On isolated smooth curves of low genera in Calabi-Yau complete intersection threefolds

Building on results of Clemens and Kley, we find criteria for a continuous family of curves in a nodal $K$-trivial threefold $Y_0$ to deform to a scheme of finitely many smooth isolated curves in a general deformation $Y_t$ of $Y_0$. As an application, we show the existence of smooth isolated curves of bounded genera and unbounded degrees in Calabi-Yau complete intersections threefolds.


Introduction
In this paper we study embeddings of complex projective curves into (smooth) Calabi-Yau complete intersection (CICY ) threefolds. Such embeddings, and Calabi-Yau threefolds in general, have in the past decades been objects of extended interest in both algebraic geometry and physics. The goal of counting such curves (especially rational) has inspired the development of quantum cohomology and led to the discovery of surprising relations between algebraic geometry and the theory of mirror symmetry.
In the paper [7], Kley developed a framework for showing existence of curves of certain genera and degrees in CICY threefolds. The paper built on the original idea in the case of genus zero curves of Clemens [1] (then used also in [6], [12] and [4]): one starts with a K3 complete intersection surface X containing a smooth rational curve C, embeds the surface in a nodal CICY of suitable intersection type Y and proves that under a general deformation Y t of Y 0 = Y , the rational curve deforms to an isolated curve in the deformation. In the higher genus case, the curve C is replaced by a complete linear system |L| of curves on the surface of dimension equal to the genus, and the idea is to prove that only finitely many of these deform to the deformation Y t and possibly also that these are smooth and isolated. The main existence result in [7] is [7,Thm. 1], claiming that for any d ≥ 3, the general CICY threefold contains smooth, isolated elliptic curves of degree d, except for degree 3 curves in the CICY of type (2,2,2,2).
A crucial point in this construction is to show that the curves on the K3 surface X do not acquire any additional deformations when considered as curves in Y , precisely that h 0 (N C/X ) = h 0 (N C/Y ) for all C ∈ |L|. Unfortunately, the proof of this step, namely [7,Thm. 3.5], contains a serious gap, which also influences the proof of its corollary [7,Cor. 3.6], which in fact cannot hold (cf. Remark 6.6 for a more detailed explanation). As a consequence, the proof of [7, Thm. 1] is incorrect. This paper has two main purposes: • We give criteria for a continuous family of curves on a regular surface in a nodal threefold Y with trivial canonical bundle to deform to a scheme of finitely many smooth isolated curves in a general deformation Y t of Y 0 = Y , using results from [2] and ideas from the unpublished preprint [8] of Kley, see Theorem 1.1. • We apply these results to prove existence of smooth, isolated curves of low genera in the various CICY threefold types, see Theorem 1.2 (of which [7, Thm. 1] is a special case). The first main result is the following. It is an improvement under slightly stronger hypotheses of a result in the preprint [8] of Kley, which has never been published, presumably because of the gap in [7].
We first state the assumptions.
Let X ⊂ Y be a smooth, regular surface (i.e. H 1 (X, O X ) = 0) and L a line bundle on X.
We remark that the genus zero case of the theorem is already known by [6,12,4]. In [8] an existence result similar to Theorem 1.2 was claimed, but only for geometrically rigid, connected curves (not necessarily smooth and isolated). But the proof of that result also relied on [7,Thm. 3.5].
The paper is divided into two parts in a natural way: The first part, consisting of Sections 2-5, is devoted to the proof of Theorem 1.1. In Sections 2-4 we study deformations of curves in a complete linear system |L| lying on a smooth surface X embedded in a nodal threefold Y with emphasis on the cases of regular surfaces in threefolds with trivial canonical bundle. Special attention is devoted to studying if the curves in |L| on X acquire additional deformations when embedded in Y , that is, to studying condition (A5). The crucial result is Proposition 4.3, which states that condition (A5) is equivalent to the condition (A5)' The set of nodes S imposes independent conditions on |L|, and the natural map γ C : an isomorphism for all C ∈ |L|. First of all, the conditions in (A5)' may be easier to check than condition (A5) directly. More importantly, however, the first of the two conditions in (A5)' assures that the locus of curves in |L| passing through at least one node is a simple normal crossing (SNC) divisor (consisting of ℓ hyperplanes). This enables us to identify a certain sheaf Q of obstructions to deformation as the locally free sheaf of differentials with logarithmic poles along an SNC divisor, cf. (5.1), (5.2) and (5.3). This is a crucial point to assure that smooth and isolated curves survive in a general The proof of Theorem 1.1 is finished in Section 5, following the proof of [8, Thm.
In the second part, consisting of Sections 6 and 7, we apply Theorem 1.1 to the case of K3 surfaces in complete intersection Calabi-Yau threefolds to prove Theorem 1.2. For each of the complete intersection types in Theorem 1.2, there is a standard construction allowing to embed a K3 surface of one (or more) of the three complete intersection types (4) in P 3 , (2, 3) in P 4 and (2, 2, 2) in P 5 into a nodal CICY threefold. This will be recalled in Section 6. We are then in the setting of Theorem 1.1 with X the K3 surface, Y the CICY , P a projective space and E the vector bundle corresponding to the complete intersection type of Y . All various complete intersection types and possible constructions are summarized in Table 1 in Section 6.
By construction and Bertini's theorem, condition (A1) and the first part of condition (A2) are satisfied. In each of the cases in Theorem 1.2, we will then need to verify the remaining conditions (A2)-(A7), and this is the reason for the various numerical conditions on d and g in the theorem. To help the reader navigate through the proof, we now briefly explain how it works.
The existence of smooth curves of certain degrees and genera on the three types of complete intersection K3 surfaces is given by the existence results in [11] and [9], cf. Theorem 7.1. We set L to be the line bundle defined by the curves and an important point is that the existence results guarantee that L and O X (1) are independent in Pic X.
Condition (A3) is automatically satisfied, as well as condition (A4), by standard arguments, cf. Lemma 6.1. Condition (A7) is also satisfied by construction, cf. Lemma 6.2. To check (A5) we will check the two conditions in (A5)'. We prove that the second one is satisfied in Proposition 6.5, and it is here that we need to use the fact that L and O X (1) are independent in Pic X.
Therefore, at the end, the conditions that will be responsible for the numerical constraints in Theorem 1.2 are conditions (A2) and (A6), as well as the first condition in (A5)', namely that the set of nodes S imposes independent conditions on the linear system |L|. This is perhaps the most tricky condition to check, and we give sufficient conditions for this to hold in Lemma 6.3.
The numerical conditions we end up with are given in Proposition 7.2 (in addition to the conditions in the existence result Theorem 7.1). Finally, a case-by-case study of these conditions will give Theorem 1.2.
Conventions and definitions. The ground field is the field of complex numbers. We say a curve C in a variety V is geometrically rigid in V if the space of embedded deformations of C in V is zero-dimensional. If furthermore this space is reduced, we say that C is isolated or infinitesimally rigid in V . From the infinitesimal study of the Hilbert scheme of V , it follows that C is infinitesimally rigid if and only if H 0 (C, N C/V ) = 0.
A K3 surface is a smooth projective (reduced and irreducible) surface X with trivial canonical bundle and such that H 1 (O X ) = 0. In particular h 2 (O X ) = 1 and χ(O X ) = 2.
A Calabi-Yau threefold Y is a projective variety of dimension 3 with trivial canonical bundle and h 1 (O Y ) = h 2 (O Y ) = 0. In this paper a Calabi-Yau threefold will be at worst nodal.  [7,8] to Calabi-Yau threefolds that are complete intersections in certain homogeneous spaces. (As a consequence, [10] has still not been published.) I must therefore thank this referee for having discovered the gap in [7] and for a very carefully written referee report where he or she explained this in detail.
Finally, I must thank the referee of this paper for a very careful reading of the manuscript and many suggestions. In particular, he or she caught a mistake in the first version and suggested a rewrite resulting in the present Sections 2-4, making the paper, I believe, much clearer and easier to read.

Curves through nodes on threefolds, local theory
be (the analytic germ of) a nodal threefold singularity in affine 4-space containing the plane Tensoring the sequence (2.1) with O X is terminally exact and so we obtain the resolution be a curve in X passing through the node (0, 0, 0, 0) of Y .
For all i ≥ 0, we define Applying Hom OX (−, O X ) to the first two columns of (2.6), we obtain From the snake lemma we obtain the exact sequence Since f is the local equation of C, we can, by tensoring with O C , deduce the short exact sequence

Curves through nodes on threefolds, global theory
In this section X will be a smooth projective surface, Y a projective threefold and P a smooth projective variety of dimension ≥ 4 such that X ⊂ Y ⊂ P . We assume that the only singularities of Y lying on X are finitely many nodal singularities and that the embedding Y ⊂ P is regular (e.g., Y is a complete intersection in P ). We denote the set of nodes of Y lying on X by S. Note that we have

by (2.4).
Let L be a line bundle on X such that |L| = ∅ and let C ∈ |L|. We define the sheaves F i C as in (2.7).
we obtain the exact sequence Applying Similarly, we also have the standard short exact sequence

Curves on regular surfaces in K-trivial threefolds with nodes
In this section we keep the hypotheses and notation from the previous section and assume further that except at the finite set S. But both sides are locally free so the isomorphism holds everywhere. Therefore, again by adjunction The main aim of this section is to find criteria for the curves C ∈ |L| not to obtain any additional deformations in Y , that is, for the equality We will need to consider, for C ∈ |L|, the composition where the left hand inclusion ι C arises from (2.8) (or, equivalently, (4.3)) and the right hand map is the connecting homomorphism of (3.3). Note that H 0 (C, N X/Y ⊗ O C ) and H 1 (C, N C/X ) are in fact dual by (4.2). Therefore, γ C is an isomorphism if and only if it is of maximal rank.
The inclusion ι C is an isomorphism for all C ∈ |L| if and only if S imposes independent conditions on |L|.
Remark 4.2. The condition that S imposes independent conditions on |L| means, precisely, that if ℓ is the number of nodes, then |L ⊗ This can also be rephrased as dim |L ⊗ J ξ1 ⊗ J ξ2 ⊗ · · · ⊗ J ξ k | = dim |L| − k for any subset {ξ 1 , . . . , ξ k } of k nodes of S, whenever k ≤ dim |L|, or, even simpler, that at most dim |L| of the nodes can lie on an element of |L|.
Note that if S imposes independent conditions on |L|, then, in particular, the points in S are different from the possible base points of |L|, so that the locus of curves in |L| passing through at least one node is an effective divisor in |L| consisting of hyperplanes. (If dim |L| = 0, this means that this locus is empty, that is, the only curve in |L| does not pass through any of the points of S.) Therefore the condition that the nodes impose independent conditions on |L| can be rephrased as saying that the locus of curves in |L| passing through at least one node is an effective, simple normal crossing (SNC) divisor consisting of hyperplanes.
to the bottom exact sequence of (3.10), we find that ) by the local to global spectral sequence for Ext. Moreover, applying Hom OC (−, ω C ) to (4.3), we obtain the short exact sequence Taking cohomology of (4.5) we therefore obtain by Serre duality and (4.2) for the last isomorphism. Therefore, ι C is an isomorphism for all C ∈ |L| if and only if is surjective for all C ∈ |L|, which precisely means that S imposes independent conditions on |L|.
The next result will be central in the rest of the paper. Since [8] has never been published, we give the whole proof of Theorem 1.1.
The setting and assumptions are as in the introduction. We observe the following: we shorten the above to By Remarks 4.2 and 5.1, condition (A5) implies that the locus of curves in |L| ≃ P dim |L| passing through the node ξ i is a hyperplane D i ⊂ |L|, and furthermore, that the locus of curves in |L| passing through at least one node, satisfies In fact, what we have done so far, is to go through the first part of the proof of [2, Thm. 3.3]. Since the notation in [2] is rather involved, we include the following translation between our notation and the notation in [2]:

Our notation
Notation in [2] In particular, the conditions in [2, Thm. 3.3] are satisfied. Moreover, by (A4), the conditions in [2,Thm. 3.5] are also satisfied, so that , the locally free sheaf of differentials with logarithmic poles along an SNC divisor consisting of ℓ hyperplanes (see e.g. [5, §2] for the definition). By [7, Thm. 1.5] and (A6) we have that p * q * E is locally free on H and |L| is the zero scheme of p * q * s 0 , so that (5.1) identifies Q as the excess normal bundle to p * q * s 0 , cf. This already finishes the proof in the case dim |L| = 0, by (A3). In the remaining cases it suffices to prove that ρ(p 0 * q 0 * s) will vanish at precisely ℓ−2 dim |L| distinct points of |L|, all corresponding to smooth, irreducible curves in |L|. Indeed, by [7, Prop. 1.4] (and its proof), if ρ(p 0 * q 0 * s) has a reduced and isolated zero at a point z ∈ |L|, then the scheme Z(p * q * (s 0 + ts)) has a reduced and isolated zero in any small enough complex analytic neighborhood of z in H for t > 0 sufficiently small. Since the curve corresponding to z is smooth, the same holds true in this neighborhood of z. This will finish the proof of the theorem.
Assume then that dim |L| > 0. The fact that ℓ ≥ dim |L| + 2 in condition (A2) implies that the locally free sheaf Q ≃ Ω 1 P dim |L| [log ℓ P dim |L|−1 ] is globally generated (see e.g. [3, Thm. 3.5] for a proof of this fact) and that Consider the standard exact sequence in the same coordinates. In the proof of [2,Thm. 3.3] it is shown that the composition is given by Therefore, the image of the composition is surjective. Thus, as Q is globally generated, it follows by (5.4) that for the general section s ∈ H 0 (P, E), we have that ρ(p 0 * q 0 * s) vanishes at precisely ℓ−2 dim |L| distinct points of |L|, all corresponding to smooth, irreducible curves in |L|, by (A3), as desired. This finishes the proof of the theorem.
Remark 5.2. Looking more closely at the proof, we see that the condition that E splits as a direct sum of line bundles is not necessary: it would suffice (when r ≥ 6) that E = F ⊕ M, where F is a vector bundle of rank r − 4 and M is a line bundle on P . Writing s 0 ∈ H 0 (P, E) as s 0 = s 0,F ⊕ s 0,M , with s 0,F ∈ H 0 (P, F ) and s 0,M ∈ H 0 (P, M), we would then have Z := Z(s 0,F ) in the setting of Theorem 1.1.

K3 surfaces embedded in nodal Calabi-Yau complete intersection threefolds
The rest of the paper is devoted to proving Theorem 1.2. We first recall the well-known construction used in [1], [6], [12] and [4] to embed a K3 surface into a nodal Calabi-Yau complete intersection (CICY ) threefold.
It is well known, and easily seen by adjunction, that there are three types of K3 complete intersection surfaces in projective space, namely the intersection types (4) in P 3 , (2, 3) in P 4 and (2, 2, 2) in P 5 . Similarly, there are five types of CICY threefolds in projective space, namely the intersection types (5) in P 4 , (3, 3) and (4, 2) in P 5 , (3, 2, 2) in P 6 and (2, 2, 2, 2) in P 7 . Let X be a K3 surface of degree 2µ − 2 in P µ that is a complete intersection of type (a 1 , . . . , a r−2 ) in some P r , for r ≥ µ. We will always assume that but we may have a r−3 = 1 or a r−2 = 1.
Then each b i ≥ 2 and we can construct a Calabi-Yau threefold Y that is a complete intersection of type (b 1 , . . . , b r−3 ) in P r as follows: Choose generators g i of degrees a i for the ideal of X. So X = Z(g 1 , . . . , g r−2 ). For general α ij ∈ H 0 (P r , (here we follow [7, Section 3], except for arranging indices in a different way). If the coefficient forms α ij are chosen in a sufficiently general way, Y has only ℓ = (2µ − 2)a r−3 a r−2 ordinary double points and they all lie on X. This can be checked using Bertini's theorem. In fact, the ℓ nodes are the intersection points of two general elements of |O X (a r−3 )| and |O X (a r−2 )| (distinct, when a r−3 = a r−2 ). As above, we denote the set of nodes by S. Moreover, for general α ij , Bertini's theorem yields that the fourfold We are therefore in the setting of Theorem 1.1 given in the introduction with P = P r , . By construction, condition (A1) and the first part of condition (A2) are satisfied.
We refer to Table 1 for all values of a j , b i , ℓ, µ and r. (This is the same table as [7, Table p. 201], except for one case, namely (b i ) = (3, 3), (a j ) = (2, 2, 2), present in [7, Table p. 201] but absent in our table, because in this case none of the two cubic hypersurfaces will be smooth along X.) Assume now that X carries a line bundle L such that the general element of |L| is a smooth, irreducible curve of degree d and genus g. It is well known, and easily seen, that such a line bundle satisfies see e.g. [13].
In the next section we will finish the proof of Theorem 1.2 by applying Theorem 1.1 and a result guaranteeing the existence of the line bundle L, cf. Theorem 7.1. In particular, we will verify that conditions (A2)-(A7) are satisfied (under certain numerical conditions, giving the different constraints in Theorem 1.2). In the rest of this section we will give some results that will be needed for the verifications of the conditions (A4), (A5) and (A7), where we in the case of (A5) will consider the equivalent condition (A5)' from Remark 5.1.
The following result, which is folklore among experts on K3 surfaces, will be needed to verify that condition (A4) is satisfied. Proof. We can assume that g = dim |L| > 0. We then have that |L| is base point free, see [13,Thm. 3.1], and dim |L| = g and L 2 = 2g − 2 ≥ 0 by (6.2).
If L 2 = 0 it is well known that |L| is an elliptic pencil, see [13,Prop. 2.6], so that we are done by (ii). (Note that |L ⊗ J x | has only one element.) If L 2 > 0 we consider the morphism ϕ L : X → P g and its Stein factorization Then α is the contraction of the finitely many smooth, rational curves Γ satisfying Γ.L = 0 [13, (4.2)] and β is finite of degree one or two [13, (4.1)]. It follows from hypothesis (i) that the base scheme of |L ⊗ J x | is finite of length at most two, whence is curvilinear. Therefore the general member of |L ⊗ J x | is smooth and irreducible by Bertini.
By this lemma, condition (A4) is satisfied for general α ij . Indeed, the reduced and irreducible curves Γ on X satisfying Γ.L = 0 are only finitely many by standard results, and the singular points of fibers in an elliptic pencil are also finitely many. Therefore, since the nodes are the points of intersection of two general (distinct) elements of |O X (a r−3 )| and |O X (a r−2 )|, we can make sure that the nodes satisfy the conditions (i) and (ii) in the lemma.
The next result shows that condition (A7) is satisfied.
Lemma 6.2. For general α ij , the image of the natural restriction map H 0 (P r , O P r (a r−3 + a r−2 )) / / H 0 (S, O S (a r−3 + a r−2 )) ≃ C ℓ has codimension one.
Proof. Recall that S = H 1 ∩ H 2 , for general members H 1 ∈ |O X (a r−3 )| and H 2 ∈ |O X (a r−2 )|. We note that the map above factorizes through the natural restrictions to X and H 1 . The restriction map H 0 (P r , O P r (a r−3 + a r−2 )) → H 0 (X, O X (a r−3 + a r−2 )) is surjective because X is a complete intersection. The restriction map H 0 (X, O X (a r−3 + a r−2 )) → H 0 (H 1 , O H1 (a r−3 + a r−2 )) is surjective from the cohomology of Finally, the cokernel of the map H 0 (H 1 , O H1 (a r−3 +a r−2 )) → H 0 (S, O S (a r−3 + a r−2 )) is one-dimensional from the cohomology of This concludes the proof.
The following result gives criteria for the first of the two conditions in (A5)' to hold: Lemma 6.3. Assume that the α ij are general, that h 0 (X, L ⊗ O X (−a r−2 )) = 0 and that Then S imposes independent conditions on |L|.
Finally, we will consider the second of the two conditions in (A5)'. We first recall the following result: of the restriction with the connecting homomorphism arising from (3.9) with P = P r is surjective. Furthermore, ker ϕ is independent of C ∈ |L|.
We now prove that, for general α ij , the second of the two conditions in (A5)' from Remark 5.1 holds if L and O X (1) are independent in Pic X. The proof is taken basically ad verbatim from the last part of the proof of [7, Thm. 3.5] and we give it not only for the sake of completeness, but also because our statement is different and the proof of [7,Thm. 3.5] has a gap (cf. Remark 6.6 below). Proof (following the proof of [7,Thm. 3.5]). We consider the commutative diagram (3.10) with P = P r . By definition of γ C and the fact that h 1 (N C/X ) = 1 for any C ∈ |L| by (4.6) and (6.2), we just need to show that the connecting homomorphism of the left hand vertical exact sequence in (3.10) is nonzero. A diagram chase in (3.10) reduces this to proving that for general α ij and all C ∈ |L|, we have is the connecting homomorphism of the right hand vertical exact sequence in (3.10) and Φ C is given in (3.10). As in [7], we define A := ⊕ r−2 j=1 O P r (a j ) and denote by A X its restriction to X. Now we have natural isomorphisms (6.5 is given by the matrix (α ij ), where α ij is the restriction of α ij to C (cf. [7,Lemma 3.4], where B : To prove (6.4), use Gaussian elimination to find a generator N = N (α ij ) of the null-space of the linear map C r−2 (αij ) / / C r−3 . Keeping track of degrees, this can be done in such a way that the ith coordinate of N is of degree a i . For example, if (b i ) = (4, 2) and (a j ) = (3, 2, 1), the vector N = (α 12 α 23 −α 13 α 22 , −α 11 α 23 , α 11 α 22 ). Since N is well-defined only up to scalar, we view it as a line in H 0 (P r , A) or a point of P (H 0 (P r , A)). Now each term of each coordinate of N is a term of a determinant of (α ij ), so that as (α ij ) varies, N hits a multiple of each element of the form (0, . . . , 0, λ 1 λ 2 · · · λ ai , 0, . . . , 0), where the λ k s are of degree 1. Since the image of the Segre embedding of P(H 0 (P r , O(1))) ×ai in P(H 0 (P r , O(a i ))) is nondegenerate, the linear span of the image of N includes the spaces and hence all of ⊕ i H 0 (P r , O(a i )). Now (6.4) follows from Lemma 6.4 since H 0 (P r , A) → H 0 (X, A X ) is surjective for any complete intersection X.
Remark 6.6. In [7,Thm. 3.5] it is claimed that H 0 (N C/X ) ≃ H 0 (N C/Y ) for all C ∈ |L| under the same hypotheses as in Proposition 6.5. However, there is a mistake in the proof: it is claimed that there is a (not necessarily surjective) map µ : N C/Y → N X/Y ⊗ O C fitting into the left bottom corner of (3.10). While this is true from 2) and (4.3), the same does not hold if C passes through some nodes of Y . In this case, N X/Y ⊗ O C is a proper subsheaf of F 0 C and one cannot conclude that there is a map as µ. Moreover, the gap in the proof of [7, Thm. 3.5] also influences the proof of [7, Corollary 3.6], which in fact does not hold. Indeed, if H 0 (N C/X ) ≃ H 0 (N C/Y ) for all C ∈ |L|, then by (3.3), (4.3) and the fact that h 1 (N C/X ) = h 1 (O C (C)) = 1 for every C ∈ |L|, we must have . It therefore follows that [7, Corollary 3.6] does not hold for the curves C ∈ |L| passing through at least one of the nodes, as it is claimed there that H 1 (C, N C/Y ) ≃ H 1 (C, O C ) for any C ∈ |L|, and it is well known that h 1 (O C ) = g for any C ∈ |L|, cf. e.g. [ Clemens and Kley consider, with different notation, the two upper cases in Table  1, that is, either (6.6) g 1 , α 12 ∈ H 0 (P 4 , O P 4 (4)) and g 2 , α 11 ∈ H 0 (P 4 , O P 4 (1)) or (6.7) g 1 , α 12 ∈ H 0 (P 4 , O P 4 (3)) and g 2 , α 11 ∈ H 0 (P 4 , O P 4 (2)) (respectively cases (4.3.1) and (4.3.2) in their example), so that both the K3 surface X := Z(g 1 , g 2 ) (which they call Y ) and the del Pezzo surface X ′ := Z(g 2 , α 11 ) (which they call S) are smooth, and such that the quintic threefold Y := Z(α 11 g 1 + α 12 g 2 ) (which they call X 0 ) has only ordinary nodes (16 and 36 respectively), given by The authors claim that for any curve C ′ ⊂ X ′ , one has To prove (6.8) they again state the existence of an exact sequence As mentioned above, one cannot a priori conclude the existence of such a sequence.
In fact, as we will see now, (6.8) does not hold in certain cases. By Proposition 4.3 a necessary and sufficient condition for (6.8) to hold for all curves C ′ in the linear system |C ′ | is that the nodes S impose independent conditions on |C ′ |, since X ′ is a del Pezzo surface, so that h 1 (N C/X ′ ) = 0 for all C ∈ |C ′ | and γ C is automatically surjective. Now in case (6.6), let C ′ be the quartic X ′ ∩ Z(g 1 ) on X ′ . Then C ′ pass through all 16 nodes but |O X ′ (C ′ )| = |O P 2 (4)| is 14-dimensional. Hence the nodes S do not impose independent conditions on |C ′ |, so that (6.8) does not hold for all curves in |C ′ | in this case. In fact, it does not hold when one considers linear systems |O P 2 (d)| with d ≥ 4. This example also shows that the assertion "If the g i and the α ij are sufficiently general, the divisor of curves passing through at least one node is a simple-normal crossing divisor consisting of hyperplanes" in [2,Example 4.3] does not hold if d ≥ 4, cf. Remark 4.2.
We now show, however, that S does impose independent conditions on |O P 2 (d)| when d ≤ 3, using the same argument as in the proof of Lemma 6.3.
In the same way one can show that the applications in the case (6.7) are correct.
7. Proof of Theorem 1.2 In this section we apply Theorem 1.1 and the results from the previous section to construct smooth, isolated curves in general CICY s of each intersection type, thus proving Theorem 1.2.
Recall the following result from [11] and [9], which guarantees the existence of smooth curves of certain degrees and genera on the three types of complete intersection K3 surfaces: For X and C as in the theorem, we let L := O X (C).
In the setting described in the previous section, we now want to find out under which circumstances the conditions (A1)-(A7) in Section 5 are satisfied, so that we can apply Theorem 1.1.  Proof. Conditions (A1) and (A3) are obviously satisfied (as is the first part of (A2)). Condition (A4) is satisfied by Lemma 6.1 and the lines following the proof of that lemma. Condition (A7) is satisfied by Lemma 6.2.
Finding, for every g, the lowest possible bound for d given by (7.4) and (7.5), yields the numerical conditions in Theorem 1.2(a).