A remark on generalized complete intersections

We observe that an interesting method to produce non-complete intersection subvarieties, the generalized complete intersections from L. Anderson and coworkers, can be understood and made explicit by using standard Cech cohomology machinery. We include a worked example of a generalized complete intersection Calabi-Yau threefold.


Introduction
Calabi-Yau varieties, in particular those of dimension three, are of great interest in string theory. Since there are not many general results yet on their classification, but see [W], the explicit construction of CY threefolds is a quite important enterprise. For example, Kreuzer and Starke classified the toric fourfolds which have CY threefolds as (anticanonical) hypersurfaces [KS], [AGHJN]. Besides generalizations to complete intersection CYs in certain ambient toric varieties, like products of projective spaces, there are various other examples of CY threefolds constructed with more sophisticated algebro-geometrical methods. Recent examples include [IMOU], [CGKK], [KK].
In the recent paper [AAGGL], L. Anderson, F. Apruzzi, X. Gao, J. Gray and S-J. Lee found a very nice method to construct many more CY threefolds. The basic idea is to take a hypersurface Y in an ambient variety P and to consider hypersurfaces X in Y . These hypersurfaces need not be complete intersections in P , that is, there need not exist two sections of two line bundles on P whose common zero locus is X. There are various generalizations of this method, but we will stick to this basis case. As in [AAGGL], we refer to these varieties as generalized complete intersections (gCIs).
A particularly interesting and accessible case that was found and studied by Anderson and coworkers is when the ambient variety is a product of two varieties, one of which is P 1 , so P = P 2 × P 1 . The variety P 2 they consider is a product of projective spaces, but this is not essential, one could consider any toric variety or even more general cases. The factor P 1 is important since there are line bundles on P 1 with non-trivial first cohomology group and this is essential to find generalized complete intersections. We review this construction in Section 1.1.
We provide a proposition, proven with standard Cech cohomology methods, that allows one, under a certain hypothesis, to find three equations (more precisely, three sections of three line bundles on P ) that define X. In Section 2 we work out a detailed example, with explicit equations, of a CY threefold which was already considered in [AAGGL]. The explicit example X has an automorphism of order two and the quotient of X by the involution provides, after desingularization, another CY threefold. More generally, we think that among the gCIs found in [AAGGL] one could find more examples of CY threefolds with non-trivial automorphisms. It might be hard though to implement a systematic search as was done in [CGL] for complete intersection CY threefolds in products of projective spaces. We did not find new CY threefolds with small Hodge numbers (see [CCM] for an update on these), but the gCICY seem to be a promising class of CYs to search for these. The recent paper [BH] by Berglund and Hübsch provides further techniques to deal with gCICYs whereas [AAGGL2] explores string theoretical aspects of gCICYs.
1. The construction of generalized complete intersections 1.1. The general setting. Let P 2 be a projective variety of dimension n and let P := P 2 ×P 1 . We denote the projections to the factors of P by π 1 , π 2 respectively. For a coherent sheaf F on P 2 and an integer d we define a coherent sheaf on P by: Recall that the only non-zero cohomology of where (z 0 : z 1 ) are the homogeneous coordinates on P 1 .
Let L be a line bundle on P 2 and assume that L[d], for some d ≥ 1, has a non-trivial global section F . Using the Künneth formula, we can write for certain sections f i ∈ H 0 (P 2 , L). Let Y = (F ) be the zero locus of F in P . We assume that Y is a (reduced, irreducible) variety, although this will not be essential in this section.
To define a codimension two subvariety of P , we consider another line bundle M on P 2 . The Künneth formula shows that M[−e] has no global sections if e ≥ 1. But upon restricting to Y , the vector space H 0 (Y, M[−e] |Y ) could still be non-trivial. In fact, from the exact sequence we deduce the exact sequence thus H 0 (Y, M[−e] |Y ) ∼ = ker(F 1 ), where we denote by F 1 the map induced by multiplication by F on the first cohomology groups. Since now the domain of F 1 is non trivial if and only if h 0 (P 2 , L −1 ⊗ M) = 0. So for suitable choices of line bundles on P 2 we might find interesting, non-complete intersection, codimension two subvarieties of P in this way. In the proof of Proposition 1.4 we explain how to compute F 1 . 1.3. Generalized complete intersections. Given a variety Y ⊂ P that is the zero locus of F ∈ H 0 (L[d]) as in Section 1.1, and given a global section τ ∈ H 0 (M[−e] |Y ), its zero locus X := (τ ) ⊂ Y is called a generalized complete intersection.
The scheme X may not be defined by two global sections σ 1 , σ 2 of line bundles L 1 , L 2 on P . However in certain cases we can find three sections of line bundles on P which define X: with d, e ≥ 1 be as above and assume that H 1 (P 2 , L −1 ⊗ M) = 0.
Then there are two global sections G, H ∈ H 0 (P, M[d−1]) such that the generalized complete intersection subscheme X of P defined by τ in Y can also be defined as The cohomology groups we consider are computed with the Künneth formula. Note that after tensoring this exact sequence by a vector space W , we obtain that For an affine open subset V ⊂ P 1 , the cohomology of the exact sequence (1) on P 2 × V gives the exact sequence, where we extend M[−e] |Y by zero to P 2 × V , The Künneth formula, combined with the assumption H 1 (P 2 , L −1 ⊗ M) = 0 and the fact that H 1 (V, F ) = 0 for any coherent sheaf F since V is affine, implies that the last group is zero.
Taking V = U 0 , U 1 , the exact sequence (1) on P 2 × V thus gives two exact sequences whose sum (term by term) is Similarly taking V = U 0 ∩ U 1 one has the exact sequence: Next we use the Cech boundary map δ to map sequence (3) to sequence (4) and we obtain a commutative diagram with three complexes as columns. The first two columns are Cech complexes for the covering {U i } i=0,1 of P 1 , their cohomology groups are respectively The zero-th cohomology group of the last column is H 0 (Y, M[−e] |Y ). So we conclude that the maps q and F 1 can be computed with the long exact cohomology sequence associated to this diagram. We observe, but will not use, that the Künneth formula implies that H 2 (P, (L −1 ⊗ M)[−d − e]) = 0 and thus the cohomology sequence of (1) gives a six term exact sequence with the zero-th and first cohomology groups. The first 5 terms are the same as those of the long exact sequence associated to the diagram, so we conclude that the first cohomology group of the last Since the first row (3) of the complex is exact, the section τ is locally given by restricting sections , the first summand lies in M[−e](P 2 × U 0 ) (where z 0 = 0) and the last summand lies in M[−e](P 2 × U 1 ), we denote these summand by τ 0 and −τ 1 respectively. The middle summand has monomials z a 0 z b 1 with both a, b < 0. Thus F q represents a class in q ′ ∈ H 1 (P, M[−e]), which is the same as the class represented by the middle summand. By definition, one has q ′ = F 1 (q) and thus q ∈ ker(F 1 ) when all coefficients r k , k = −e + 1, . . . , −1, are zero.
Since q ∈ ker(F 1 ) this middle summand is zero, so that F q = τ 0 − τ 1 as desired. Now we define G := z d+e−1 1.5. Example. With the choices of P 2 , L, M as in Example 1.2, and if X is a smooth variety (of dimension n − 1), then H 1 (P 2 , L −1 ⊗ M) = H 1 (P n , O P n (l)) = 0, for any l, if n > 1. The adjunction formula implies that X has trivial canonical bundle if we choose l = n + 1 − 2k and d = 3. In that case P = P n × P 1 and F is homogeneous of bidegree (k, 3) whereas G, H have bidegree (n + 1 − k, 2). 1.6. A fibration on X. Given X as in the proposition, the projection π 2 : P 2 × P 1 → P 1 restricts to X to give a fibration denoted by π 2 : X → P 1 . For a point p = (z 0 : z 1 ) ∈ P 1 , we denote by F p ∈ H 0 (P 2 , L), H p ∈ H 0 (P 2 , M) the restrictions of F and H to the fiber X p . The equation AF = z d+e−1 1 G + z d+e−1 0 H shows that if z 1 = 0 then F p and H p define the fiber X p , which is thus a complete intersection in P 2 . 1.7. Example. This example illustrates that X, as in Proposition 1.4, might be reducible, even if h 0 (Y, M[−e] |Y ) is rather large. The example is taken from [AAGGL, Table 4], third item (with i = 2) where it is in fact observed that no smooth varieties arise in that case. We take P 2 := P 2 × P 1 × P 1 , L := O(0, 1, 1), M := O(3, 1, 1) , d = 4, e = −2 .
To see that indeed h 0 (O S (1, 1, −2)) = 1 for a general equation f , take a smooth (genus one) curve C of bidegree (2, 2) in P 1 ×P 1 and choose eight distinct points on C which are not cut out by another curve of bidegree (2, 2). As curves of bidegree (1, 4) depend on 2·5 = 10 parameters, we can find two polynomials g 0 , g 1 of bidegree (1, 4) such that g 0 = g 1 = 0 consists of these eight points on C. Take f = x 0 g 0 + x 1 g 1 with (x 0 : x 1 ) ∈ P 1 , the first copy of P 1 in (P 1 ) 3 , and the g i on the last two copies of P 1 . The surface S ⊂ (P 1 ) 3 defined by f is thus the blow up of P 1 × P 1 in the eight points where g 0 = g 1 = 0. The adjunction formula shows that the line bundle O S (1, 1, −2) is the anticanonical bundle of S. The effective anticanonical divisors are the strict transforms of bidegree (2, 2)-curves on passing through these eight points. Hence the strict transform of C in S will be the unique effective anticanonical divisor on S and therefore h 0 (O S (1, 1, −2)) = 1.
2. An example: a generalized complete intersection Calabi-Yau threefold 2.1. We illustrate the use of Proposition 1.4 (and its proof) for the generalized complete intersection Calabi Yau discussed in [AAGGL,Section 2.2.2]. We also consider an explicit example which has a non-trivial involution and we compute the Hodge numbers of a desingularization of the quotient threefold which is again a CY.
2.2. The varieties P 2 and Y . We consider the case that P 2 = P 4 , we choose the line bundle L := O P 4 (2) and we let d = 3. Then the line bundle L[d] = O P (2, 3) is very ample on P = P 4 × P 1 and thus a general section F will define a smooth fourfold Y of P . To obtain a CY threefold in Y , we consider global sections of the anticanonical bundle of Y . By adjunction, ω Y = (O P (−5, −2) ⊗ O P (2, 3)) Y = O Y (−3, 1). Thus we take M = O P 4 (3) and e = 1, so that As the H 1 of any line bundle on P 4 is trivial, we can use (the proof of) Proposition 1.4 to find polynomials G, H ∈ H 0 (P, O P (3, 2)) which together with F define a generalized complete intersection X.
To find explicit elements of H 0 (O Y (3, −1)), we write the defining equation of Y as F = P 0 z 3 0 + P 1 z 2 0 z 1 + P 2 z 0 z 2 1 + P 3 z 3 1 (∈ H 0 (P, O P (2, 3))) , with P i ∈ H 0 (P 4 , O(2)) homogeneous polynomials of degree two in y = (y 0 : . . . : y 4 ). As (−4)), a basis of this 5 · 3 = 15 dimensional vector space are the products of one of y 0 , . . . , y 4 with one of z −3 0 z −1 1 , z −2 0 z −2 1 , z −1 0 z −3 1 . Thus any class q ∈ H 1 (O P (1, −4)) has a representative with linear forms Q i ∈ H 0 (P 4 , O(1)). As in the proof of Proposition 1.4 we must write: 2.3. The base locus of | − K Y |. In Section 2.2 we showed how to find the global sections of ω −1 Y = O Y (3, −1) explicitly, locally such a section is given by the polynomials G and H. From the formula for F we see that if x ∈ P 4 and P 0 (x) = . . . = P 3 (x) = 0, then the curve {x} × P 1 lies in Y . This curve also lies in the zero loci of G and H, for any choice of Q 0 , Q 1 , Q 2 ∈ H 0 (O P 4 (1)), hence it lies in the base locus of anticanonical system | − K Y |. Since the four quadrics P i = 0 in P 4 intersect in at least 2 4 points, counted with multiplicity, we see that this base locus is non-empty. Thus we cannot use Bertini's theorem to guarantee that there are smooth CY threefolds X ⊂ Y , but we resort to an explicit example, see below.
Its fixed point locus has two components, one defined by y 3 = y 4 = 0 and the other by y 0 = y 1 = y 2 = 0 in X. The first is a curve in P 2 × P 1 ⊂ P , which is smooth, irreducible and reduced of genus 8 according to Magma. Similarly, the other component is a genus 2 curve in P 1 (y 3 :y 4 ) × P 1 (z 0 :z 1 ) ⊂ P . In fact, only F = 0 provides a non-trivial equation for this curve since y 0 = y 1 = y 2 = 0 implies Q 0 = Q 1 = Q 2 = 0 and hence G = H = 0 on this P 1 × P 1 . As F = 0 defines a smooth curve of bidegree (2, 3) in P 1 × P 1 , this curve has genus (2 − 1)(3 − 1) = 2.
In particular, the singular locus of the quotient X/ι consists of two curves of A 1 -singularities. Since the fixed point locus X ι consists of two curves, we conclude that locally on X the involution is given by (t 1 , t 2 , t 3 ) → (−t 1 , −t 2 , t 3 ) in suitable coordinates. Hence ι acts trivially on the nowhere vanishing holomorphic 3-form on the CY threefold X. Thus the blow up Z of X/ι in the singular locus will again be a CY threefold.
We determine the Hodge numbers of Z. To do so, it is more convenient to consider the blow upX of X in the fixed point locus X ι . The involution extends to an involutionι onX, the fixed point set ofι consists of the two exceptional divisors and the quotientX/ι is the same Z. Moreover, H i (Z, Q) ∼ = H i (X, Q)ι, theι-invariant subspace.