Special Ulrich bundles on regular surfaces with non–negative Kodaira dimension

Let S be a regular surface endowed with a very ample line bundle OS(hS)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal O_S(h_S)$$\end{document}. Taking inspiration from a very recent result by D. Faenzi on K3 surfaces, we prove that if OS(hS)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal O}_S(h_S)$$\end{document} satisfies a short list of technical conditions, then such a polarized surface supports special Ulrich bundles of rank 2. As applications, we deal with general embeddings of regular surfaces, pluricanonically embedded regular surfaces and some properly elliptic surfaces of low degree in PN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {P}^{N}$$\end{document}. Finally, we also discuss about the size of the families of Ulrich bundles on S and we inspect the existence of special Ulrich bundles on surfaces of low degree.


Introduction
Let P N be the projective space of dimension N over an algebraically closed field k of characteristic 0. If X ⊆ P N is a variety, i.e. an integral closed subscheme, then it is naturally endowed with the very ample line bundle O X (h X ) := O P N (1) ⊗ O X . We say that a sheaf E on X is Ulrich (with respect to O X (h X )) if h i X, E(−ih X ) = h j X, E(−( j + 1)h X ) = 0, for each i > 0 and j < dim(X ).
Ulrich bundles on a variety X have many properties: we refer the interested reader to [23], where the authors also raised the following questions. Questions. Is every variety (or even scheme) X ⊆ P N the support of an Ulrich sheaf? If so, what is the smallest possible rank for such a sheaf?
When C is a curve, i.e. a smooth variety of dimension 1, the above questions have very easy answers: indeed if g is the genus of C and L ∈ Pic g−1 (C) satisfies h 0 C, L = 0, then L(h C ) is an Ulrich line bundle.
At present, no general answers to the above questions are known when X has dimension greater than 1, though a great number of partial results have been proved: without any claim of completeness, we recall [2,9,13,14,18,22,[24][25][26]31,32,38,39]. The interested reader can also refer to the recent survey [10] for further results. In this paper we study the case of surfaces, i.e. smooth varieties of dimension 2. In particular, following the argument used by D. Faenzi for K 3 surfaces in [24] we deal with surfaces S with q(S) = 0, partially extending analogous results proved in [9,13,14] when q(S) = p g (S) = 0.
In order to state our main result, we quickly recall a few facts and definitions. Recall that a coherent sheaf G on S is called simple if Hom S G, G ∼ = k. There exists a (possibly non-separated) coarse moduli space Spl S (r ; c 1 , c 2 ) parameterizing simple coherent torsion-free sheaves on S with fixed rank r and Chern classes c 1 , c 2 (see [1]).
Moreover, if we set μ(G) := c 1 (G)h S /rk(G), the coherent torsion-free sheaf G is μ-stable (with respect to O S (h S )) if μ(K) < μ(G) for each subsheaf K with 0 < rk(K) < rk(G). There exists a quasi-projective scheme M S (r ; c 1 , c 2 ) parameterizing μ-stable coherent torsion-free sheaves on S with fixed rank r and Chern classes c 1  Finally recall that a line bundle L is non-special if h 1 S, L = 0.
We are now able to state the main result of the paper. If h 2 S + 4 ≥ h S K S , then S supports special simple Ulrich bundles E of rank 2. If h 2 S > h S K S and the complement S 0 of the union of smooth rational curves is dense in S, then the aforementioned bundle E can be chosen μ-stable.
The point corresponding to E in either Spl S (2; c , is smooth and lies in a unique component of dimension h 2 S − K 2 S + 5χ(S). Before dealing with the structure of the paper we make some comments about the hypothesis in the statement above.
If p g (S) = 0 the existence of special Ulrich bundles on S has been proved with the same construction in [13] without the hypothesis h 0 S, Nevertheless, the proof of the μ-stability of E given here cannot be extended to the case p g (S) = 0 when the Kodaira dimension κ(S) is negative, because it rests on the hypothesis S 0 = ∅, which necessarily imply κ(S) ≥ 0.
The μ-stability of E when κ(S) = −∞ has been proved in [14] with a different argument, under the additional hypothesis that k is uncountable.
A priori, it could happen that S 0 is not open even when κ(S) ≥ 0. E.g., in [5] the authors give an example of a K 3 surface containing infinitely many smooth rational curves.
In particular, we cannot deduce the μ-stability of E without further information. E.g. if S is a surface of general type, i.e. κ(S) = 2, then S contains at most a finite number of smooth rational curves by [36], hence S 0 is certainly dense in this case.
We now deal with the content of the paper. In Sect. 5 we prove the above theorem via a finite induction. We first construct a vector bundle of rank 2 such that c 1 (F) = h S + K S , h 1 S, F = 0 and h 0 S, F = p g (S) using standard techniques (see Sect. 3). Then, via a classical result due to I.V. Artamkin (see [3]), we construct inductively a sequence of vector bundles As pointed out above, it follows that E := F p g (S) (h S ) is a special Ulrich bundle of rank 2: in the paper we show that it has all the properties listed in the statement of Theorem 1.1.
It is natural to ask whether surfaces as in Theorem 1.1 actually exist. On the one hand, each surface S with q(S) = 0 can be endowed with many very ample line bundles satisfying the hypothesis of Theorem 1.1 (see Example 6.1). On the other hand, there are several interesting polarized surfaces satisfying the hypothesis of Theorem 1.1 besides the case of K 3 surfaces described in [24]. For instance, each regular surface of general type S ⊆ P N with O S (h S ) ∼ = O S (nK S ) for an n ≥ 3 (see Example 6.2). Some properly elliptic surfaces in P 4 provide a second interesting example (see Example 6.4).
In Sect. 7 we make some comments about the size of families of Ulrich bundles on the surfaces described in Examples 6.1, 6.2 and 6.4.
Finally, in Sect. 8, we deal with surfaces of low degree in P N , extending some results from [15].

Some preliminary facts
In this section we list some results which will be used in the paper. For all the other necessary results we refer the reader to [28].
If G and H are coherent sheaves on the surface S, then the Serre duality holds (see [29,Proposition 7.4]: see also [6]). Thus q(S) If G is a vector bundle on S, then the Riemann-Roch theorem for G on S is (2.3) We finally recall the Cayley-Bacharach construction of vector bundles on S. Let Z ⊆ S be a 0-dimensional locally complete intersection subscheme and let L ∈ Pic(S). Recall that Z satisfies the Cayley-Bacharach condition with respect to L if To prove this assertion we first notice that It follows that D = A, because they have the same degree with respect to any very ample line bundle on S, i.e. the class of each D ∈ D in the Néron-Severi group NS(S) contains exactly one smooth rational curve.
We deduce from the above discussion that S 0 is certainly non-empty and dense if k is uncountable (see [28,Exercise V.4.15 (c)]). Nevertheless, as pointed out in the introduction, in many cases S 0 is actually open and non-empty, without any restriction on the cardinality of k (see [36]).

The base case
As explained in the introduction, the proof of Theorem 1.1 is by induction. In this section we deal with its base case.
Let S be a surface with q(S) = 0 and O S (h S ) a non-special very ample line bundle.
In particular O S (h S ) induces an embedding S ⊆ P N . Let X be any scheme. In what follows X [N +2] denotes the Hilbert scheme of 0-dimensional subschemes of degree N + 2 inside X . Since S is integral and non-degenerate in P N , it follows the existence of an open non-empty subset Z ⊆ S [N +2] whose points correspond to schemes Z of N + 2 points in general linear position inside P N . Each scheme Z corresponding to a point in Z satisfies the hypothesis of Theorem 2.1 with respect to O S (h S ), hence there is a rank 2 vector bundle F fitting into Notice that Thus the cohomology of the exact sequence i.e. F is uniquely determined by Z .
for the vector bundle F obtained from a scheme Z as in Construction (3.1).
In the next proposition we deal with the properties of the point corresponding to F in the moduli spaces Spl S (2; To this purpose, we denote by Z 0 the open subset of Z of points corresponding to schemes Z such that h 0 S, ∩Z. As pointed out in Remark 2.3, Z 1 could be empty, but if k is uncountable and κ(S) ≥ 0, then it is certainly dense inside S [N +2] .

Proposition 3.3. Let S be a surface with q(S) = 0 and p g (S) ≥ 1, endowed with a non-special very ample line bundle
Then the following properties hold for the vector bundle F obtained from a scheme Z ∈ Z as in Construction (3.1) Proof. In order to prove assertion (1), applying Hom S F, − to Sequences (3.3) tensored by O S (h S ) and (3.2), taking into account of Lemma 3.2 we obtain Tensoring the cohomology of Sequence The choice of Z and the hypothesis p g (S) ≥ 1 imply Let us prove assertion (2). The choice of Z implies h 0 S, The cohomology of the same exact sequence tensored by Assertion (2) The sheaf M is trivially torsion-free and it is also normal (see [ The Nakai criterion then implies We claim that Ah S ≥ N + 1. Assuming the claim, Equality (3.1) yields a contradiction. We deduce that a sheaf M as above does not exist, hence F is μ-stable. It remains to prove the claim. To this purpose, let C 1 , . . . , C s be the integral components of A intersecting Z and B their union. Since Z ⊆ S 0 , it follows that : by combining these remarks with an easy induction on s we then deduce On the one hand, the cohomology of Thus the Riemann-Roch theorem for the curve B and Inequality (3 We then deduce that Ah S ≥ Bh S ≥ N + 1, hence the claim is proved and the proof of assertion (3) is complete.
Remark 3.4. Construction 3.1 makes sense also in the case p g (S) = 0. Indeed it is the method we used in [13,14] for proving the existence of special Ulrich bundles when p g (S) = q(S) = 0. Notice that in this case the inequality h 2 S > h S K S is for free.
On the one hand, the proof of Lemma 3.2 can be carried over word by word to this case. On the other hand, the proof of assertion (1) of Proposition 3.3 cannot be extended to the case p g (S) = 0. Moreover, the proof of assertion (3) is alternative to [13,14, Theorem 1.2] when Z 1 = ∅: in particular, it certainly needs κ(S) ≥ 0.
One of the hypothesis of [13,14, Theorem 1.2] is that k is uncountable. Thus, the above proof and [36] extend such a result when κ(S) = 2 also to the case of a countable base field. When p g (S) = 0 and κ(S) ≤ 1, as in the case p g (S) ≥ 1, the condition Z 1 = ∅ is not immediate: e.g. there exist Enriques surfaces containing infinitely many rational curves (see [19]).

The inductive step
In this section we explain the inductive step of the proof of Theorem 1.1.

Construction 4.1. Let S be a surface endowed with a very ample line bundle
Each ϕ as above is surjective and we have the exact sequence where F ϕ := ker(ϕ). Notice that for the sheaf F ϕ obtained from F as in Construction (4.1). To this purpose there exists a non zero s ∈ H 0 S, F . Its zero locus is the union of a 0-dimensional scheme X and a divisor E on S. In particular we have an exact sequence of the form we obtain the exact sequence We have because we compute them locally on X and p / ∈ X ∪ E. The exact sequence of the low degree terms of the spectral sequence The commutativity of the diagram then implies that the same is true for ϕ.

Proposition 4.3. Let S be a surface endowed with a very ample line bundle O S (h S ).
Assume that F is a vector bundle of rank 2 with h 0 S, F ≥ 1.
Then the following properties hold for a sheaf F ϕ obtained from F as in (4.1) Proof. In order to prove assertion (1), we notice that the quotient map O S → O p induces by duality an inclusion 0 Thus the functor Hom S O p , − applied to Sequence (4.1) we obtain On the one hand, each map in Hom S F, F ϕ induces a map in Hom S F, F by composing with the inclusion F ϕ ⊆ F: it follows that the composed map is never surjective. On the other hand, each non-zero element in Hom S F, F must be a homothety because F is simple, hence it is an isomorphism. We deduce that Hom S F, F ϕ = 0.
Thus we obtain Hom S F ϕ , F ϕ ⊆ Ext 1 S O p , F ϕ , by applying Hom S (−, F ϕ to Sequence (4.1). If we combine this inclusion with Equality (4.3) and (4.4), we finally obtain that F ϕ is simple, i.e. assertion (1) is proven.
We prove assertion (2). Since F is a vector bundle, the following obvious equalities hold, hence Ext 2 S F, F ϕ ∼ = Ext 2 S F, F , by applying Hom S F, − to Sequence (4.1). Equality (2.1) and Hom S F ϕ , − applied to Sequence (4.1) tensored by The statement follows from the above inclusion and Inequality (2.2). Consider now assertion (3). Since F ϕ ⊆ F and μ(F ϕ ) = μ(F), it follows that each subsheaf destabilizing F ϕ also destabilizes F. Thus assertion (3) is proven.

The proof of Theorem 1.1
In this section we put together the above results with the following classical theorem for proving Theorem 1.1 stated in the introduction.

1) implies that E is Ulrich if and only if
Let F be the bundle defined in Construction 3.1. If p g (S) ≥ 1, then E := F(h S ) satisfies all the conditions for being a special Ulrich bundle, but the vanishing h 0 S, E(−h S ) = h 0 S, F = 0. Nevertheless, F can be viewed as a good approximation of a special Ulrich bundle. For this reason we introduce the following definition. As explained above, if F is a 0-good bundle, then E := F(h S ) is a special Ulrich bundle of rank 2.
If q(S) = 0, then the bundle F obtained from a scheme Z as in Construction 3.1 is p g (S)-good by Lemma 3.2: in particular if p g (S) = 0, then F(h S ) is a special Ulrich bundle.
We are now able to prove Theorems 1.1 stated in the introduction.
Proof of Theorem 1.1.. We will prove the statement by descending induction, showing for each d = p g (S), . . . , 0 the existence of a simple (resp. μ-stable) . For the base step let F p g (S) be the bundle F defined in Construction (3.1) from Z ∈ Z (resp. Z ∈ Z 1 ) which is p g (S)-good and simple when h 2 S + 4 ≥ h S K S (resp. μ-stable when h 2 S > h S K S ), thanks to Lemma 3.2 and Proposition 3.3. Now let p g (S) ≥ d ≥ 1 and assume the existence of a simple (resp. μ-stable) . It follows from Construction 4.1 the existence of a simple (resp. μ-stable) (d − 1)-good sheaf F ϕ with dim Ext 2 S F ϕ , F ϕ = p g (S), thanks to Lemmas 3. Since F d satisfies the hypothesis of Theorem 5.1 and F ϕ is locally free on S \ { p }, up to shrinking B we can assume that F b is locally free at p for each The smoothness of B implies that the tangent space at b ∈ B has dimension

Examples
In this section we give some applications of Theorem 1.1. We first show that Ulrich bundles are quite common on regular surfaces. Then we give examples of polarized surfaces fulfilling the hypothesis of Theorem 1.1. Since H is ample, it follows that there is n 1 such that n 1 H is very ample and non-special. Again the ampleness of H implies H 2 ≥ 1, hence there is n 2 such that An example of the above set up is the case of an elliptic fibration with a section: in this case there are trivially no multiple fibres. This case has been inspected in [39] with a more direct approach.
In [24] the author proved the existence of special Ulrich bundles of rank 2 on each polarized K 3 surface, extending some earlier results (see [2,18]). The case of Enriques surfaces was examined in [9,13]. We list below some other interesting examples. Indeed, in this case Z 1 = ∅ (see [36]) and K S is ample, hence S is minimal: then h 1 S, O S (h S ) = 0 thanks to [7,Proposition VII.5.3]. Moreover, the conditions h 2 S > h S K S and h 0 S, O S (2K S − h S ) = 0 are trivially satisfied in this case. The above result cannot be extended to the cases 1 ≤ n ≤ 2 using the same argument. Indeed, h 0 S, O S (2K S − h S ) ≥ 1 in these cases. In particular, we cannot apply Theorem 5.1 in the base case of the induction in the proof of Theorem 1.1.
Notice that the case n = 2 could be within reach: indeed it would be sufficient to check that h 0 S, F ⊗ I Z |S < h 0 S, F in the proof of Proposition 3.3, because h 0 S, O S (2K S − h S ) = 1 (for some similar results in this direction see Example 8.4).
On the other hand, the case n = 1 seems to be out of reach with our methods, It is classically known (see [11] and the references therein) that non-degenerate surfaces S of degree d = 2N − 2 + s in P N are geometrically ruled or K 3 when s ≤ 0. In [11] the author gives a description of surfaces with 1 ≤ s ≤ N − 3 when O S (h S ) is non-special. In all these cases s ≡ h S K S (mod 2), Then S supports special Ulrich bundles of rank 2 with respect to O S (h S ). Since k ∼ = C, it follows that Z 1 = ∅ hence such bundles can be taken μ-stable. The classification given in [11, Propositions 2.1, 2.2, 2.3] and [11, Lemma 1.10] imply that S is always regular in these cases, hence the assertion follows from [13,14] if p g (S) = 0. Thus, from now on, we will assume p g (S) ≥ 1. Notice that the inequality = 0 thanks to the Nakai criterion. In both the cases the assertion then follows from Theorem 1.1.
Example 6.4. Let S ⊆ P N be a non-degenerate regular surface with κ(S) ≤ 1, N ≥ p g (S)−2 and such that O S (h S ) is non-special. Then S supports simple special Ulrich bundles of rank 2 with respect to O S (h S ). If Z 1 = ∅ and N ≥ p g (S) + 1 such bundles can be taken μ-stable.
Indeed, in this case, we already know that The inequalities N ≥ p g (S) − 2 and N ≥ p g (S) + 1 are respectively equivalent to h 2 S + 4 ≥ h S K S and h 2 S > h S K S . Thus the assertions follows from Theorem 1.1. As a more concrete example, assume that k ∼ = C and let S ⊆ P 4 be a surface with κ(S) = 1 and degree d = 7, 8. In [41,42] it is shown that the surfaces S are exactly the ones linked to a plane in either a quadro-quartic or a cubo-cubic complete intersection inside P 4 .
The results in [44] imply that the minimal free resolutions of I S|P 4 look like respectively. The cohomology of the exact sequence In both the cases S is determinantal. Nevertheless, S is not defined by a matrix with linear entries, hence we cannot use the results proved in [31,32] for deducing the existence of an Ulrich bundle.
Similarly, we cannot use the results in [39]. Indeed, in [35, Theorem III.4.2 and Observation III.3.5], the author proves that for a surface S as above which is also very general, then Pic(S) is generated by h S and K S . If the canonical map ψ : S → P 1 has a section σ : P 1 → S, then C := im(σ ) is a rational curve linearly equivalent to xh S + y K S for some integers x, y. Since C is the image of a section of the canonical map, it follows that 1 = C K S = xh S K S which is impossible, because the last number is a multiple of 3 or 4 (according with the two cases h 2 S = 7, 8).

Ulrich-wildness
In this very short section we deal with the size of the families of Ulrich bundles supported on the surfaces we are interested in.
A variety X is called Ulrich-wild if it supports families of dimension p of pairwise non-isomorphic, indecomposable, Ulrich sheaves with respect to O X (h X ) for arbitrary large p.

Lemma 7.1. Let X be a smooth variety endowed with a very ample line bundle
If E is a simple Ulrich bundle on X such that dim Ext 1 S E, E ≥ 3, then X is Ulrich-wild.
Proof. It is an immediate consequence of [26, Theorem A, Corollary 2.1 and Remark 1.6 v)], because every Ulrich bundle is semistable by [12,Theorem 2.9] and each non-zero automorphism of a simple sheaf is an isomorphism.
We deduce the following criterion for the surfaces we are interested in. Proof. Consider the bundle F defined in Construction (3.1). We know from Proposition 3.3 that F is simple and dim Ext 1 S F, F = p g (S). Then, Equality (2.3) yields The same argument applied to the Ulrich bundle E defined in Theorem 1.1, then yields Thus the statement follows immediately from Lemma 7.1.
The above proposition extends [13,Theorem 1.3] to the case p g (S) ≥ 1. Then S is Ulrich-wild.
Proof. The statement is a trivial consequence of Proposition 7.2.

Ulrich bundles on surfaces of low degree
In [15, Theorems 1.4 and 1.5] the author deals with the existence of special Ulrich bundles on surfaces of low degree on surfaces S ⊆ P N . We start this section by improving [15,Theorem 1.4]. We work over the complex field C, hence S 0 is dense for each surface S with κ(S) ≥ 0 by Remark 2.3.
Then the following assertions holds.
(2) If κ(S) = 2 and S is general, then S supports special Ulrich bundles.
Proof. Thanks to [15,Theorem 1.4] we know that if κ(S) = 1, then S supports Ulrich bundles E of even rank r sp Ulrich as in [15, Table A] and such that In particular r sp Ulrich = 2 if either κ(S) ≤ 0 or κ(S) = 2 and S is general. Thus such an Ulrich bundle E is special in these cases.
If κ(S) = 1, taking into account of the classification in [15, Table A] and the results in [41,42], we know that S ⊆ P 4 is a properly elliptic surface of degree either 7 or 8. The existence of special Ulrich bundles on such surfaces has been proved in Example 6.4.
We now extend [15,Theorem 1.5]. Here π(S) denotes the genus of a general plane section of S.  [13,14], because S is neither a rational normal scroll nor a plane by degree reasons. In what follows we will briefly deal with the case p g (S) ≥ 1. We The so-called Severi Theorem (see [46] Moreover, the adjunction formula returns h S K S = 2π(S) − 11. By combining such two equalities we obtain χ(O S ) = π(S) − 5. Finally, the inequality 9 > h S K S forces π(S) ≤ 9.
Thanks to the above discussion and to the classification in [4, Theorem 0.1 and its proof] we have to deal only with the following cases.
• A minimal properly elliptic surface S with h S K S = 3, K 2 S = 0, p g (S) = 1. • A minimal surface S of general type with h S K S = 5, K 2 S = 1, p g (S) = 2. • A surface S linked with a possibly singular/reducible cubic scroll Y via a cuboquartic complete intersection: in this case h S K S = 7, K 2 S = 2, p g (S) = 3. In the first case case we trivially have h 0 S, O S (2K S − h S ) = 0, hence Theorem 1.1 yields the existence of μ-stable special Ulrich bundles on S.
In the remaining cases the vanishing h 0 S, O S (2K S − h S ) = 0 is not evident as above. E.g., let us examine the second case.   [4,Section (2.13)] the authors show that the general hyperplane section C of the surface Y linked to S is aCM with p a (C) = 0. This fact and the Riemann-Roch theorem on C implies that the ideal of C, hence of Y , is generated by the minors of a 3 × 2 matrix of linear forms. Thus there exists an exact sequence of the form Thanks to [44] there is a resolution of the form If q(S) = p g (S) = 0, then S is not a rational normal scroll by degree reasons, hence it supports special Ulrich bundles which are μ-stable from [13,14].
From now on we will assume that q(S) and p g (S) do not vanish simultaneously. The argument of Example 8.3 yields h S K S = 2χ(O S ) and h S K S = 2π(S) − 12, hence χ(O S ) = π(S) − 6 and π(S) ≤ 10. Thus the results in [45, Theorem 0.1 and its proof in Section 9] lead us to deal only with the following cases.
• An abelian surface. where the A i 's are disjoint curves such that A 2 i = −2 and A i K S = 0. The existence of special Ulrich bundles on Abelian or bielliptic surfaces has been proved in [8,10]. Such bundles can be taken μ-stable: see [8] for abelian surfaces and [16,17] for bielliptic surfaces.
In the other cases h 0 S, O S (2K S − h S ) = 1, hence Theorem 1.1 does not give any information on the existence of special Ulrich bundles on such an S. hence the condition h S K S < h 2 S = d yields χ(O S ) ≤ 4. The double point formula (see [28], Example A. 4

.1.3) is
The Hodge index theorem for the divisors h S , K S , Equality (8.2) and the hypothesis h S K S < d then yields d ≤ 12.
Taking into account the bound 1 ≤ χ(O S ) ≤ 4, computing K 2 S , h S K S , applying again the Hodge index theorem for the divisors h S , K S and Equality (8.2) one easily checks that the case d = 12 cannot occur. If d = 11, the same argument yields the χ(O S ) = 4, K 2 S = 7, h S K S = 9. If S is not minimal, then K S = K 0 + E where K 0 E = 0, K 2 0 ≥ 8 and h S K 0 ≤ 8. The Hodge index theorem for the divisors h S , K 0 , then yields a contradiction. It follows that S is minimal. If q(S) ≥ 1, then p g (S) ≥ 4, hence we should have K 2 S ≥ 8, thanks to [21, Théorè eme 6.1], a contradiction. We deduce that S is minimal and regular. We