L^⊗(m−2) ⊗ OY(∆)) = 0and m >max{m(L)−^L.∆_L2 ,⌈^L.K^Y^+2−L.∆_L2 ⌉+ 1}. Then Y is nonextendable.

Proof. We apply Corollary 2.2 withD₀ =LandH =L^⊗m⊗ OY(∆).

By hypothesis the general D ∈ |L| is smooth and irreducible of genus g(D) = ¹₂L.(L+K_Y)+1. SinceH¹(H−2L) = 0, we haveV_D =H⁰((H− L)_|D). Also (H−L).D= (m−1)L²+L.∆≥L.(L+K_Y)+2 = 2g(D) by hypothesis, whence |(H−L)_|D|is base-point free and birational (as D is not hyperelliptic) andµ_V_D_,ω_D is surjective by [1, Thm. 1.6]. Moreover H¹((H−L)_|D) = 0, whence also H¹(H−L) = 0.

The surjectivity ofµ_V_D_,ω_D_(L) follows by [15, Cor. 4.e.4], as degω_D(L)

≥2g(D)+1 becauseL² ≥3. Indeed, ifL² ≤2 we have thath⁰(L_|D)≤1 as D is not hyperelliptic, whence h⁰(L) ≤ 2, contradicting the hypotheses on L. The surjectivity of Φ_H_D_,ω_D follows by the inequality m > m(L)−^L.∆_L2 and well-known results about Gaussian maps (see e.g.

[31, Prop. 1.10], [23, Prop. 2.9, Prop. 2.11 and Cor. 2.10], [4, Thm. 2]).

q.e.d.

We can be a little bit more precise in the case of pluricanonical embeddings:

Proof of Corollary 1.2. We apply Corollary 3.3 with L = OY(K_Y) and H=OY(mK_Y + ∆) and prove thatH¹(OY((m−2)K_Y + ∆)) = 0.

If ∆ is nef this follows by Kawamata-Viehweg vanishing. Now suppose that ∆ is reduced and K_Y is ample. Again H¹(OY((m−2)K_Y)) = 0, whenceH¹(O_Y((m−2)K_Y + ∆)) = 0, sinceh¹(O_∆((m−2)K_Y + ∆)) =

h⁰(O∆(−(m−3)K_Y)) = 0. q.e.d.

Remark 3.4. Consider the 5-uple embedding X of P³ into P⁵⁵ (respectively, the 4-uple embedding of a smooth quadric hypersurface in P⁴ into P⁵⁴). A general hyperplane section Y of X is embedded with 5K_Y (resp. 4K_Y) and K_Y² = 5 (resp. K_Y² = 8). Thus, in Corollary 1.2, the conditions onK_Y² and mcannot, in general, be weakened.

We can be even more precise in the case of adjoint embeddings.

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Corollary 3.5. LetY ⊂PV be a minimal surface of general type with base-point free and nonhyperelliptic canonical bundle and V ⊆H⁰(L⊗ OY(K_Y + ∆)), where L is a line bundle on Y and ∆≥0 is a divisor.

Suppose that Y is regular or linearly normal, that H¹(L ⊗ OY(∆− K_Y)) = 0 and that

L.K_Y+K_Y.∆>











14 if K_Y² = 2;

20 if K_Y² = 5 and the general divisor in|K_Y|is a plane quintic;

2K_Y² + 9 if 3≤K_Y² ≤11 andK_Y is trigonal;

3K_Y² −ε(K_Y) otherwise.

Proof. Similar to the proof of Corollary 3.3 with D₀ =OY(K_Y) and

H =L⊗ OY(KY + ∆). q.e.d.

To state the pluriadjoint case, given a big line bundleLon a smooth surfaceY, we define

ν(L) =











12

L² + 1 if L.(L+K_Y) = 4;

15

L² + 1 if L.(L+K_Y) = 10 and the general divisor in|L|is a plane quintic;

L.KY+18

2L² +³₂ if 6≤L.(L+K_Y)≤22 and L is trigonal;

L.KY−ε(L)

L² + 2 otherwise.

Corollary 3.6. LetY ⊂PV be a smooth surface withV ⊆H⁰(L^⊗m⊗ OY(K_Y + ∆)) where L is a base-point free, big and nonhyperelliptic line bundle on Y with L.(L+K_Y) ≥ 4 and ∆ ≥ 0 is a divisor such that H¹(L^⊗(m−2) ⊗ OY(K_Y + ∆)) = 0. Suppose that Y is regular or linearly normal and that m > max{2 + _L¹2, ν(L)} − ^L.∆_L2 . Then Y is nonextendable.

Proof. Similar to the proof of Corollary 3.3 with D₀ = L and H =

L^⊗m⊗ OY(K_Y + ∆). q.e.d.

4. Basic results on line bundles on Enriques surfaces Definition 4.1. Let S be an Enriques surface. IfD is a divisor on S we will denote by Hⁱ(D) the cohomology Hⁱ(OS(D)). We denote by ∼(respectively ≡) the linear (respectively numerical) equivalence of divisors (or line bundles) onS. A line bundleLisprimitiveifL≡hL^′ for some line bundleL^′and some integerh, impliesh=±1. An effective line bundle L is quasi-nef [21] if L² ≥ 0 and L.∆ ≥ −1 for every ∆ such that ∆>0 and ∆² =−2.

Anodalcurve is a smooth rational curve. Anodal cycleis a divisor R >0 such that (R^′)² ≤ −2 for any 0< R^′ ≤R. Anisotropic divisor

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F is a divisor such that F² = 0 and F 6≡0. An isotropic k-sequence is a set {f1, . . . , f_k} of isotropic divisors such thatf_i.f_j = 1 for i6=j.

We will often use the fact that ifRis a nodal cycle, thenh⁰(OS(R)) = 1 and h⁰(O_S(R+K_S)) = 0.

Let L be a line bundle on S with L² > 0. Following [11] we define φ(L) = inf{|F.L| : F ∈ PicS, F² = 0, F 6≡ 0}. One has φ(L)² ≤ L² [11, Cor. 2.7.1] and, ifLis nef, then there exists a genus one pencil|2E| such that E.L=φ(L) [10, 2.11]. Moreover we will extensively use the fact that if L is nef, then it is base-point free if and only if φ(L) ≥ 2 [11, Prop. 3.1.6, 3.1.4 and Thm. 4.4.1].

A line bundleL >0 with L²≥0 onS has a (nonunique) decomposition L≡a₁E₁+. . .+a_nE_n, wherea_i are positive integers, and each E_i is primitive, effective and isotropic, cf. e.g. [22, Lemma 2.12]. We will call such a decomposition an arithmetic genus 1 decomposition.

Definition 4.2. An effective line bundle L with L² ≥ 0 is said to be of small type if either L = 0 or if in every arithmetic genus 1 decomposition of Las above, all a_i= 1.

The next result is an easy (computational) consequence of [21, Lemma 2.1] and [22, Lemma 2.4].

Lemma 4.3. LetLbe an effective line bundle on an Enriques surface with L² ≥ 0. Then L is of small type if and only if it is of one of the following types (where E_i >0, E_i² = 0 and E_i primitive): (a) L = 0;

(b) L² = 0, L ∼ E₁; (c) L² = 2, L ∼ E₁ +E₂, E₁.E₂ = 1; (d) L² = 4, φ(L) = 2, L ∼ E₁+E₂, E₁.E₂ = 2; (e) L² = 6, φ(L) = 2, L∼E₁+E₂+E₃,E₁.E₂ =E₁.E₃ =E₂.E₃= 1;(f)L²= 10,φ(L) = 3, L∼E₁+E₂+E₃, E₁.E₂ = 1, E₁.E₃ =E₂.E₃ = 2.

Among all arithmetic genus 1 decompositions of an effective line bun- dleL withL² >0, we want to choose the most convenient for our pur- poses. For any line bundleL >0 which isnot of small typewithL²>0 and φ(L) =F.L for someF >0 with F² = 0, define

(8) α_F(L) = min{k≥2|(L−kF)² ≥0 and if (L−kF)² >0, then there existsF^′ >0 with (F^′)²= 0, F^′.F >0 and F^′.(L−kF)≤φ(L)}.

By [22, Lemma 2.4], it is easy to see thatα_F(L) exists and that one ob- tains an equivalent definition by replacing the last inequality byF^′.(L− kF) =φ(L−kF).

If L²= 0 and L isnot of small type, then we defineα_F(L) to be the maximal integer k ≥2 such that there exists an isotropic F such that L≡kF. The next result is an easy computation.

Lemma 4.4. Let Lbe an effective line bundle not of small type with L² > 0 and (L², φ(L)) 6= (16,4), (12,3), (8,2), (4,1). Then (L − α_F(L)F)² >0.

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We will also use the following consequence of [11, Prop. 3.1.4], [21, Cor. 2.5] and [22, Lemma 2.3]:

Lemma 4.5. Let L be a nef and big line bundle on an Enriques surface and letF be a divisor satisfyingF.L <2φ(L)(respectivelyF.L= φ(L) and L is ample). Then h⁰(F) ≤ 1 and if F > 0 and F² ≥ 0 we have F² = 0, h⁰(F) = 1, h¹(F) = 0 and F is primitive and quasi-nef (resp. nef ).

5. Main results on extendability of Enriques surfaces It is well-known that abelian and hyperelliptic surfaces are nonextendable [14, Rmk. 3.12]. The extendability problem is open for K3’s, but answers are known for general K3’s [7,8,3]. Let us deal now with Enriques surfaces.

We start with a simplification of Corollary 2.2 that will be central to us.

Proposition 5.1. Let S ⊂ P^r be an Enriques surface and H its hyperplane bundle. Suppose there is a nef and big (whence effective) line bundleD₀ on S withφ(D₀)≥2, H¹(H−D₀) = 0 and such that the following conditions are satisfied by the general element D∈ |D0|:

(i) the Gaussian map ΦHD,ωD is surjective;

(ii) the multiplication map µ_V_D_,ω_D is surjective, where V_D := Im{H⁰(S, H−D₀)→H⁰(D,(H−D₀)_|D)};

(iii) h⁰((2D₀−H)_|D)≤ ¹₂D₀²−2.

Then S is nonextendable.

Proof. Apply Corollary 2.2 and Remark 2.1, using that D₀ is base-

point free sinceφ(D₀)≥2. q.e.d.

Our first observation will be that, for many line bundles H, a line bundle D₀ satisfying the conditions of Proposition 5.1 can be found with the help of Ramanujam’s vanishing theorem.

Proposition 5.2. Let S ⊂ P^r be an Enriques surface such that its hyperplane section H is not 2-divisible in NumS. Suppose there exists an effective divisor B on S satisfying:

(i) B² ≥4 and φ(B)≥2,

(ii) (H−2B)² ≥0 and H−2B ≥0,

(iii) H² ≥64 if B² = 4 and H² ≥54 if B² = 6.

Proof. We first claim that there is a nef divisor D^′ > 0 satisfying (i)-(iii) and with D^′ ≤ B, (D^′)² = B², φ(D^′) = φ(B). Indeed, if Γ is a nodal curve, define the Picard-Lefschetz reflection on PicS as π_Γ(L) := L + (L.Γ)Γ. Then π_Γ preserves intersections, effectiveness

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[5, Prop. VIII.16.3] and the function φ. Now if B is not nef, there is a nodal Γ such that Γ.B < 0. Since 0 < π_Γ(B) < B, we see that π_Γ(B) satisfies (i)-(iii). If π_Γ(B) is not nef, we repeat the process, which must end, as π_Γ(B) < B, and we get the desired nef D^′. Since H−D^′≥H−B > H−2B ≥0 and (D^′)²>0, we haveD^′.(H−D^′)>0.

Now define the following set, which is nonempty, by what we just saw, Ω(D^′) = {M ∈PicS :M ≥D^′, M is nef, satisfies (i)-(ii) and

M.(H−M)≤D^′.(H−D^′)}.

For any M ∈ Ω(D^′) we have H−2M > 0, whence H.M is bounded.

Let then D₀ be amaximal divisor in Ω(D^′), that is, such that H.D₀ is maximal. We want to show that h¹(H−2D₀) = 0.

Set R := H −2D₀. If h¹(R) > 0, then by Ramanujam vanishing [5, Cor. II.12.3] we could write R+K_S ∼ R₁ +R₂, for R₁ > 0 and R₂ > 0 with R₁.R₂ ≤ 0. We can assume that R₁.H ≤ R₂.H. If D₁:=D₀+R₁ is nef, thenφ(D₁) is calculated by a nef divisor, whence φ(D₁) ≥φ(D^′) ≥2 and D₁² ≥D₀² ≥(D^′)² ≥4 (since D₁ ≥D₀ ≥ D^′).

Moreover (H −2D₁)² = R² −4R₁.R₂ ≥ R² ≥ 0, and since (H − 2D₁).H = (R₂ −R₁).H ≥ 0, we get by Riemann-Roch and the fact that H is not 2-divisible in NumS, that H−2D₁ > 0. Furthermore, D₁.(H−D1) =D₀.(H−D0)+R₁.R₂≤D₀.(H−D0), whenceD₁ ∈Ω(D^′) withH.D₁ > H.D₀, contradicting the maximality ofD₀.

HenceD₁ cannot be nef and there exists a nodal curve Γ with Γ.D₁ <

0 (whence Γ.R₁ <0). SinceHis ample, we have Γ.(H−D1)≥ −Γ.D1+ 1 ≥ 2. Since Γ.R₁ < 0, we have D₂ := D₁−Γ ≥ D₀, whence, if D₂ is nef, we have as above that φ(D₂) ≥ φ(D^′) ≥ 2 and D₂² ≥ D²₀ ≥ (D^′)² ≥ 4. Moreover H −2D₂ > H−2D₁ > 0 and (H −2D₂)² = (H −2D₁)² −8 + 4(H−2D₁).Γ ≥ (H−2D₁)²+ 4 >0. Furthermore D₂.(H−D₂)< D₁.(H−D₁)≤D₀.(H−D₀), whence D₂ 6=D₀. If D₂ is nef, thenD₂∈Ω(D^′) with H.D₂> H.D₀, a contradiction. HenceD₂ is not nef, and we repeat the process, with a nodal Γ₁≤R₁−Γ. As the process must end, we get h¹(H−2D₀) = 0.

Note that since D₀² ≥(D^′)² =B², then D₀ also satisfies (iii) above.

FurthermoreD₀is base-point free since it is nef withφ(D₀)≥φ(D^′)≥2.

Since h¹(H −2D0) = 0 and h¹(OD(H −D₀)) = 0 for reasons of degree, we findh¹(H−D₀) = 0. To prove the proposition, we only have left to show, by Proposition 5.1, that Φ_H_D_,ω_D is surjective.

From (H−2D₀).D₀ ≥2 again, we get degH_D ≥4g(D)−2, whence Φ_H_D_,ω_D is surjective if Cliff(D) ≥2 by [4, Thm. 2]. This is satisfied if D²₀ ≥8 by [22, Cor. 1.5 and Prop. 4.13].

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If D₀² = 6, then g(D) = 4, whence Φ_H_D_,ω_D is surjective if we have h⁰(OD(3D₀+K_S −H)) = 0 by [31, Prop. 1.10]. Since H² ≥ 54, we get by Hodge index that H.D ≥ 18 with equality if and only if H ≡ 3D₀. If H.D₀ > 18, we get degOD(3D₀ +K_S −H) < 0. If H ≡ 3D₀, then we may assumeH ∼3D₀, possibly after exchangingD₀ with D₀ +K_S, so that h⁰(O_D(3D₀ +K_S −H)) = h⁰(O_D(K_S)) = 0. If D²₀ = 4, theng(D) = 3, whence Φ_H_D_,ω_D is surjective by [31, Prop. 1.10]

as h⁰(OD(4D₀ −H)) = 0. Indeed, since H² ≥ 64, we get by Hodge index that H.D≥17, whence degOD(4D0−H)<0. q.e.d.

We now improve Proposition 5.2 in the cases B² = 4 and 6, using [23].

Proposition 5.3. Let S ⊂P^r be an Enriques surface such that its hyperplane section H is not 2-divisible in NumS. Suppose there exists an effective divisor B on S satisfying: (i) B² = 6 and φ(B) = 2, (ii) (H−2B)² ≥0 andH−2B ≥0, (iii) h⁰(3B−H) = 0or h⁰(3B+K_S− H) = 0. Then S is nonextendable.

Proof. Argue exactly as in the proof of Proposition 5.2 and let D^′, D₀ and D be as in that proof, so that, in particular, D₀² ≥(D^′)² = 6.

If D₀² ≥ 8, we are done by Proposition 5.2. If D²₀ = 6 write D₀ = D^′ +M with M ≥ 0. Since both D₀ and D^′ are nef we find 6 = D²₀ = (D^′)²+D^′.M +D₀.M ≥6, whence D^′.M =D₀.M = 0, so that M² = 0. ThereforeM = 0 andD₀=D^′, whence 3D₀−H∼3D^′−H≤ 3B−H. It follows that eitherh⁰(3D0−H) = 0 orh⁰(3D0+K_S−H) = 0. Possibly after exchanging D₀ with D₀ +K_S, we can assume that h⁰(3D₀+K_S−H) = 0. Ash¹(2D₀+K_S−H) =h¹(H−2D₀) = 0, we get h⁰(O_D(3D₀+K_S−H)) = 0, whence Φ_H_D_,ω_D is surjective by [23, Thm(ii)]. The map µ_V_D_,ω_D is surjective as in the previous proof. q.e.d.

Proposition 5.4. Let S ⊂P^r be an Enriques surface such that its hyperplane section H is not 2-divisible in NumS. Suppose there exists an effective divisorB on S satisfying: (i) B is nef, B² = 4and φ(B) = 2, (ii) (H −2B)² ≥ 0 and H −2B ≥ 0, (iii) H.B > 16. Then S is nonextendable.

Proof. Argue as in the proof of Proposition 5.2 and let D^′, D₀ and D be as in that proof. By (i) we have D^′ =B, and since D₀ ≥D^′, we get H.D₀ >16. If D₀² ≥8, we are done by Proposition 5.2. If D²₀ = 6, then D₀ > D^′ = B, so that H.D₀ ≥ 18 whence (3D₀ −H).D₀ ≤ 0.

If 3D₀ −H > 0, it is a nodal cycle, whence either h⁰(3D₀ −H) = 0 or h⁰(3D0 +K_S −H) = 0 and we are done by Proposition 5.3. If D²₀ = 4, then D₀ =D^′ =B and degOD(4D₀−H)<0 as in the proof of Proposition 5.3, whence Φ_H_D_,ω_D is surjective by [23, Thm(i)] and so is µ_V_D_,ω_D, as in the proof of Proposition 5.2. q.e.d.

In several cases the following will be very useful:

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Lemma 5.5. Let S ⊂ P^r be an Enriques surface with hyperplane section H ∼ 2B +A, for B nef, B² ≥ 2, A² = 0, A > 0 primitive, H² ≥28 and satisfying one of the following conditions:

(i) A is quasi-nef and (B², A.B)6∈ {(4,3),(6,2)};

(ii) φ(B)≥2 and(B², A.B)6∈ {(4,3),(6,2)};

(iii) φ(B) = 1, B²= 2l, B ∼lF₁+F₂, l≥1, F_i >0, F_i²= 0, i= 1,2, F₁.F₂= 1, and either

(a) l≥2, F_i.A≤3 for i= 1,2 and (l, F₁.A, F₂.A)6= (2,1,1); or (b) l= 1, 5≤B.A≤8, F_i.A≥2 for i= 1,2 and

(φ(H), F₁.A, F₂.A)6= (6,4,4).

Proof. Possibly after replacing B with B +K_S if B² = 2 we can, without loss of generality, assume that B is base-component free.

We first prove the lemma under hypothesis (i).

One easily sees that D₀ :=B+Ais nef, sinceAis quasi-nef andH is ample. Moreover, D₀² =B²+ 2B.A≥6, as 2A.B =A.H ≥φ(H) ≥3, since H is very ample. If φ(D₀) = 1 = F.D₀ for some F > 0 with F² = 0, we get F.B = 1, F.A = 0 and the contradiction F.H = 2.

Hence φ(D₀)≥2.

One easily checks that (i) implies D²₀ ≥ 12. Since h⁰(2D₀ −H) = h⁰(A) = 1 by [21, Cor. 2.5], we have that Φ_H_D_,ω_D is surjective by [23, Thm(iv)]. Also h¹(H−2D₀) = h¹(−A) = 0, again by [21, Cor. 2.5], so that V_D = H⁰(OD(H −D₀)). As H−D₀ = B is base-component free and |D0| is base-point free and birational by [11, Lemma 4.6.2, Thm. 4.6.3 and Prop. 4.7.1], also V_D is base-point free and is either a complete pencil or birational. Hence µ_V_D_,ω_D is surjective by the base- point free pencil trick [1,§1] and [1, Thm. 1.6]. ThenSis nonextendable by Proposition 5.1.

Therefore the lemma is proved under the assumption (i) and, in particular, the whole lemma is proved with the additional assumption that A is quasi-nef.

Now assume that A is not quasi-nef. Then there is a ∆ > 0 with

∆² = −2 and ∆.A ≤ −2. We have ∆.B ≥ 2 by the ampleness of H.

Furthermore, among all such ∆’s we will choose a minimal one, that is, such that no 0 < ∆^′ < ∆ satisfies (∆^′)² =−2 and ∆^′.A ≤ −2. Then one easily proves that B₀ := B + ∆ is nef. Moreover, B²₀ ≥ 2 +B², and φ(B₀) ≥ φ(B). We also note that H−2B₀ ∼A−2∆ >0 and is primitive by [22, Lemma 2.3] with (H−2B0)² ≥0.

Under the assumptions (ii), we have φ(B₀) ≥ 2. Then S is nonextendable by Proposition 5.2 if B₀² ≥ 8. If B₀² = 6, we have B² = 4 and ∆.B = 2, so that ∆.A=−2 or−3 by the ampleness of H. Hence H ∼ 2B₀ +A^′, with B₀² = 6 and A^′ ∼ A−2∆ satisfies (A^′)² = 0 or 4. In the first case we are done by conditions (i) if A^′ is quasi-nef, and if not we can just repeat the process and find that S is nonextendable

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by Proposition 5.2. In the case (A^′)² = 4 we have A^′.B₀ ≥5 by Hodge index. Therefore (3B₀−H).B₀ = (B₀−A^′).B₀≤1< φ(B₀), so that if 3B₀−H >0, then it is a nodal cycle. Hence eitherh⁰(3B₀−H) = 0 or h⁰(3B₀+K_S−H) = 0 and S is nonextendable by Proposition 5.3. We have therefore shown that S is nonextendable under conditions (ii).

Now assume (iii). Set k := −A.∆. By [22, Lemma 2.3] we have that A₀ := A−k∆ is primitive, effective and isotropic. In case (iii- a) we deduce k = 2 and F₁.∆ = F₁.A₀ = 1. Then H ∼ 2B₀ +A₀ satisfies conditions (ii) andS is nonextendable. Now consider case (iii- b), so that F_i.A ≤ 6 for i = 1,2. If ∆.F₁ ≤ 0, then F₂.∆ ≥ 2. As 6≥F₂.A=F₂.A₀+kF₂.∆, we getk=F₂.∆ = 2, so that ∆.F₁ = 0 and F₂.A≥ 4. Then F₁.B₀ = 1, so that B₀ ∼ 2F₁ +F₂^′, where F₂^′ ∼F₂+

∆−F1>0 and (F₂^′)²= 0. AlsoF₁.A₀=F₁.A≤4, and equality implies F₂.A = 4, F₂ ≡ A₀ and the contradiction F₁.A₀ = F₁.F₂ = 1. Hence F₁.A₀≤3. MoreoverF₂^′.A₀= (F₂+ ∆−F₁).A₀ = (F₂−F₁).A−2≤2 and it cannot be that (F₁.A₀, F₂^′.A₀) = (1,1), for thenF₁.A= 1. Then H ∼2B₀+A₀ satisfies the conditions in (iii-a) andS is nonextendable.

We can therefore assume ∆.F₁ >0, and by symmetry, also ∆.F₂ >0.

Hence φ(B₀)≥2. Ifk≥3, thenF_i.A=F_i.A₀+kF_i.∆≥4 for i= 1,2, and we get k = 3, F_i.A = 4 and F_i.∆ = F_i.A₀ = 1. Then B.A = 8 and H² = 40, so that φ(H)≤5 by hypothesis. LetF be isotropic with F.H =φ(H). Now (A^′)²= 4 and 5≥F.H = 2F.B₀+F.A^′ ≥5, so that F.H = 5, F.A^′ = 1, (A^′ −2F)² = 0, A^′−2F > 0 and (A^′ −2F).H = (A−2∆−2F).H = 4, a contradiction. Hence k = 2, A²₀ = 0 and B₀.A₀ = (B+ ∆).A₀ ≥3. Then the conditions (ii) are satisfied and S is nonextendable, unless possibly if B₀² = 4 and B.A₀ = 1. But then B.∆ = 2 and A₀ ≡F_i, fori= 1 or 2. Hence ∆.B= ∆.(F₁+F₂) = 3, a

contradiction. q.e.d.

We also have the following helpful tools to check surjectivity ofµ_V_D_,ω_D when h¹(H−2D₀)6= 0. The first lemma holds on any smooth surface.

Lemma 5.6. Let S be a smooth surface, L a line bundle on S and D₁>0 andD₂>0divisors on S not intersecting the base locus of |L|, such that h⁰(OD1) = 1 and h⁰(OD1(−L)) = h⁰(OD2(−D1)) = 0. For any divisor B > 0 on S set V_B := Im{H⁰(S, L) → H⁰(B, L_|B)}. If µ_V_D

1,ωD1 and µ_V_D

2,ωD2(D1) are surjective, then µ_V_D_,ω_D is surjective for general D∈ |D₁+D₂|.

Proof. LetD^′ =D₁+D₂. We have two surjective maps π_i :V_D′ → V_D_i, fori= 1,2, and an exact sequence

0 ^/^/H⁰(ω_D₁) ^/^/H⁰(ω_D^′) ^ψ ^/^/H⁰(ω_D₂(D₁)) ^/^/0,