aa r X i v : . [ m a t h . AG ] J un CALABI–YAU THREEFOLDS IN P GRZEGORZ KAPUSTKA, MICHA L KAPUSTKA
Abstract.
We study the geometry of 3-codimensional smooth subvarieties of the complex pro-jective space. In particular, we classify all quasi-Buchsbaum Calabi–Yau threefolds in projective6-space. Moreover, we prove that this classification includes all Calabi–Yau threefolds contained ina possibly singular 5-dimensional quadric as well as all Calabi–Yau threefolds of degree at most 14in P . Introduction
It is conjectured that, when 2 n ≥ N , there is a finite number of smooth families of smooth n -dimensional subvarieties of P N that are not of general type. This conjecture was inspired by[EP] where the statement was formulated and proven in the case of surfaces in P . In [BOS],the conjecture was proven in the case of threefolds in P . Moreover, Schneider in [Sch] provedthat the statement is true when 2 n ≥ N + 2. In this context, it is a natural problem to classifyfamilies of smooth n -folds of small degree in P N for chosen n, N ∈ N satisfying 2 n ≥ N . In thecase of codimension 2 subvarieties, this problem was addressed by many authors (see [BSS], [EFG],[DES],[AR]).The next step is to study codimension 3 subvarieties in P . In this case, standard tools such asthe Barth-Lefschetz theorem do not apply. However, some general structure theorems were recentlydeveloped. We say that a submanifolds X ⊂ P n is subcanonical when ω X = O X ( k ) for some k ∈ Z .A codimension 3 submanifold X is called Pfaffian if it is the first nonzero degeneracy locus of askew-symmetric morphism of vector bundles of odd rank E ∗ ( − t ) ϕ −→ E where t ∈ Z . In this case, X is given locally by the vanishing of 2 u × u Pfaffians of an alternating map ϕ from the vectorbundle E of odd rank 2 u + 1 to its twisted dual. More precisely, if X is Pfaffian then we have:(1.1) 0 → O P n ( − s − t ) → E ∗ ( − s − t ) ϕ −→ E ( − s ) → I X → s = c ( E ) + ut . Moreover, from [O, § ω X = O X ( t + 2 s − n − . Since the choice of an alternating map ϕ is equivalent to the choice of a section σ ∈ H ( V E ( t )),we will use the notation Pf( σ ) for the variety described by the Pfaffians of the map correspondingto σ .Answering a question of Okonek (see [O]), Walter, in [W], proved that if n is not divisibleby 4 then a locally Gorenstein codimension 3 submanifold of P n +3 is Pfaffian if and only if it issubcanonical. In the case when n = 4 k , the last statement is not true, however, there is anotherstructure theorem (see [EPW]).The non-general type subcanonical threefolds in P are either well understood Fano threefolds orthreefolds with trivial canonical class. A very natural class of varieties among varieties with trivialcanonical class are Calabi–Yau threefolds i.e. smooth threefolds X with K X = 0 and H ( X, O X ) =0. In the paper, we shall sometimes also consider singular Calabi–Yau threefolds by which we meancomplex projective threefolds with Gorenstein singularities, ω X = 0 and with h ( O X ) = 0. Mathematics Subject Classification.
Primary: 14J32.
Key words and phrases.
Calabi-Yau threefolds, Pfaffian varieties. or Calabi–Yau threefolds the theory of Pfaffians is more specific. For instance, Schreyer, fol-lowing [W] shows that if X is Pfaffian, h i ( O X ) = 0 for 0 < i < dim X and E is any vector bundlesuch that there exists σ ∈ H ( V E ( t )) with X = Pf( σ ) then, keeping the notation above, E ( − s )appears as a sum of the sheafified first syzygy module Syz ( HR ( X )) of the Hartshorne–Rao module HR ( X ) = ⊕ Nk =1 H ( I X ( k )) (seen as a S = C [ x , . . . , x ] module) and a sum of line bundles. If weadd the assumption that X has trivial canonical class then, by considering an appropriate twist,we can choose a bundle E such that t = 1 and s = 3. More precisely, if X is a Calabi–Yau threefoldthen there exists a bundle E of rank 2 u + 1 such that 3 = c ( E ) + u and X = Pf( σ ) for some σ ∈ H ( V E (1)). Moreover, if we denote by M the Hartshorne–Rao module of X with gradationshifted by 3 then the chosen bundle E is obtained as a sum of Syz ( M ) with a sum of line bundles.Let us point out that all threefolds can be smoothly projected to P . It is, moreover, knownfrom [BC, rem. 11] that Calabi–Yau threefolds embedded in P are complete intersections; eitherof two cubics, or of a quadric and a quartic, or of a quintic and a hyperplane. Having this inmind, we study nondegenerate Calabi–Yau threefolds i.e. such Calabi–Yau threefolds which arenot contained in any hyperplane.Nondegenerate Calabi–Yau threefolds in P were already studied in [R], [T], [Be], [K], and [KK1],where examples of degree 12 ≤ d ≤
17 were constructed. It is not hard to see that the degree ofsuch a threefold is bounded between 11 and 41 (see Corollary 2.2) but we expect a sharper bound(see [KK]). Okonek proposed the following problem:
Problem 1.1.
Classify the Calabi–Yau threefolds in P . The central result of the paper is a full classification of quasi-Buchsbaum Calabi–Yau threefoldsin P i.e. Calabi–Yau threefolds in P such that their higher cohomology modules have trivialstructure (see Definition 3.1). The classification is given in Theorem 3.2. The proof that thisclassification includes all Calabi–Yau threefolds in P of degree d ≤
14 and a classification ofCalabi–Yau threefolds contained in 5-dimensional quadrics is our main result. The classification isgiven by providing a list of vector bundles { E i } i ∈ I such that the considered Calabi–Yau threefoldsare exactly the smooth threefolds which appear as Pfaffians Pf( σ ) for some σ ∈ H ( V E i (1)) and i ∈ I . Let us point out that our list contains two distinct vector bundles corresponding to degree14 Calabi–Yau threefolds.In Section 2, we prove basic general results concerning the classification of Calabi–Yau threefoldsin P . In particular, we observe that a Calabi–Yau threefolds in P must be linearly normal. We,moreover, prove the finiteness of the classification problem 1.1.Theorem 3.2 is the main theorem of Section 3. It presents the classification of Calabi–Yauthreefolds that are quasi-Buchsbaum. As a consequence, we find that the examples that are arith-metically Cohen-Macaulay (see Definition 3.1) are of degrees 12 ≤ d ≤ Theorem 1.2. If ( X, Q r ) is a pair consisting of a nondegenerate Calabi–Yau threefold X ⊂ P ofdegree d X and a 5-dimensional quadric Q r of corank r in P such that X ⊂ Q r , then r ≤ and ≤ d X ≤ . For the proof, we consider case by case the possible coranks of the quadrics containing theCalabi–Yau threefolds. For low corank quadrics (i.e. r = 0 , ,
2) we consider hyperplane sectionsof our Calabi–Yau threefold and so work with canonically embedded surfaces of general type; see[Ca] for more information on such surfaces. For instance, on a smooth quadric in P containinga canonically embedded surface of general type S , we can apply the double point formula to getthe bound for the degree d S of S to be 12 ≤ d S ≤
14. Inputting an additional assumption on S ,stating that it is a section of some Calabi–Yau threefold X contained in a smooth quadric in P , eads to the result d X = d S = 12 or d X = d S = 14. For canonically embedded surfaces of generaltype contained in quadrics of rank 5 in P , we obtain the same bound 12 ≤ d ≤
14 working onthe resolution of this quadric. The latter resolution is the projectivization of a vector bundle ofrank 2. The last step is the proof that there are no Calabi–Yau threefolds contained in quadricsof corank ≥ P . It is worth noticing that there is no similar result for canonically embeddedsurfaces of general type contained in quadrics of corank ≥ P . In particular, we present, inPropositions 4.9, 4.10 examples of nodal Calabi–Yau threefolds contained in quadrics of rank 4which have degree 11 and 15. Their general hyperplane sections are canonical surfaces of respectivedegrees 11 and 15 which are contained in 4-dimensional quadrics of rank 4.The classification of Calabi–Yau threefolds of degree d ≤
14 in P in terms of vector bundlesassociated to them by the Pfaffian construction is completed in Sections 5 and 6. More precisely,we prove that all Calabi–Yau threefolds of degree at most 14 in P are quasi-Buchsbaum and usethe classification of the latter threefolds contained in Section 3 (see Theorem 3.2).By Theorem 1.2, the classification of Calabi–Yau threefolds of degree d ≤
14 in P provides alsoa classification of all Calabi–Yau threefolds contained in 5-dimensional quadrics. We, moreover,observe that there are two types of Calabi–Yau threefolds of degree d = 14. Calabi–Yau threefoldsof the first type are not contained in any quadric whereas Calabi–Yau threefolds of the second typeare.Finally, in Section 7, we perform a classification of Calabi–Yau threefolds of degree d ≤
14 in P up to deformation. Since this type of classification is weaker than the classification in terms ofvector bundles and stronger than the classification by degree, the only remaining ingredient is theproof that there is a unique maximal flat family of Calabi–Yau threefolds of degree 14. Throughoutthe paper we study three families of Calabi–Yau threefolds of degree 14. The first is the family C of degree 14 Calabi–Yau threefolds contained in a smooth quadric Q . To define it we think ofthe smooth 5-dimensional quadric Q as a homogenous variety with respect to the standard actionof the simple Lie group G . Then Q admits a natural bundle C called a Cayley bundle which ishomogeneous with respect to this action. The family C is the family of all smooth threefoldsappearing as zero loci of sections of a twist C (3). To confirm that the family is not complete,we compute that these threefolds have more deformations then obtained by varying the section of C (3). In fact, by Corollary 5.4, we deduce that C is part of a larger family B of threefolds givenas Pfaffian varieties associated to the bundle E = Ω P (1) ⊕ O P (1). Then we prove a technicalresult (Proposition 7.2) on deformation of Pfaffian varieties implying that any threefold B ∈ B appears as a smooth degeneration of the family T of Calabi–Yau threefolds defined by 6 × × Corollary 1.3.
There is one family of Calabi–Yau threefolds in P in each degree d ≤ B and B of degrees 14 and 15 respectively that we consider in thispaper were already constructed in [Be]; however, our results stay in contradiction with [Be, Prop4.3.] and the results in subsections 4.2.2 and 4.2.3 therein. In particular, we prove that both theexamples of degree 14 and 15 are flat deformations of families of Calabi-Yau threefolds constructedin [T]. This means that if to each member of a family of deformations of Calabi-Yau threefoldsin P we associate the minimal degree of hypersurfaces containing it then, unlike in the case ofcomplete intersections, this number can drop for special members. We, moreover, prove that theexamples of degree 15 constructed in [Be] are not smooth but admits three ordinary double points. cknowledgments We would like to thank Ch. Okonek for initiating the project, motivation, and help. We wouldlike to thank L. Gruson and C. Peskine for mathematical inspiration. We would also like to thankA. Bertin, J. Buczynski, S. Cynk, D. Faenzi, P. Pragacz for helpful comments and the referee forhelpful remarks and suggestions. The first author was supported by MNSiW, N N201 414539, thesecond by the Forschungskredit of the University of Zurich and Iuventus Nr IP2011 005071 “Ukladylinii na zespolonych rozmaitosciach kontaktowych oraz uogolnienia”.2.
Preliminaries
Let us first discuss some general properties of Calabi–Yau threefolds embedded in P . We callsuch a threefold nondegenerate if it is not contained in any hyperplane. The degenerate Calabi–Yauthreefolds (those which are not nondegenerate) in P are known to be complete intersections either X , ⊂ P ⊂ P or X , ⊂ P ⊂ P or X ⊂ P ⊂ P (see [BC, rem. 11]). Proposition 2.1.
Let X ⊂ P be a Calabi–Yau threefold; then X is linearly normal i.e. the naturalrestriction map H ( O P (1)) → H ( O X (1)) is surjective.Proof. It follows from [F, thm. 2.1] that there are only three families of non-linearly normal three-folds in P . These families have degrees 6 , (cid:3) Next, we show an a priori bound on the degree of the Calabi–Yau threefolds contained in P . Corollary 2.2.
The degree d of a nondegenerate Calabi–Yau threefold X ⊂ P is bounded between11 ≤ d ≤ Proof.
Observe that a generic hyperplane section S ⊂ X is a canonically embedded surface ofgeneral type. It was already remarked in [T] that using the Castelnuovo inequality for surfaces ofgeneral type we can deduce that d ≥
11. Next, from the Riemann–Roch theorem for line bundleson X and Proposition 2.1, we deduce that 7 = h ( O X (1)) = χ ( O X (1)) = S.c ( X ) + d . It isa classical result on Calabi–Yau threefolds, contained in [M], that H.c ( X ) ≥ H on X . Thus we infer 7 ≤ d .Moreover, from [M] we also know that, for any ample divisor H , we have H.c ( X ) = 0 if andonly if X is a finite ´etale quotient of an abelian threefold (this implies in particular c ( X ) = 0).Let us now show that d = 42 is impossible. By the above, in this case, it is enough to considerCalabi–Yau threefolds with trivial c ( X ). Those were classified in [OS, thm. 0.1]. There are twopossibilities and in each of them we have χ top ( X ) = 0. On the other hand, from the double pointformula (cf. [T]) we get(2.1) χ top ( X ) = − d + 49 d −
588 = − (42) + 49 · − = 0 . We thus obtain a contradiction proving that d = 42 and in consequence d ≤ (cid:3) Remark 2.3.
It is a natural problem to ask if other smooth threefolds with K X = 0 (i.e. withoutassuming that h ( O X ) = 0) can be embedded in P . Note that, in [V], it is proven that there areno Abelian threefolds in P .Corollary 2.2 implies, in particular, that there is a finite number of families of Calabi-Yau three-folds in P ; three families of degenerate examples and a finite number of nondegenerate. Indeed, bythe Riemann–Roch theorem, H.c ( X ) and the degree H determine the Hilbert polynomial of a po-larized Calabi–Yau threefold ( X, H ). Moreover, if (
X, H ) is a Calabi–Yau threefold in P polarizedby its hyperplane section then, again by the Riemann–Roch theorem, H.c ( X ) is determined by H and h ( O X ( H )) = 7. It follows that the Hilbert polynomial of X is determined by the degree f X . Hence, in each degree, there is a finite number of families. This means that having a boundon the degree implies finiteness.A slightly sharper bound on the degree could be obtained for Calabi–Yau threefolds withrk Pic( X ) = h , ( X ) = 1. In this case, using the double point formula 2.1, we obtain(2.2) 2 ≥ h , ( X ) − h , ( X )) = χ top ( X ) = − d + 49 d − . We then infer that either d ≤
21 or d ≥
28. Let us, however, point out that there exist examplesof Calabi–Yau threefolds X ⊂ P with h , ( X ) > Quasi-Buchsbaum Calabi–Yau threefolds
Recall the following definitions.
Definition 3.1.
Let X ⊂ P n be a subvariety in a projective space. Let us, for each i ∈ N ≥ , denoteby H i ∗ ( I X ) the i -th cohomology module L j ∈ Z H i ( P n , I X ( j )). We say that X is arithmeticallyCohen–Macaulay (aCM for short) if and only if H i ∗ ( I X ) = 0 for 1 ≤ i ≤ dim( X ) −
1. Moreover, X is called quasi-Buchsbaum if and only if, for 1 ≤ i ≤ dim( X ) −
1, we have: H i ∗ ( I X ) is annihilatedby the maximal ideal of the structure ring of P n . Finally, X is arithmetically Buchsbaum if eachof its linear sections is quasi-Buchsbaum.It is part of the mathematical folklore that the aCM Calabi–Yau threefolds in P are only theones listed in [T] up to degree 14. However, since we have not found a proper proof of this fact inthe literature, we provide it below as a consequence of a more general result which will be importantfor the rest of the paper. More precisely, we provide a classification of all quasi-Buchsbaum Calabi–Yau threefolds in P . In particular, this also gives a classification of all arithmetically BuchsbaumCalabi–Yau threefolds in P . Theorem 3.2.
Let X be a quasi-Buchsbaum Calabi–Yau threefold in P . Then X = P f ( ϕ ) forsome ϕ ∈ H ( P , V E (1)) where E is a vector bundle such that one of the following holds: (1) E = L u +1 i =1 O P ( a i ) with: (a) u = 1 , a = − , a = 2 , a = 2 , and X is a complete intersection of type , , ; (b) u = 1 , a = − , a = 1 , a = 2 , and X is a complete intersection of type , , ; (c) u = 1 , a = 0 , a = 0 , a = 2 , and X is a complete intersection of type , , ; (d) u = 1 , a = 0 , a = 1 , a = 1 , and X is a complete intersection of type , , ; (e) u = 2 , a i = 0 , for i ∈ { . . . } , a = 1 , and X is a degree 13 Calabi–Yau threefolddescribed in [T] ; (f) u = 3 , a i = 0 , for i ∈ { . . . } , and X is a degree 14 Calabi–Yau threefold described in [T, R] ; (2) E = Ω P (1) ⊕ L v +1 i =1 O P ( a i ) with: (a) v = 0 , a = 1 , and X is a degree 14 Calabi–Yau threefold from the family B describedin [Be] (see also Example 5.3); (b) v = 1 , a = 0 , a = 0 , a = 0 , and X is a degree 15 Calabi–Yau threefold described in [T] .Proof. Take X a quasi-Buchsbaum Calabi–Yau threefold in P . Then, by definition, the Hartshorne–Rao module HR ( X ) = H ∗ ( I X ) is annihilated by the maximal ideal of the structure ring of P . Itfollows that the resolution of HR ( X ) is given by a direct sum of twisted Koszul complexes. Thusthe sheafification Syz ( HR ( X )) of the first syzygy module of HR ( X ) is L n b i =1 Ω ( b i −
2) for some b . . . b n b ∈ Z . We, moreover, claim that b i ≤ i = 1 . . . n b . Indeed, from Proposition 2.1 andthe exact sequence: 0 → I X ( q ) → O P ( q ) → O X ( q ) → . e have H ( I X ( q )) = 0 for q ≤
1. To prove the claim, it is now enough to observe that in theabove notation we have H ( I X (2 − b i )) > i = 1 . . . n b .Let, now, M be the module obtained by shifting the gradation in the Hartshorne–Rao module HR ( X ) by 3. As observed in the introduction, there exists a vector bundle E = Syz ( M ) ⊕ L n a j =1 O P ( a i ) with a . . . a n a ∈ Z and a section ϕ ∈ H ( P , V E (1)) such that X = P f ( ϕ ).Without loss of generality, we assume that E is a bundle that has minimal rank among bundlesfor which there exists such a ϕ ∈ H ( P , V E (1)) that X = P f ( ϕ ). The rank of the bundle E is2 u + 1 for some u ∈ N . Observe that there is a decomposition E = n b M i =1 Ω ( b i + 1) ⊕ n a M j =1 O P ( a i ) . We fix such a decomposition for the rest of the proof. The bundles appearing in this decomposition,treated both as subbundles and as quotient bundles, will be called components of E . We also havean induced decomposition E ∗ ( −
1) = n b M i =1 (Ω ( b i + 1)) ∗ ( − ⊕ n a M j =1 O P ( − a i −
1) = n b M i =1 (Ω (5 − b i )) ⊕ n a M j =1 O P ( − a i − . One can, now, think of ϕ as of a matrix consisting of blocks of the following types • ϕ ai,j = π aj ◦ ϕ | O P ( − a i − : O P ( − a i − → O P ( a j ), for i, j ∈ { , . . . , n a } ; • ϕ a,bi,j = π bj ◦ ϕ | O P ( − a i − : O P ( − a i − → Ω P ( b j + 1), for i ∈ { , . . . , n a } , j ∈ { , . . . , n b } ; • ϕ a,bi,j = π aj ◦ ϕ | Ω P (6 − b i ) : Ω P (5 − b i ) → O P ( a j ) for i ∈ { , . . . , n b } , j ∈ { , . . . , n a } ; • ϕ bi,j = π bj ◦ ϕ | Ω P (6 − b i ) : Ω P (5 − b i ) → Ω P ( b j + 1) for i, j ∈ { , . . . , n b } ;where π aj and π bj are the projections onto the components of E i.e. O P ( a j ) and Ω P ( b j + 1)respectively for each j . We then have ϕ ai,j = − ( ϕ aj,i ) ∗ , ϕ a,bi,j = − ( ϕ b,aj,i ) ∗ and ϕ bi,j = − ( ϕ bj,i ) ∗ .Equivalently, we have a decomposition:(3.1) H ( ^ E (1)) = M ≤ i Under the above notation, either n neg = n pos = 0 , or both the following hold: • n neg < n pos , • l i + k i +1 ≥ for each i ∈ . . . n neg . Before we pass to the proof of Lemma 3.3, let us finish the proof of Theorem 3.2 assuming thelemma.Keeping our notation, each component F of E such that w ( F ) < O P ( l i ( F ) ) for some i ( F ) ∈ { . . . n neg } or a bundle of twisted first differentials Ω P ( l i ( F ) ) with l i ( F ) = · · · = l i ( F )+6 for some i ( F ) ∈ { . . . n neg } . Now, to each component F of E with w ( F ) < A ( F ) of E in the following way. A ( F ) := ( O P ( k i ( F )+1 ) for F = O P ( l i ( F ) ) L j =1 O P ( k i ( F )+ j ) for F = Ω P ( l i ( F ) )By Lemma 3.3, the bundle A ( F ) is well defined for every component F of E with w ( F ) < • A ( F ) is a sum of components of positive degree not involving the component O P ( k ); • if F ∩ F = 0 then A ( F ) ∩ A ( F ) = 0; • rk A ( F ) = rk F ; • w ( F ) + w ( A ( F )) ≥ rk F .It follows that we get a decomposition of E : E = M F component of Ew ( F ) < ( F ⊕ A ( F )) ⊕ M F component of Ew ( F ) > F ∩ L w ( G ) < A ( G )=0 F into bundles with positive w (observe that w ( F ) = 0 when F is a component of E ) including theline bundle of maximal degree O P ( k ). We then easily list all possibilities. Indeed, we have:(1) b i = 0 for i ∈ { . . . n b } , because for any c ≤ 0, we have w (Ω P ( c ) + A (Ω P ( c ))) ≥ > n b ≤ 1, because w (Ω P (1)) = 2;(3) if n b = 1 then there is no negative a i , because otherwise w ( O P ( l ) ⊕ O P ( k ) ⊕ O P ( k )) ≥ ; 4) if n b = 0 then l ≥ − 2, because when l ≤ − 3, we have w ( O P ( l ) ⊕O P ( k ) ⊕O P ( k ) ≥ ;(5) if l = − E = O P ( − ⊕ O P (2), because w ( O P ( l ) ⊕ O P ( k ) ⊕ O P ( k ) ≥ andequality holds only if k = k = 2;(6) if l = l = − E = 3 O P (1) ⊕ O P ( − w ( O P ( l ) ⊕ O P ( l ) ⊕ O P ( k ) ⊕O P ( k ) ⊕ O P ( k ) ≥ and equality holds only if k = k = k = 1;(7) if l = − a i are nonnegative then we have two possibilities:(a) E = O P ( − ⊕ O P (1) ⊕ O P (2)(b) E = O P ( − ⊕ O P (1) ⊕ O P and the constant terms in ϕ are 0(8) if n b = 0 and all a i are non-negative we have 4 possibilities for E as in the assertion.To conclude, we need to exclude two cases which do not appear in the assertion: • E = 3 O P (1) ⊕ O P ( − • E = O P ( − ⊕ O P (1) ⊕ O P and the constant terms in ϕ ∈ V E (1) are 0; we easily seethat P f ( ϕ ) either does not exist (i.e. the degeneracy locus is of codimension ≤ 2) or mustbe contained in a hyperplane.To complete the proof of Theorem 3.2 we, hence, need only to prove Lemma 3.3 and Proposition4.9. (cid:3) Proof of Lemma 3.3. If n neg = 0 then the assertion is trivial. Assume, hence, by contradiction that n neq > i ∈ { . . . n neg } such that i + 1 > n pos or that k i +1 + l i < 0. Let E = M { j | a j ≤ l i } O P ( a j ) ⊕ M { j | b j ≤ l i } Ω P (1 + b j )considered as a subbundle of E . Observe that, by definition of the sequence ( l i ) i ∈{ ...n neg } , we haverk E ≥ i . Let, moreover, E = (L { j | a j ≤ k i +1 } O P ( a j ) ⊕ L { j | b j ≤ k i +1 } Ω P (1 + b j ) when i + 1 ≤ n pos L { j | a j ≤ } O P ( a j ) ⊕ L { j | b j ≤ } Ω P (1 + b j ) when i + 1 > n pos also considered as a subbundle of E . Similarly as for E we have rk E ≥ u + 1 − n pos when i + 1 > n pos and rk E ≥ u + 1 − i . In any case:(3.4) rk E + rk E ≥ rk E = 2 u + 1 . We also clearly have E ⊂ E . Moreover, the following lemma proves that ϕ E ,E = 0. Lemma 3.4. In the notation of the proof of Theorem 3.2, we have: • ϕ ai,j = 0 when a i + a j ≤ − • ϕ a,bi,j = 0 and ϕ b,aj,i = 0 when a i + b j ≤ − • ϕ bi,j = 0 when b i + b j ≤ Proof. Using the following vanishing • Hom ( O P ( − a − , O P ( b )) = H ( O P ( b + a + 1)) = 0 when a + b ≤ − • Hom ( O P ( − a − , Ω P ( b + 1)) = H (Ω P ( a + b + 2)) = 0 when a + b ≤ − • Hom ((Ω P ) ∗ ( − a − , Ω P ( b + 1)) = H (Ω P ( a + b + 3)) = 0 when a + b ≤ − • ϕ ai,j = 0 when a i + a j ≤ − • ϕ a,bi,j = 0 and ϕ b,aj,i = 0 when a i + b j ≤ − • ϕ bi,j = 0 when b i + b j ≤ t remains to prove that ϕ ai,j = 0 when a i + a j = − 1. Assume the contrary, then ϕ ai,j , for some i, j ∈ { . . . l } , is a non-zero section of O P thus a non-zero constant which we assume to be 1 byre-scaling. Denote by E ′ the subbundle of E such that E = E ′ ⊕ O P ( a i ) ⊕ O P ( a j ). The section ϕ is then decomposed as a sum ϕ = (1 , ϕ O P ( a i ) ,E ′ , ϕ O P ( a j ) ,E ′ , ϕ E ′ ,E ′ ) ∈ H ( O P ⊕ E ′ ( a i + 1) ⊕ E ′ ( a j + 1) ⊕ ^ E ′ (1))Consider ϕ ′ = (1 , , , ϕ E ′ ,E ′ + ( ϕ O P ( a i ) ,E ′ ∧ ϕ O P ( a j ) ,E ′ )) . We claim that Pf( ϕ ) = Pf( ϕ ′ ) = Pf( ψ ), where ψ = ϕ E ′ ,E ′ +( ϕ O P ( a i ) ,E ′ ∧ ϕ O P ( a j ) ,E ′ ) ∈ H ( V E ′ (1)).Indeed, Pf( ϕ ) = Pf( ϕ ′ ) follows from the fact that locally under a trivialization of E respecting thedecomposition E = E ′ ⊕ O P ( a i ) ⊕ O P ( a j ) we have ϕ ′ is obtained by row and column operationsfrom ϕ . On the other hand, the equality Pf( ϕ ′ ) = Pf( ψ ) is clear. The claim being proven, we get acontradiction with the minimality of E . This shows that ϕ ai,j = 0 when a i + a j = − (cid:3) We conclude the proof of Lemma 3.3 by obtaining a contradiction of the above with the followinglemma. Lemma 3.5. Let E ⊂ E ⊂ E be subbundles of E given by some sums of its components. Consider ϕ E ,E : E ∗ ( − → E as defined above. If ϕ E ,E = 0 then rk( E ) + rk( E ) < u + 1 .Proof. Consider the map ϕ E ,E c , where E c is the subbundle of E being the sum of those compo-nents of E which are not contained in E . Under our assumptions, we clearly have D u − ( ϕ ) ⊃ D (rk( E ) − ( ϕ E ,E c ). Now, since D u − ( ϕ ) is not the whole space, we have 2 u + 1 − rk E = rk E c ≥ rk E − E ) + rk( E ) ≤ u + 2.To exclude rk( E ) + rk( E ) = 2 u + 2, we observe that in such case we have D (rk( E ) − ( ϕ E ,E c ) iseither empty or of codimension at most 2. It cannot be of codimension at most 2 as it is containedin a Pfaffian variety, hence, it must be empty. If it is empty then ϕ E ,E c induces an embedding ofvector bundles E ∗ ( − → E c . This means that we have an exact sequence0 → E ∗ ( − → E c → L → , where L is a line bundle on P . Now, since E is a direct sum of line bundles and of twisted firstdifferentials on P , we have Ext P ( L, E ∗ ( − E c ≃ E ∗ ( − ⊕ L . It follows that ϕ E ,E c consists of blocks of the form ϕ ai,j for some i, j ∈ Z with constant entries which by minimalityof E and Lemma 3.4 are zero. This leads to a contradiction with ϕ E ,E c being an embedding andproves that rk( E ) + rk( E ) = 2 u + 2.To exclude rk( E ) + rk( E ) = 2 u + 1, we shall prove that, in this case, we have Y = Pf( ϕ ) ∩ D (rk( E ) − ( ϕ E ,E c ) is contained in Pf( ϕ ) and is of codimension at most 2 in P . To see the latter, wedescribe Y as a degeneracy locus of a map between vector bundles of expected codimension 2. Moreprecisely, we claim that Y = D (rk( E ) − ( κ ) , where κ = ( κ , κ ) : E ∗ ( − → (det E )( u − rk E ) ⊕ E c with κ = ϕ E ,E c and κ = ϕ E ,E ∧ ( ϕ E ,E ) ∧ ( u − rk( E )) ∈ H (( E ⊗ det( E ))( u + 1 − rk E )) . Indeed, the claim follows directly from the following observation:(3.5) ϕ ∧ u = ( κ ∧ ( ϕ E ,E c ) ∧ (rk( E ) − , ( ϕ E ,E c ) ∧ rk E ∧ γ ) ∈ H ((det E ⊗ det( E ) ⊗ rkE − ^ E c )( u )) ⊕ H ((det( E ) ⊗ rk E − ^ E ⊗ det E c )( u )) ⊂ H ( u ^ E ( u )) or some γ ∈ V rk E − E .Now, if D (rk( E ) − ( ϕ E ,E c ) is non-empty then it is a hypersurface in P and in consequence Y = Pf( ϕ ) ∩ D (rk( E ) − ( ϕ E ,E c ) is also nonempty. The codimension of Y is then at most 2 givinga contradiction with Y ⊂ Pf( ϕ ). If D (rk( E ) − ( ϕ E ,E c ) is empty we have E c ≃ E ∗ ( − 1) and we geta contradiction with the minimality of E as in the case rk( E ) + rk( E ) = 2 u + 2. (cid:3) Inequality 3.4, holding under the assumption that there exists an i ∈ { . . . n neg } such that i + 1 > n pos or that k i +1 + l i < 0, together with Lemma 3.4 gives a contradiction with Lemma 3.5and thus finishes the proof of Lemma 3.3. (cid:3) The following is a straightforward Corollary from Theorem 3.2. Corollary 3.6. All quasi Buchsbaum Calabi–Yau threefolds in P are arithmetically Buchsbaum.Moreover, arithmetically Cohen Macaulay Calabi–Yau threefolds in P are exactly those whichcorrespond to case (1) in Theorem 3.2.4. The degrees of Calabi–Yau threefolds in quadrics The approach to the classification of Calabi–Yau threefold in P by considering the minimaldegrees of hypersurfaces that contain the Calabi–Yau threefolds is inspired by [H]. In this section,we shall work on quadrics of different ranks and dimensions. We shall use the following notation.For k ≤ 5, we shall denote by Q rk ⊂ P k +1 a k -dimensional quadric hypersurface with corank r . Proposition 4.1. Let Q ⊂ P be a smooth 4-dimensional quadric hypersurface. If S is a nonde-generate canonically embedded surface of general type of degree d contained in Q , then ≤ d ≤ .Proof. Take S a canonical surface of degree d contained in a smooth 4-dimensional quadric Q . Bythe analog of the double point formula for quadrics (see [Fu, thm. 9.3]), the second Chern classof the normal bundle c ( N S | Q ) = S · S . Now, the Chow group A ( Q ) of Q has two generatorscorresponding to two families of planes on Q . We shall denote them θ and θ . We have θ = θ = 1and θ · θ = 0. By definition we know that S · ( θ + θ ) = d , hence S ∼ aθ + ( d − a ) θ for some a ∈ Z . It follows that c ( N S | Q ) = 2 a + d − da. On the other hand, from the exact sequence0 → T S → T Q | S → N S | Q → , we find that c ( N S | Q ) = 5 h and c ( N S | Q ) = 12 h − c ( S ) = 12 d − c ( S ) , where h = K S is the restriction of the hyperplane class from P to S . It follows that c ( S ) = 12 d − a − d + 2 ad. On the other hand, using the Riemann-Roch formula we have7 = ( d + c ( S )), thus c ( S ) = 84 − d . By comparing both formulas, we infer(4.1) 84 − d = 12 d − a − d − ad. The only integers d for which this equation has a solution are d = 12 , , (cid:3) Let us now consider a similar problem for a singular quadric. Proposition 4.2. The degree d of a nondegenerate canonically embedded surface of general type S ⊂ P contained in a quadric cone Q ⊂ P is either or or . roof. Take S a canonical surface of degree d contained in a 4-dimensional quadric Q of corank1. Let us consider the projective bundle g : F := P ( O Q ⊕ O Q (1)) → Q , of the 1-dimensional quotients of the vector bundle F := O Q ⊕ O Q (1) . The linear system of itstautological line bundle O F (1) defines a morphism π : F → Q ⊂ P . Observe that π is the blowup of Q in the singular point. Denote by ξ and h the pull back to F by π and g respectively ofthe hyperplane sections of Q and Q . Let S ′ be the proper transform of S ⊂ Q on F . By theGrothendieck relation, we have ξ = hξ so that A ( F ) is generated by ξ and h . We can, hence,write the class of S ′ as [ S ′ ] = aξ + bh where a, b ∈ Q .Observe that S ′ is smooth and either isomorphic to S or the blow up of S in the center of thecone Q (when S contains this center). From the double point formula, we infer c ( N S ′ | F ) = ( aξ + bh ) = ξ ( a + 2 ab ) = 2 a ( d − a ) , since d = [ S ′ ] ξ = 2 a + 2 b. Now, let us compute this Chern class using the exact sequence0 → T S ′ → T F | S ′ → N S ′ | F → . We deduce that c ( N S ′ | F ) = c ( F ) − c ( S ′ ) − c ( S ′ ) c ( N S ′ | P ) . Next, we use, 0 → O F → g ∗ F ∗ (1) → T F | Q → → T F | Q → T F → g ∗ T Q → . We obtain c ( N S ′ | F ) = 12 d − 84 + 2 a using the fact that c ( S ′ ) = ( − ξ + h ) | S ′ and the Noetherformula c ( S ′ ) = 84 − h [ S ′ ] = 84 − a . We infer the equation: a + a (1 − d ) + 6 d − 42 = ( a − a − d + 7) = 0 . We thus deduce that either a = 6 and b = d − a = d − b = 7 − d. Now, the exceptional divisor Ξ ⊂ F of π is in the linear system | ξ − h | . Since S is smooth, wehave one of the following: • either Ξ | S ′ = 0 and S ′ is isomorphic to S ; • or (Ξ | S ′ ) = − S ′ is the blow up of S in one point.In the first case, we deduce that b = 0 so either d = 12 or d = 14. In the second case, b = so d = 13. (cid:3) Theorem 4.3. The degree d of a nondegenerate Calabi–Yau threefold X embedded in a quadric in P of corank ≤ is bounded by ≤ d ≤ .Moreover, the degree of a nondegenerate Calabi-Yau threefold contained in a smooth quadriccannot be 13.Proof. First, remark that a generic hyperplane section of X ⊂ P is a canonically embedded surfaceof general type contained either in a smooth quadric Q or in a corank 1 quadric Q ⊂ P . Weobtain the bound on the degree from Propositions 4.2 and 4.1.To prove the second part, observe that A ( Q ) has only one generator θ and that the restriction θ | H to a hyperplane section H = Q is θ + θ in the notation of Proposition 4.2. It follows that,in this case, in the proof of Proposition 4.1, we may additionally assume a = b . It is now enoughto observe that the equation (4.1) has no solution for d = 13 giving a = b . In fact, when d = 13,we must have S ∼ θ + 7 θ . (cid:3) To conclude the proof of Theorem 1.2, it is enough to prove the following: roposition 4.4. If X is a smooth Calabi–Yau threefolds contained in a quadric Q r of corank r ≥ in P then X is contained in a linear space contained in Q r .Proof. The proof will consist of two steps which will be formulated as lemmas. Lemma 4.5. Let X be a smooth threefold contained in a quadric Q r of corank r ≥ in P . Thenone of the following holds: (1) X is contained in a linear space L ⊂ Q r (2) X is fibered by surfaces contained in linear spaces contained in Q r .Proof. If the quadric Q r is of corank ≥ X being smoothmust be contained in one of them.If the quadric Q r has corank 4 then it is a cone Q with center some P ≃ P and spanned overa smooth conic Q . Consider the hyperplane H l spanned by P and any line in the plane spannedby Q . Observe that H l ∩ Q decompose as a sum of two linear spaces L l and L l . It follows that H l ∩ X is either the whole X for some l or is a divisor on X decomposed as S L = S l + S l with S l ⊂ L l and S l ⊂ L l . In the first case the assertion follows as H l ∩ Q is a union of two linearspaces and X is irreducible. We, hence, need only to consider X having a 1-dimensional linearsystem given by surfaces S il which are clearly all in the same system independently on i ∈ { , } .Take two distinct points q, q ′ ∈ Q . For a 1-dimensional linear system on a smooth threefold, wehave two possibilities: • the system is base point free, in which case we have a fibration of X given by surfaces S q giving the assertion, • or the system has a base curve C .In the latter case, we claim that P must be the space tangent to X in every point of this curve.Indeed, all hyperplanes containing P intersect X in the union of two surfaces passing through C ,hence, the intersection is singular in every point of C . It follows that the tangent space to X inany of the points of C is contained in P thus is equal to P . We now use the famous Zak theoremon tangencies, which for smooth varieties is formulated as follows. Theorem 4.6 (Zak [Z]) . Let X n ⊂ P n + a be a subvariety not contained in a hyperplane. Fix anylinear space L = P n + k ⊂ P n + a , ≤ k ≤ a − . Then dim { x ∈ X | ˜ T x X ⊂ L } ≤ k, where ˜ T x X denotes the embedded tangent space to X in x . Zak’s theorem implies that a tangent projective space to a threefold in P cannot be tangent toit in a curve. In particular P cannot be tangent to X in C . The contradiction concludes the prooffor r = 4. Let us point out that from now on we shall consider ˜ T x X as standard notation for theembedded tangent space.Finally, assume that r = 3 i.e. the quadric Q r is of rank 4. Then Q is a cone with center aplane P and spanned over a smooth quadric Q of dimension 2. Consider the family { X q } q ∈ Q consisting of intersections of X with hyperplanes H q spanned by P and planes ˜ T q Q tangent to Q parametrized by the tangency points. Observe that ˜ T q Q ∩ Q consists of two lines l q and l q with q = l q ∩ l q , hence, H q ∩ Q is the union of two 4-dimensional linear spaces L q and L q spanned by P and lines l q and l q respectively. If X q = X for some q ∈ Q then X , being irreducible, is containedin one of the two linear spaces L pi for i = 1 or 2 and the assertion follows. If X q = X for all q ∈ Q then X q is decomposed as X q = S q ∪ S q . We thus have two 1-dimensional linear systems on X andas in the previous case we have the following possibilities • one of the linear systems is base point free giving us a fibration by surfaces as in the assertion • both systems contain a base curve. n the latter case, let us denote the base curves by C and C . Then C ∩ C = ∅ as these areplane curves. Let us choose any point p ∈ C ∩ C . Now, for every q , we have H q ∩ X = S q ∪ S q is a reducible surface which is clearly singular in p ∈ C ∩ C ⊂ S q ∩ S q . It follows that ˜ T p X ⊂ H q for all q ∈ Q . Thus ˜ T p X ⊂ T q ∈ Q H q = P . As ˜ T p X is of dimension 3 and P is a plane, we get acontradiction excluding the last case and finishing the proof. (cid:3) Lemma 4.7. There are no smooth Calabi–Yau threefolds contained in a quadric Q r of corank ≤ r ≤ which admit a fibration by surfaces contained in linear spaces contained in Q r .Proof. Assume the contrary. Let X be a smooth Calabi–Yau threefold contained in Q r such that X admits a fibration f : X → Y with fibers being surfaces contained in linear spaces contained in Q r . As usual we use the notation for which Q r will be a cone with center a linear space P andspanned over a smooth quadric Q − r of dimension 5 − r . As X is a Calabi–Yau threefold, we have Y ≃ P and the fiber X y for a generic point y ∈ Y is either a K3 or an abelian surface (smooth inboth cases by Bertini theorem). Observe moreover that the maximal dimensional linear spaces in Q r are projective spaces of dimension 4. In both cases, we have only 1-dimensional families of P contained in Q r and no two fibers of the fibration can be contained in the same P as otherwise theywould have to intersect. It follows that we can choose a family { L y } y ∈ Y such that L y ≃ P ⊂ Q r and L y ∩ L ′ y = P for any distinct y, y ′ ∈ Y such that for a generic y ∈ Y the fiber X y is either aK3 or an abelian surface contained in L y . Claim. We claim that such a fiber X y must be one of the following: • a complete intersection of a quadric and a cubic in L y ≃ P • a complete intersection of a hyperplane and a quartic in L y ≃ P • the zero locus of the Horrocks Mumford bundle on L y ≃ P Indeed, fix a generic y ∈ Y . The fiber X y is then a smooth K3 or abelian surface contained in L y = P . The theorem of Severi, saying that the only surface in P which is not linearly normalis the Veronese surface, implies that X y is linearly normal. Let us denote its degree by d X y . Thefollowing computation applies:By the double point formula, c ( N X y | P ) = [ X y ] = d X y . On the other hand, by the exactsequence: 0 → T X y → T P | S → N X y | P → , and the vanishing c ( T X y ) = 0, we have: c ( N X y | P ) = c ( T P | X y ) − c ( T X y ) = ( d X y − 24 when X y is a K3 surface10 d X y when X y is an abelian surfaceComparing the two equations, we have d X y = 4 or 6 and X y is a K3 surface or d X y = 10 and X y is an abelian surface. Finally, we use the classification (cf. [DP, table 7.3]) of non-general typesurfaces in P with d X y = 4 , , 10 to obtain our claim.From now on, we shall consider the cases of rank r = 3 and r = 4 separately. For r = 3, wehave P is a 3-dimensional space and Q is a cone centered at P and spanned over a conic Q . Ageneric fiber X y meets P in a curve C y and all these curves must be disjoint. It follows that X ∩ P contains a surface S fibered by the curves C y . Observe, moreover, that since the fibers X y cover X it follows that S = X ∩ P is of pure dimension 2. Now, by the Zak tangencies theorem S is smoothin codimension 1 (since any singular point of S is a tangency point of P with X ) and since it is ahypersurface it must be irreducible with isolated singularities. Observe now that it follows that thehyperplane class H on X is linearly equivalent to S + 2 X y . We shall compute the degree of S interms of the degrees of the fiber X y in two ways. The first will be based on the adjunction formulaon S . Since C y is smooth we can compute its canonical class: On one hand C y is a hyperplane section of X y hence a canonical curve of degree d = 4 , K C y ) = deg( X y ) . • On the other C y is a fiber of a fibration on the hypersurface S ⊂ P = P hence deg( K C y ) =deg( S ) − X , we have ω S = S | S = ( H − X y ) | S = H | S − C y . But, on the otherhand, on P we have ω S = (deg( S ) − H | S . It follows that deg( C y ) = (5 − deg( S )) deg( S )) and,by comparing with deg( C y ) = deg( K C y ) = deg( S ) − 4, we get a contradiction with deg( S ) beingan integer. This concludes the proof for r = 3.Let us pass to the case r = 4. We then have P is a plane and Q is a cone centered in P andspanned over a quadric surface Q . In this case the generic fiber X y meets P in a finite set of points.It follows that X ∩ P is a curve C . Consider for a generic p ∈ Q the hyperplane H p spanned by P and the tangent ˜ T p Q . The intersection H p ∩ Q decomposes into two 3-dimensional linear spaces L and L such that L contains a fiber S . It follows that H p ∩ X decomposes into a fiber X y ofthe fibration and a surface T such that T contains C and T = X ∩ L . Denote the blow up of Q in L by π : P → Q and the proper transform of X by the blow up by ˜ X . Since X ∩ L is aCartier divisor it follows that π : ˜ X → X is an isomorphism. Our aim is to prove that this is infact impossible and in this way finish the proof. We achieve our aim by performing a computationon the resolution P proving that ˜ X must contain a fiber of the projection π . The computation ispresented below.First, observe that P ≃ P (2 O P (1) ⊕ O P ) admits a natural fibration g : P → P . The Chowgroup of P is generated by the pullback ξ = π ∗ ( H ) of the hyperplane section H of Q and thefiber h = g ∗ ([ pt ]) with the relation ξ = 2 ξ · h and h = 0. It follows that the class of X has adecomposition [ X ] = d X y ξ + ( d − d X y ) hξ . Claim. We claim that one of the following holds: • d X y = 6 and [ ˜ X ] = 6 ξ + ( d − hξ • [ ˜ X ] = 10 ξ − hξ • [ ˜ X ] = 4 ξ + 3 hξ The claim follows from a computation with the double point formula analogous to the proof ofTheorem 4.2. Indeed, consider a section of X by a generic element from the system ξ . It is asurface of general type G embedded in a variety P ′ ≃ P (2 O P (1) ⊕ O P ) with generators of theChow group ξ ′ = ξ | P ′ and h ′ = h | P ′ such that [ G ] = d X y ξ ′ + ( d − d X y ) h ′ ξ ′ . By the double pointformula, we have c ( N G | P ′ ) = [ G ] = 2 dd X y − d X y . On the other hand, by the exact sequence0 → T G → T P ′ → N G | P ′ → G , we obtain another formula c ( N G | P ′ ) = 2 d X y + 12 d − Theorem 4.8 (Serre’s construction (for this version see [A, thm 1.1])) . Let X be a smooth varietyof dimension ≥ and L a line bundle on X such that h ( L − ) = 0 . Let Y be a locally completeintersection subscheme of pure codimension . Then Y ⊂ X is the zero locus of a section of a ranktwo vector bundle E on X if and only if ω Y = ( ω X ⊗ L ) | Y . Moreover in such a case we have thefollowing exact sequence → O X → E → I Y ⊗ L → . By the Serre’s construction, ˜ X is obtained as the zero locus of a section of a rank 2 vector bundle E on P . We have c ( E ) = 5 ξ and c ( E ) = [ ˜ X ]. Consider first the case d S = 6. We see that the estriction of E to any fiber F t of g decomposes as O F t (2) ⊕ O F t (3). Indeed, the zero locus of thesection defining X restricted to F t is Gorenstein of codimension 2 as a divisor on X , hence, is acomplete intersection. This means that the restricted bundle decomposes over every fiber and thisdecomposition is the same as for the generic fiber i.e. E| F t = O F t (2) ⊕ O F t (3).It follows that g ∗ E ( − ξ ) is a line bundle and that R i g ∗ E ( − ξ ) = 0 for i = 1 , 2. This implies that χ ( g ∗ E ( − ξ )) = χ ( E ( − ξ )). The latter is computed by the Hirzebruch-Riemann-Roch theorem on P to be 13 − d .Hence, g ∗ E ( − ξ ) = O P (12 − d ). Now, by the projection formula we have E ( − ξ + ( d − h ) hasa section. Since c ( E ( − ξ + ( d − h )) = 0 and E ( − ξ + ( d − h ) ⊗ O ( − D ) has no section fornon-trivial effective divisors D , the section does not vanish. It follows that we have the followingexact sequence:(4.2) 0 → O P (3 ξ − ( d − h ) → E → O P (2 ξ + ( d − h )Let us now restrict E to the exceptional locus Ξ ≃ P × P of the blow up π and denote π ′′ = π | Ξ the projection onto P . We observe that under the assumption that ˜ X contains no fiber of theprojection π , we have E ′ = E| E restricted to every fiber of the projection π ′′ is a bundle of rank 2with trivial first Chern class and either a non-vanishing section or a section vanishing in one pointi.e. it is either O P ⊕ O P or O P ( − ⊕ O P (1). It follows that π ′′∗ ( E ′ ) is a vector bundle of rank 2and R i π ′′∗ E ′ = 0. We compute c ( π ′′∗ ( E ′ )) . By restricting the exact sequence 4.2 to Ξ, we get0 → O E (3 ξ ′ − ( d − h ′ ) → E ′ → O E (2 ξ ′ + ( d − h ′ ) . Taking the push-forward by π ′′ , we have:0 → π ′′∗ ( O E (3 ξ ′ − ( d − h ′ )) → π ′′∗ E ′ → π ′′∗ O E (2 ξ ′ + ( d − h ′ ) → R π ′′∗ ( O E (3 ξ ′ − ( d − h ′ )) → R π ′′∗ E ′ · · · We now observe that π ′′∗ ( O E (3 ξ ′ − ( d − h ′ )) = 0 as O E (3 ξ ′ − ( d − h ′ ) restricted to any fiber of π has no nontrivial section. Moreover, R π ′′∗ E ′ = 0 as E ′ restricted to any fiber of π is either 2 O P or O P ( − ⊕ O P (1). Finally, by projection formula and base change π ′′∗ O Ξ (2 ξ ′ + ( d − h ′ ) = O P (2) ⊗ H ( P , O P ( d − d − O P (2)and R π ′′∗ ( O Ξ (3 ξ ′ − ( d − h ′ )) = O P (3) ⊗ H ( P , O P (12 − d )) == O P (3) ⊗ H ( P , O P ( d − d − O P (3) , and we get 0 → π ′′∗ E ′ → ( d − O P (2) → ( d − O P (3) → . From the exact sequence, we compute the second Chern class of the rank 2 bundle π ′′∗ E ′ : c ( π ′′∗ E ′ ) = 12 d − d + 108 . The latter is nonzero for d ∈ Z . This means that every section of π ′′∗ ( E ′ ) vanishes in some point on P ,hence, every section of E ′ vanishes on some fiber of π ′′ . Finally this implies that ˜ X contains a fiberof the blow up π giving a contradiction which completes the proof in the case [ ˜ X ] = 6 ξ +( d − hξ .The proof in the case [ ˜ X ] = 4 ξ + 3 ξ · h is completely analogous leading to c ( π ′′∗ ( E ′ )) = 1. Thecase [ ˜ X ] = 10 ξ − ξ · h is excluded because it has negative intersection with [Ξ] ξ = ( ξ − ξh ) ξ = ξ − ξ h . (cid:3)(cid:3) e have proved in Theorem 1.2 that there are no smooth Calabi–Yau threefolds of degrees 11and 15 contained in a quadric. For canonical surfaces of general type, the same bounds apply onlyif we restrict the rank of the quadric to be at least 5. In fact, the proof of Lemma 4.7 suggests thatone may construct nodal Calabi–Yau threefolds contained in quadrics of rank 4.Below, we recall examples of nodal Calabi–Yau threefolds and in consequence also smooth canon-ical surfaces of general type of degrees 11 and 15 contained in quadrics of rank 4. Proposition 4.9. Let E = 3 O P (1) ⊕ O P ( − and let σ ∈ H (( V E )(1)) be a generalsection. Then X = Pf( σ ) is well defined and is a singular Calabi–Yau threefold with singularlocus consisting of one ordinary double point.Proof. Since E is a decomposable bundle the variety X is scheme theoretically defined byPfaffians of a skew-symmetric matrix with entries being general polynomials of degrees of theshape: − where − corresponds to the zero entry. It is hence contained in a 2-dimensional system of quadricsdefining a cone over P × P . This system of quadrics is generated by the 2 × × Q . Then Q is a cone with vertex a plane P and spanned over a smooth quadric surface Q . The quadric Q has two fibrations giving twosystems of Weil divisors on X . The generic element of one of them is a complete intersection ofa hyperplane and a quartic obtained as a Pfaffian of the form the generic element of the other is a surface of degree 7 given by 2 × (cid:18) (cid:19) . It follows now from the end of proof of Lemma 4.7 that in the notation of this proof we have[ ˜ X ] = 4 ξ + 3 ξ · h leading to c ( π ′′∗ ( E ′ )) = 1. Hence X contains a singular locus of degree at least1. An example with one node is given by a Macaulay2 ([M2]) calculation. (cid:3) Let φ : 10 O P → O P (1) be a general map. Let E = ker( φ ) ⊕ O (1). Let σ ∈ H (( V E )(1))be a generic section. Proposition 4.10. Under above assumptions the variety B = Pf( σ ) is well defined and is asingular Calabi–Yau threefold with singular locus consisting of three ordinary double points.Proof. Similarly as in the proof for d = 11, we prove that c ( π ′′∗ ( E ′ )) = 3 hence B contains asingular locus of degree at least 3. The bound 3 is reached by Macaulay2 ([M2]) calculation. (cid:3) Remark 4.11. We saw that both Calabi-Yau threefolds X and B admit two birational smoothresolution of Picard rank 2. The extremal rays of these resolutions consist of the small contractionof lines and one K3 fibration and one elliptic fibration. This means that the two birational modelsgive all Calabi–Yau birational models of these varieties. . Classification of Calabi-Yau threefolds contained in five-dimensional quadrics In this section, we classify all nondegenerate Calabi–Yau threefolds contained in 5-dimensionalquadrics. Let X be such a Calabi–Yau threefold. We know from previous sections that 12 ≤ deg( X ) ≤ 14. Moreover, from the Riemann Roch theorem, we know that any nondegenerateCalabi–Yau threefold of degree at most 13 in P is contained in a quadric hence our classificationcontains all Calabi–Yau threefolds contained in P of degree at most 13. Corollary 5.1. A Calabi–Yau threefold X ⊂ P has degree d ≥ 12. Moreover, the degree 12threefold is a complete intersection of two quadrics and a cubic and the degree 13 threefold is givenby the 4 × × Proof. We deduce from the Riemann–Roch theorem that χ ( O X ( m )) = 16 md ( m − 1) + 7 m. Thus by Serre duality and Kodaira vanishing for d ≤ 13 we have h ( O X (2)) ≤ χ ( O X (2)) ≤ < 28 = h ( O P (2)). It follows that X has to be contained in a quadric so the first part follows fromTheorem 1.2. In the case deg X = 12 the threefold X is contained in a pencil of quadrics. FromProposition 4.4, all the quadrics containing X have rank ≥ X . In particular, a general quadric containing X issmooth (by the Bertini theorem) and the intersection Q of two quadrics containing X is smoothalong X . We now apply the Lefschetz hyperplane theorem to deduce that the Picard group of Q is generated by the hyperplane section from P . Since X does not pass through the singular locusof Q , it is a Cartier divisor, hence, X is cut out by a hypersurface in P of degree three. It followsthat X is a complete intersection of two quadrics and a cubic.In the case d = 13, we deduce from Theorem 4.3 that X is not contained in a smooth quadric.Moreover, from the proof of Theorem 4.3, the singular locus of a quadric containing X is containedin X . We know also from Theorem 1.2 that the ranks of the quadrics containing X are ≥ X ⊃ S ⊃ C be a general surface hyperplane section and a general curve linear section of X .It follows that C ⊂ Q where Q is a smooth quadric in P .Since C ⊂ Q is subcanonical, by the Serre construction, it is the zero locus of a section of arank 2 bundle E on Q . If we now denote by ζ the hyperplane section and by ζ the class of a lineon Q , we infer c ( E ) = 5 ζ and c ( E ) = 13 ζ . Thus for F = E ( − 3) we compute c ( F ) = − ζ and c ( F ) = ζ . Claim. We claim that F is aCM i.e. h ( F ( n )) = 0 for all n ∈ Z .Indeed, from [AS, p.205] we know that the spinor bundle is the only stable bundle with theseinvariants and the spinor bundle is arithmetically Cohen–Macaulay. Hence, the claim is proved forstable F .Suppose now that F is not stable i.e. we have h ( F ) = 0. Clearly, F cannot be decomposablesince a generic section of F (3) = E defines a codimension 2 curve of odd degree. This means thata general section of F vanishes in codimension 2 along a curve of degree c ( F ) = 1. Let us denotethis line by l . From the exact sequences0 → O Q ( n ) → F ( n ) → I l | Q ( n − → → O P ( n − → I l ( n ) → I l | Q ( n ) → h ( F ( n )) = h ( I l | Q ( n − h ( I l ( n − n ∈ Z . Since the line l is aCM,we have h ( F ( n )) = 0 for all n ∈ Z . This proves the claim. pplying the claim to the cohomology of the exact sequence0 → O Q ( n ) → F ( n + 3) → I C | Q ( n + 5) → , it follows that h ( I C | Q )( n ) = 0 for all n ∈ Z . Now, from the exact sequence0 → O P ( n − → I C ( n ) → I C | Q ( n ) → , we infer that h ( I C ( n )) = 0 for all n ∈ Z .Consider now the following exact sequences:(5.1) 0 → I X ( n − → I X ( n ) → I S ( n ) → → I S ( n − → I S ( n ) → I C ( n ) → . We know that h ( I X ( n )) = 0 and, from Proposition 2.1, that h ( I X (1)) = 0. From the long exactsequences of cohomology corresponding to 5.1, we deduce that h ( I S (1)) = 0. Since h ( I C ( n )) = 0for all n ∈ Z it follows by induction that the long exact sequence constructed from (5.2) implies h ( I S ( n )) = 0. Then, by the sequence (5.1), we have h ( I X ( n )) = 0, so X ⊂ P is aCM. Theassertion follows from Corollary 3.6. (cid:3) To complete the classification of Calabi–Yau threefolds contained in quadrics, we lack only theconsideration of Calabi–Yau threefolds of degree 14. For this, let us analyze more precisely Calabi–Yau threefolds contained in smooth 5-dimensional quadrics. Let X ⊂ Q be a Calabi–Yau threefold.Since X is subcanonical, from the Serre construction we obtain that X is the zero locus of a sectionof a rank 2 vector bundle E i.e.(5.3) 0 → O Q → E → I X | Q ( c ( E )) → . Since K X = 0 we obtain c ( E ) = 5 ζ where ζ ⊂ Q is the hyperplane section of Q . Moreover,we have c ( E ) = deg( X ) ζ , where ζ is a general codimension two linear section of Q . Finally H i ( E ( k )) = H i ( I X | Q ( c ( E ) + (5 + k )).It follows, by applying the Beilinson type spectral sequence as in [S], that the only arithmeticallyCohen-Macaulay bundles (i.e with vanishing intermediate cohomology) on Q n with 5 ≥ n ≥ Q , that are not aCM, are twists of Cayleybundles. Recall that a Cayley bundle C is defined in [Ot] by the exact sequences0 → O Q → S → G → , → O Q → G (1) → C (1) → , where G is obtained by choosing a general section of the spinor bundle S . It means that Calabi-Yau threefolds given as zero loci of sections of arithmetically Buchsbaum bundles on Q are eitherdegree 12 complete intersections or zero loci of twists of Cayley bundles. We saw in Corollary 5.1that in degree 12 the complete intersections form a unique family of examples. We prove a similarresult in the case of degree 14. Let us first construct two families of Calabi–Yau threefolds of degree14. Proposition 5.2. The zero locus of a generic section of the homogenous bundle C (3) is a Calabi–Yau threefold of degree contained in Q .Proof. We have c ( C ( k )) = 2 k − k = 3. Then c ( C (3)) = 14. We know from [Ot, thm. 3.7] that C (2) is globally generated. We deduce that thezero locus of a general section C (3) is a smooth Calabi–Yau threefold of degree 14. (cid:3) The following family of Calabi-Yau threefolds of degree 14 was introduced in [Be]. roposition 5.3. Let E = Ω P (1) ⊕ O P (1) . Then the degeneracy locus D ( φ ) of a generic skewsymmetric map φ : E ∗ ( − → E is a Calabi-Yau threefold of degree 14.Proof. Let B be a general variety obtained by this construction. By the Bertini type theorem forPfaffian subvarieties given in [O, § 3] to prove that B is a smooth threefold it is enough to provethat the bundle V E (1) is globally generated. Now, V E (1) = Ω P (3) ⊕ Ω P (3). The Eulersequence gives us: 0 → Ω P (1) → O P → O P (1) → . Taking its second and third wedge powers we have:0 → Ω P (2) → (cid:18) (cid:19) O P → Ω (2) → → Ω P (3) → (cid:18) (cid:19) O P → Ω P (3) → . It follows that Ω P (2) and Ω P (3) are globally generated, hence, Ω P (3) and Ω P (3) ⊕ Ω P (3) arealso globally generated. It follows that B is a Pfaffian variety associated to the bundle E =Ω P (1) ⊕ O P (1), with t = 1. The vanishing of the dualizing sheaf is then given by the adjunctionformula (1.2) for Pfaffian varieties and the vanishing of the cohomology by the Pfaffian sequence1.1. (cid:3) We shall denote by C and B the families of Calabi–Yau threefolds obtained in Proposition5.2 and Proposition 5.3 respectively. Observe that a generic B ∈ B is contained in a quadric.Indeed, among the equations defining B we have a section of the bundle ( V (Ω P (1)))(3) = O P (2)corresponding to the Pfaffian of a map (Ω P (1)) ∗ ( − → Ω P (1).We can now classify all Calabi–Yau threefolds of degree 14 contained in a 5-dimensional quadric. Corollary 5.4. If X is a Calabi–Yau threefold of degree 14 contained in a smooth quadric Q then X is obtained as the zero locus of a section of the twisted Cayley bundle C (3). Moreover, if X isa Calabi–Yau threefold contained in any quadric it is given by the Pfaffian construction applied to E = Ω P (1) ⊕ O P (1) . Proof. Let X be a Calabi–Yau threefold of degree 14 contained in the smooth quadric Q . By theSerre construction, we know that X is given as the zero locus of a section of a rank 2 vector bundle E such that c ( E ( − − ζ and c ( E ( − ζ . It follows from the main theorem in [Ot] that E ( − 3) is a Cayley bundle if we assume it is stable. To prove that E ( − 3) is stable is the same as toprove that h ( E ( − h ( I X | Q (2)) = 0 (see (5.3)). To prove that X is not contained in a secondquadric (i.e. that h ( I X | Q (2)) = 0), we argue as in the proof of Corollary 5.1. More precisely, if E ( − 3) is not stable then its general section defines a degree 2 threefold W contained in Q . Since Q contains no 3-dimensional linear space, W must be a 3-dimensional quadric. Hence, W is aCMimplying E is aCM and in consequence X is aCM. Thus X is not contained in any quadric and thisgives a contradiction.To prove the second part, we first show that h ( I X ( k )) = 0 for k = 2 and h ( I X (2)) = 1. Fromthe long cohomology exact sequence obtained from 5.1 and 5.2, we infer h ( I X ( n )) = h ( I Z ( n ))where C ⊂ P is a codimension 2 linear section of X ⊂ P . But we know that X is contained ina quadric of rank ≥ C is contained in a smooth quadric. It follows that C is given as thezero locus of a section of a rank 2 vector bundle B (3) such that c ( B ) = − ζ and c ( B ) = ζ . Itfollows that the zero locus of a general section of B is a sum t of two skew lines. We find thatthe cohomologies of h ( I t | P ( k )) are 0 for k = 2 and 1 for k = 2. Since all nonzero elements ofthe Hartshorne–Rao module of X are of the same weight, it follows that X is quasi-Buchsbaum. ence, we deduce from Theorem 3.2 that X = Pf( σ ) for some σ ∈ H ( V E (1)) where E =Ω P (1) ⊕ O P (1). We thus get the assertion. (cid:3) As a direct consequence of the above, we have the following. Corollary 5.5. The family C is contained in B as a dense subset.6. Classification of degree 14 Calabi-Yau threefolds in P The aim of this section is the classification of all degree 14 Calabi–Yau threefolds in P .From the Riemann-Roch theorem, we deduce that if X ⊂ P is a Calabi-Yau threefold of degree14 then h ( I X (2)) = h ( I X (2)) and h ( I X (3)) + 7 = h ( I X (3)) (recall that h ( I X (1)) = 0). Wehave two possibilities: • X is contained in a quadric, • X is not contained in any quadric.The first case is solved by Corollary 5.4. Assume, hence, that X is not contained in any quadric.Then there is an at least 7-dimensional space of cubics vanishing along X . We, moreover, claimthe following. Proposition 6.1. The singular locus of a generic cubic from H ( I X (3)) has dimension ≤ . Before we pass to the proof of Proposition 6.1, let us prove the following. Lemma 6.2. Suppose that a Calabi–Yau threefold in P is not contained in any quadric. Let X ⊃ S ⊃ C ⊃ F be general linear sections of X of codimension 1,2,3 respectively. Then, we have h ( I X (3)) = h ( I S (3)) = h ( I C (3)) = h ( I F (3)) and h ( I X (2)) = h ( I S (2)) = h ( I C (2)) = h ( I F (2)) .Proof. First, we have h i ( I X ( k )) = 0 for i = 2 , k ∈ Z . Furthermore, we assumed 0 = h ( I X ( m )) = h ( I X ( m )) for m = 1 , 2. From the exact sequence0 → I X → I X (1) → I S (1) → • h i ( I S ( m )) = 0 for i = 0 , , m = 0 , • h i ( I S (2)) = 0 for i = 1 , • H ( I X (3)) ≃ H ( I S (3)).We conclude by repeating this procedure for a hyperplane section and a codimension 2 linearsection. (cid:3) We can now pass to the proof of Proposition 6.1 Proof of Proposition 6.1. We keep the notation from Lemma 6.2. By Lemma 6.2, it is enough toprove that the general cubic hypersurface in P containing C is smooth. For this, we need tostudy the scheme-theoretic intersection of all cubics containing C or more generally the scheme-theoretic intersection of cubics containing X . Let Λ be the scheme-theoretic intersection of thecubics containing X . Then either X is a component of Λ (necessarily of multiplicity 1 for degreereason), or Λ has a component Y ⊂ P of codimension 2 such that X ⊂ Y .In the first case, we use the excess intersection formula. We choose C , . . . , C generic cubicscontaining X . Then we find ( C . . . . .C ) X the equivalences of X in the intersection C . . . . .C .Indeed, from [Fu, prop. 9.1.1], we infer( C . . . . .C ) X = ( X i =1 c ( N C i | P | X ) c ( T P | X ) c ( T X )) = c + 15 h − h.c = 3 − hat is an element from A ( X ). The equivalences of all the distinguished components [Fu, & 6.1]of C ∩ · · · ∩ C sum up to C . . . . .C . Moreover, by the refined Bezout theorem [Fu, thm. 12.3],the equivalences of the distinguished components are positive numbers of degree bigger than thedegree of this component and all the irreducible components of T C i are among the distinguishedcomponents. It follows that there can be at most one such component whose support is notcontained in X and this component have degree 1 with multiplicity 1 in the intersection of C i for1 ≤ i ≤ 6. Since the equivalences of linear spaces of dimension ≥ X in the intersection T i =1 C i .Let us now consider in this case a general codimension 2 linear section C of X ⊂ P . Wecan choose it not to pass through the additional point. We then have that C is set theoreticallydefined by the cubics containing it. Moreover, C is a component of multiplicity one of the schemetheoretical intersection of these cubics. It follows that C is scheme theoretically defined by thecubics containing it outside possibly a finite set P of point on C . Then we know from [DH, thm2.1] that the generic cubic containing C is smooth outside the set P . Suppose that there is apoint from P such that the generic cubic containing C is singular in it. Since the choice of thecodimension 2 linear section giving C was generic, it means that the intersection of the singularloci of all cubics containing X contains a surface U . Then each cubic containing X must containthe secant variety s ( U ) of the surface U . As we already saw that set theoretically the intersectiondefines X plus possibly one point we infer s ( U ) ⊂ X . Since X is a Calabi–Yau threefold, we have s ( U ) = X and, hence, s ( U ) must be a surface. It follows that U is a plane contained in X . Butthen the fiber of the projection from U ⊂ P → P intersects the cubics containing X in linearspaces outside U . Thus either X is rational or X is covered by lines. This is a contradiction in anycase.Assume now that Λ has a component Y ⊂ P of codimension 2 such that X ⊂ Y . We still keepthe notation from Lemma 6.2. Observe that, in this case, all the cubics in P containing F containalso a fixed curve D ⊂ P being the codimension 3 linear section of Y . Moreover, since X ⊂ Y , wehave F ⊂ D . By Lemma 6.2, the projective linear system of cubics containing D is of dimensionat least 6. Moreover, since F is not contained in any quadric, the restrictions of these cubics toa general hyperplane in P form a projective linear system of cubics on P of the same dimension6. It follows that the intersection of D with a generic hyperplane is a scheme of length at most3. Hence, D is a curve of degree 3 in P . Such a curve is either contained in a hyperplane or in aquadric hypersurface. The latter is a contradiction with F ⊂ D and h ( I F (2)) = 0. (cid:3) Corollary 6.3. A Calabi–Yau threefold of degree 14 in P that is not contained in any quadric isdefined by the 6 × × Proof. Let us consider a generic codimension 2 linear section C of X ⊂ P . From Proposition 6.1,the curve C is contained in a smooth cubic threefold W . Let H be the class of the hyperplanesection of W . Since K C = 2 H and K T = − H , we deduce that C ⊂ W is subcanonical. From theKawamata-Viehweg vanishing theorem, we have h ( O W ( − H )) = 0 and h ( O T ( − H )). Thus wecan apply the Serre construction and find a rank 2 vector bundle E on W with a section vanishingalong C ⊂ W . More precisely, we obtain an exact sequence(6.1) 0 → O W → E → I C | W (4 H ) → c ( E ) = 4 H and H · c ( E ) = 14. We compute that c ( E ( − H · c ( E ( − E is stable if and only if h ( E ( − O ( − H ), we deduce that h ( E ( − = 0 implies that h ( I C | W (2)) = 0. This contradicts the factthat C is not contained in any quadric. Thus E is stable. By Serre duality, 0 = h ( E ( − h ( E ( − E ( − 2) is stable then h i ( E ( − − i )) = 0 for i ≥ 1. Itfollows that h ( E ( n )) = 0 for n ∈ Z . Now, from the long exact sequence of cohomology associated o the exact sequence (6.1), we infer h ( I C | W (4 − k )) ≤ h ( O W ( − k )) = h ( O W ( − k )). From theKodaira vanishing theorem, the last number is 0 so h ( I C | W ( k ))) = 0. Next, from the long exactsequence of cohomology associated to the sequence0 → O P ( − → I C | P → I C | W → , we obtain h ( I C | P ( k ))) = 0 . Finally, arguing as in Lemma 6.2 we infer h ( I X ( k )) = 0 for k ∈ Z thus X ⊂ P is aCM. Theassertion follows from Corollary 3.6. (cid:3) Classification up to deformations By the classification of Section 5 for d ≤ 13, the Hilbert scheme of Calabi–Yau threefolds ofdegree d has a unique irreducible component. In this section, we prove that the statement is alsovalid for d = 14. To do this, we compare the families of Calabi–Yau threefolds of degree 14 in P appearing in the classification above. We show that all such varieties are smooth degenerations ofthe family of Calabi–Yau threefolds of degree 14 defined by the 6 × × T the family of Calabi-Yau threefold defined by 6 × × T ∈ T is not contained in any quadric. Onthe other hand, we have two families C ⊂ B of Calabi–Yau threefolds contained in a quadric.We compute the dimension of C using Bott formula. From the computation and Corollary 5.5,it follows that the dimension of the component of the Hilbert scheme of Calabi–Yau threefolds ofdegree 14 in P containing B is bigger than the dimension of B . In fact, we prove the following. Theorem 7.1. For a generic Calabi-Yau threefold B belonging to B , there exists a smoothmorphism P × ∆ ⊃ X → ∆ to the complex disc ∆ such that for λ = 0 the fiber X λ is a smoothsubvariety in P belonging to the family T whereas the central fiber X = B . The theorem is a straightforward consequence of the Euler sequence and the following Proposition7.2.To formulate the proposition let us start with two vector bundles E , F of ranks 2 u and 2 u + 1on P forming an exact sequence.(7.1) 0 → E → F → O P (1) → O P (1):(7.2) 0 → ( ^ E )(1) → ( ^ F )(1) → E (2) → , and its associated cohomology sequence:(7.3) 0 → H (( ^ E )(1)) η −→ H (( ^ F )(1)) δ −→ H ( E (2)) → H (( ^ E )(1)) . Assume that the map δ is surjective. Assume moreover that the Pfaffian varieties X σ and X σ ′ associated to generic sections σ ∈ H (( V E )(1) ⊕ O P (1)) and σ ′ ∈ H (( V F )(1)) are irreducibleof codimension 3 as expected. Proposition 7.2. Let E , F be vector bundles as above. Then for a generic section s ∈ H (( V ( E ⊕O P (1)))(1)) , there exists a family of sections σ ′ λ ∈ H (( V F )(1)) parametrized by λ ∈ C \ { } suchthat the family X λ = ( Pf( σ ′ λ ) for λ = 0Pf( σ ) for λ = 0 s a flat family of 3-dimensional subvarieties of P .Proof. Let σ ∈ H (( V ( E ⊕ O P (1)))(1)). We have σ = ( σ , σ ) with σ ∈ H ( V E )(1)) and σ ∈ H ( E (2)). By the assumption made, there exists a pair ( σ ′ , σ ′ ) of sections σ , σ ∈ H ( V F )(1)such that η ( σ ) = σ ′ , δ ( σ ′ ) = σ . Let σ ′ λ = σ ′ + λσ ′ ∈ H (( V F )(1)).Consider the variety ∆ × P with projection π ∆ and π P . Consider now the subscheme ˜ X ⊂ ∆ × P defined by the vanishing of the section ∆ × P ∋ ( λ, x ) (( λ, x ) , σ ′ λ ( x ) ( ∧ r ) ) ∈ π ∗ P (( V uF )( u )) , andconsider X its irreducible component dominating ∆. By [H1, Prop. 9.7], the family π ∆ : X → ∆ isflat. We clearly see that the fiber X λ for λ = 0 is X λ . It is, hence, enough to prove that the fiber X over 0 ∈ ∆ is equal to X We have ( σ ′ + λσ ′ ) ∧ u = ( σ ′ ) ∧ u + λγ for some γ ∈ H (( V u F )( u )). It follows that X iscontained in the zero locus Z (( σ ′ ) ∧ u ) of the section ( σ ′ ) ∧ u ∈ H ( V u F ). Moreover, by the exactsequence: 0 → ( u ^ E )( u ) → ( u ^ F )( u ) ψ −→ ( u − ^ E )( u + 1) → , we infer Z (( σ ′ ) ∧ u ) = Z ( σ ∧ u ). Furthermore, knowing that ψ (( σ ′ + λσ ′ ) ∧ u ) = λσ ∧ u − ∧ σ + λ γ for some γ ∈ H (( V u F )( u )) and that Z (( σ ′ + λσ ′ ) ∧ u ) ⊂ Z ( ψ (( σ ′ + λσ ′ ) ∧ u )) , we see that X ⊂ Z ( σ ∧ u − ∧ σ ). Putting everything together, we get: X is contained in the zerolocus of( σ ∧ u , σ ∧ u − ∧ σ ) ∈ H (( u ^ ( E ⊕ O P (1)))( u ) = H (( ^ uE )( u )) ⊕ H ( u − ^ E ( u + 1) . The latter zero locus is precisely the variety X = Pf(( σ , σ )). It follows that X ⊂ X . But, since π ∆ | X : X → ∆ is flat, we know that X is of codimension 3 in P which implies by assumption on X that X = X . (cid:3) Proof of Theorem 7.1. The Euler sequence gives:0 → Ω P (1) → O P → O P (1) → . By Proposition 7.2, for any B ∈ B we obtain a flat family of manifolds defined as Pfaffiansassociated to the bundle 7 O degenerating to B . Since smoothness is an open condition in flatfamilies, we conclude that any smooth B is a smooth degeneration of a family of smooth Calabi-Yau threefolds from T . (cid:3) Remark 7.3. Similarly, we prove using Proposition 7.2 that any Calabi–Yau threefold B fromProposition 4.10 is a flat degeneration of the family of Calabi–Yau threefolds of degree 15 definedas Pfaffians of the bundle Ω P (1) ⊕ O P . References [A] Arrondo, E., A home-made Hartshorne-Serre correspondence . Rev. Mat. Complut. 20 (2007), no. 2, 423–443.[AS] Arrondo, E., Sols, I., Classification of smooth congruences of low degree J. Reine Angew. Math. 393 (1989),199–219.[AR] A. Aure, K. Ranestad, The smooth surfaces of degree 9 in P . Complex Projective Geometry, Lond. Math.Soc. L. N. S. 179 32–46.[BC] Ballico, E., Chiantini, L., On smooth subcanonical varieties of codimension in P n , n ≥ . Ann. Math.Pura Appl. 135, 99–117 (1983)[BHVV] Ballico, E., Malaspina, F. ,Valabrega, P., Valenzano, M., On Buchsbaum bundles on quadric hypersurfaces arXiv:1108.0075 [math.AG] BSS] Beltrametti, M., Schneider, M., Sommese, A. J., Threefolds of degree 9 and 10 in P5. Math. Ann. 288 (1990),no. 3, 413–444.[Be] Bertin, M.A., Examples of Calabi-Yau 3-folds of P with ρ = 1. Canad.J.Math. 61 (2009), no. 5, 1050–1072.[Bo] B¨ohm, J., Mirror symmetry and tropical geometry, Ph.D. Thesis Universitat des Saarlandes (2008),arXiv:0708.4402v1 [math.AG].[BOS] Braun, R., Ottaviani, G., Schneider, M., Schreyer, F.O., Boundedness for nongeneral-type 3-folds in P5. Complex analysis and geometry, 311–338, Univ. Ser. Math., Plenum, New York, 1993.[Ca] Catanese, F., Homological algebra and algebraic surfaces . Algebraic geometry, Santa Cruz 1995, 3–56, Proc.Sympos. Pure Math., 62, Part 1, Amer. Math. Soc., Providence, RI, (1997).[DES] Decker, W., Ein, L., Schreyer, F.O, Construction of surfaces in P4. J. Alg. Geom., 2, 185–237, (1993).[DP] Decker, W., Popescu, S., On surfaces in P4 and 3-folds in P5. Vector bundles in algebraic geometry (Durham,1993), 69–100, London Math. Soc. Lecture Note Ser., 208, Cambridge Univ. Press, Cambridge, 1995.[DH] Diaz, Steven, Harbater, D., Strong Bertini theorems. Trans. Amer. Math. Soc. 324 (1991), no. 1, 73–86.[Dr] Druel, S., Espace des modules des faisceaux de rang 2 semi-stables de classes de Chern c = 0 , c = 2 et c = 0 sur la cubique de P . Internat. Math. Res. Notices 2000, no. 19, 985–1004.[EP] Ellingsrud, G., Peskine, C. Sur les surfaces lisses de P4. Invent. Math. 95 (1989), no. 1, 1–11.[EPW] Eisenbud, D., Popescu, S., Walter, C., Lagrangian subbundles and codimension 3 subcanonical subschemes. Duke Math. J. 107 (2001), no. 3, 427–467.[EFG] Ellia, P., Franco, D., Gruson, L., Smooth divisors of projective hypersurfaces. Comment. Math. Helv. 83(2008), no. 2, 371–385.[F] Fujita, T., Projective threefolds with small secant varieties. Sci. Papers College Gen. Ed. Univ. Tokyo 32(1982), no. 1, 33–46.[Fu] W. Fulton, Intersection theory Second edition. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge.A Series of Modern Surveys in Mathematics, Springer-Verlag, Berlin, 1998. xiv+470 pp.[M2] Grayson, D. R., Stillman, M. E., Macaulay2, a software system for research in algebraic geometry. Genre de courbes alg´ebriques dans l’espace projectif (d’apr`es L. Gruson et C. Peskine). (French) [Genus of algebraic curves in projective space (after L. Gruson and C. Peskine)] Bourbaki Seminar,Vol. 1981/1982, pp. 301313, Ast´erisque, 92-93, Soc. Math. France, Paris, 1982.[H1] Hartshorne, R., Algebraic geometry. Graduate Texts in Mathematics, No. 52. Springer-Verlag, New York-Heidelberg, 1977. xvi+496 pp. ISBN: 0-387-90244-9.[K] Kapustka, G., Projections of del Pezzo surface and Calabi–Yau threefolds arXiv:1010.3895 [math.AG].[KK] Kapustka, G., Kapustka, M., Del Pezzo surfaces in P and Calabi–Yau threefolds in P . arXiv:1310.0774[math.AG][KK1] Kapustka, G., Kapustka, M., A cascade of determinantal Calabi-Yau threefolds. Math. Nachr. 283 (2010),no. 12, 1795–1809.[M] Miyaoka, Y., The Chern class and Kodaira dimension of a minimal variety , Adv. Stud. Pure Math. vol. 10,449–476.[OS] Oguiso, K., Sakurai, J., Calabi-Yau threefolds of quotient type. Asian J. Math. 5 (2001), no. 1, 43–77.[O] Okonek, Ch., Note on subvariety of codimension in P n Manuscripta Math. 84 (1994), no. 3-4, 421–442.[Ot] Ottaviani, On Cayley bundles on the five-dimensional quadric. Boll. Un. Mat. Ital. A (7) 4 (1990), no. 1,87–100.[R] Rødland, E. A., The Pfaffian Calabi–Yau, its mirror, and their link to the Grassmannian G (2 , On spinor bundles. J. Pure. Appl. Algebra 35 (1985), 85–94.[Sch] Schneider, M., Boundedness of low-codimensional submanifolds of projective space. Internat. J. Math. 3(1992), no. 3, 397–399.[ST] Schreyer,F.-O., Tonoli,F., Needles in a haystack: special varieties via small fields. Computations in algebraicgeometry with Macaulay 2, 251–279, Algorithms Comput. Math., 8, Springer, Berlin, 2002.[T] Tonoli, F. Construction of Calabi-Yau 3-folds in P . J. Algebraic Geom. 13 (2004), no. 2, 209–232.[V] Van de Ven, A., On the embedding of abelian varieties in projective spaces . Ann. Mat. Pura Appl. (4) 103(1975), 127–129.[W] Walter, Ch., Pfaffian subscheme J. Algebraic Geom. 5 (1996), no. 4, 671–704.[Z] Zak, F., Tangents and secants of algebraic varieties. Translations of Mathematical Monographs, 127. Amer-ican Mathematical Society, Providence, RI, 1993. viii+164 pp. nstitut f¨ur MathematikMathematisch-naturwissenschaftliche Fakult¨atUniversit¨at Z¨urich, Winterthurerstrasse 190, CH-8057 Z¨urichDepartment of Mathematics and Informatics,Jagiellonian University, Lojasiewicza 6, 30-348 Krak´ow, Poland.Institute of Mathematics of the Polish Academy of Sciences,ul. ´Sniadeckich 8, P.O. Box 21, 00-956 Warszawa, Poland. E-mail address: [email protected] E-mail address: [email protected]@uj.edu.pl