On smooth and isolated curves in general complete intersection Calabi-Yau threefolds

On smooth and isolated curves in general complete intersection Calabi-Yau threefolds

Xun Yu Department of Mathematics, The Ohio State University, Columbus, OH, 43210-1174, USA yu@math.ohio-state.edu
Abstract.

Recently Knutsen found criteria for the curves in a complete linear system on a smooth surface in a nodal K-trivial threefold to deform to a scheme of finitely many smooth isolated curves in a general deformation of . In this article we develop new methods to check whether the set of nodes of imposes independent conditions on . As an application, we find new smooth isolated curves in complete intersection Calabi-Yau threefolds.

0. Introduction

Calabi-Yau threefolds have two related and interesting properties: a) is unobstructed; b) the expected dimension of the deformation space of any l.c.i. curve lying in is zero.

However, it is difficult to show even the existence of a rational curve of given degree and genus on , let alone that such a curve is geometrically rigid. Thus, there is interest in measuring the known families of geometrically rigid curves of given degree and genus on general . For genus zero it is known that there exist rigid rational curves of given degree on the general member of many complete intersection Calabi-Yau (CICY) threefolds . For higher genus Knutsen has provided many examples in [5]. Knutsen’s technique is to construct a curve of given degree and genus lying in a linear system on a smooth surface lying in a nodal complete intersection Calabi-Yau threefold . He then uses deformation theory to show that only a finite number of the deform when is deformed generally.

In [5] he lists and proves a set of conditions sufficient to ensure this construction.

To introduce Knutsen’s criterion, we first state the assumptions.

Let be a smooth projective variety of dimension and a vector bundle of rank on that splits as a direct sum of line bundles .

Let be a regular section, where for . Set and
(where if ).

Let be a smooth , regular surface (i.e. ) and a line bundle on .

We make the following additional assumptions:
(A1) has trivial canonical bundle;
(A2) is smooth along and the only singularities of which lie in are nodes . Furthermore


(A3) and the general element of is a smooth, irreducible curve;
(A4) for every if , then its general member is nonsingular at ;
(A5) for all ;
(A6) for all ;
(A7) the image of the natural restriction map

has codimension one.

Then Knutsen’s criterion is the following:

Theorem 0.1.

([5, Theorem1.1]). Under the above setting and assumptions (A1)-(A7), the members of deform to a length

scheme of curves that are smooth and isolated in the general deformation of . In particular, contains a smooth, isolated curve that is a deformation of a curve in .

Using Knutsen’s criterion, we find some new smooth and isolated curves in general Calabi-Yau complete intersection (CICY) threefolds in this paper.

To apply Theorem 0.1, we need to choose appropriate surfaces and line bundles on . Then we need to show that we can find a nodal Calabi-Yau threefold containing such that all the conditions (A1)-(A7) are satisfied. [5, Proposition 4.3] shows that under certain assumptions, (A5) is equivalent to (A5’). Actually, all cases in this paper satisfy those assumptions and we will always check (A5’) instead of (A5). (A5’) consists of two parts:

The set of nodes imposes independent conditions on ; ()

The natural map (cf.[5] (4.4)) is an isomorphism for all .

The part (1) of (A5’) is critical to this paper, and we denote it as .

In §1 we develop the tools we will need to prove in our cases. In §2 we treat cases in which is a K3 surface. Finally in §3 we treat the cases in which is a rational surface.

Our main results are Theorem 2.9, Theorem 2.12 and Theorem 3.1.

: Let and be integers. Then in any of the following cases the general Calabi-Yau complete intersection threefold of a particular type contains an isolated, smooth curve of degree and genus :

(a) and ; and ; and ; and ; and .

(b) and ; and ; and ; and ; and ; and ; and ; and ; and ; and ; and ; and .

(c) and .

(d) and .

(e) and ; and .

: Let and be integers. Then in any of the following cases the general Calabi-Yau complete intersection threefold of a particular type contains an isolated, smooth curve of degree and genus :

(a) and ; and ; and .

(b) and ; and ; and ; and ; and ; and ; and ; and ; and .

(c) and .

(d) and ; and .

In order to use Theorem 0.1 to prove Theorem 2.9 and Theorem 2.12, the surfaces are complete intersection surfaces with . (cf.[6, Theorem 1.1]), where is the hyperplane section of and is a smooth irreducible curve of desired genus and degree. The K3 surfaces used in the proof of Theorem 2.9 do not have -2 divisors, but the K3 surfaces used in the proof of Theorem 2.12 have -2 divisors. Furthermore, we define the line bundle to be . Let be nodes of general CICY containing . By the construction of and , , where is a general member of and . The integers and vary case by case. The good news is that for all cases in Theorem 2.9 and Theorem 2.12, all the conditions [5] (A1)-(A7) except the part (1) of (A5’), say (), can be easily verified by using results in [5, §6 & §7] (Lemma 6.1, Lemma 6.2, Prop.6.5 and the proof of Prop 7.2). The bad news is that () is hard to verify. Actually, in [5, Proposition 7.2] Knutsen uses the condition (7.2) to guarantee (), but all cases in Theorem 2.9 and Theorem 2.12 do not satisfy the condition (7.2). For this reason, in §1 and Appendix A we develop two new methods which can be used to verify ().

Actually, we can appy Corollary 1.8 to show for all cases in Theorem 2.9 and Theorem 2.12. In order to apply Corollary 1.8, we will need to show that . If the K3 surfaces do not have -2 divisors like in Theorem 2.9, we can just apply Lemma 2.8 to show . Roughly speaking, Lemma 2.8 gives a numerical criterion for the vanishing of the first cohomology group of line bundles on a K3 surface without -2 divisors. Essentially, Lemma 2.8 comes from the fact that on a K3 surface without -2 divisors all complete linear system are base points free. On the other hand, if the K3 surfaces have -2 divisors, we first compute the closed cone of curves and the nef cone and then we can prove by using the information about these cones.

: Let and be integers. Then in any of the following cases the general Calabi-Yau complete intersection threefold of a particular type contains an isolated, smooth curve of degree and genus :

(i) and .

(ii) and .

The proof of Theorem 3.1 is a quite different story since in this case we use complete intersection rational surfaces instead of complete intersection K3 surfaces. For case (i) of Theorem 3.1 we choose blow-up at six general points and , where the hyperplane class proper transform of a cubic through the six points and is the pull-back of ; for case (ii) we choose .

To prove Theorem 3.1, we will also check all conditions (A1)-(A7) for each case. Notice that all these rational surfaces used are Fano varieties, which makes verifying the conditions (A1)-(A7) much easier since some cohomology groups in question vanish by applying Kodaira Vanishing to -.

(A1) and (A2) are easily verified. Because the line bundles are very ample, (A3) and (A4) are easily verified. The way to check (A7) is similar to [5, Lemma 6.2]. The way to check (A6) is similar to certain parts of the proof of [5, Proposition 7.2]. Like before we check (A5’) instead of (A5). The second part of (A5’) is easily checked since it’s easy to show . In order to show , we just apply Corollary 1.8.

Because this paper cites many results from reference [5] and some of them are not described here, the reader probably need a copy of reference [5] in order to read the rest of this paper.

In the Appendix C, a summary for the existence of smooth isolated curves in general CICY threefolds known so far is given by putting results in [5] and this paper together.

Acknowledgments

I would like to thank my advisor Herb Clemens for his insight, advice and continuous support. This paper grew out of numerous conversations with him. I would also like to thank Andreas Knutsen for reading the first draft of this paper and for giving very helpful comments. One conversation with David Morrison was also very helpful, and is gratefully acknowledged.

1. Definitions and Lemmas

It is convenient to consider the first half of [5] (A5’) i.e. “the nodes imposes independent conditions on ”, say (), in a general setting.

Let be a smooth surface in projective space and let be a line bundle on such that . Let

Definition 1.1.

Let be a reduced 0-cycle on . We say imposes independent conditions on if , the natural evaluation map is surjective.

Clearly, imposes independent conditions on if and only if subset , the natural restriction map of maximal rank.

Remark 1.2.

Note that if imposes independent conditions on , then, in particular, the points in are different from the possible base points of , so that the locus of curves in passing through at least one point of is an effective divisor in consisting of hyperplanes. Therefore the condition that imposes independent conditions on can be rephrased as saying that the locus of curves in passing through at least one point of is an effective, simple normal crossing divisor consisting of hyperplanes.

If the cardinality , then the condition that imposes independent conditions on is equivalent to the condition that at most points of can lie on an element of .

For positive integers let

We will assume through out that

(1)
Definition 1.3.

Let be a smooth irreducible curve. We say the pair can impose independent conditions on if there exists such that is a set of distinct points and imposes independent conditions on

Remark 1.4.

In later sections, the surface will be a smooth and complete intersection surface (K3 or rational). The nodes of a general CICY threefold containing will be a complete intersection on , say where and . The positive integers and are determined by complete intersection types of and . (cf. Table 1 in section 2). We want to show imposes independent conditions on the chosen complete linear system on . Actually, our goal is to show that there is a dense subset of such that for any the intersection is a reduced 0-cycle on and it imposes independent conditions on . Clearly this is reduced to show the following statement: for any fixed smooth irreducible member there exists a member such that is a set of distinct points that impose independent conditions on . This simple observation motivates the above definition, and this reduction will allow us to bring to bear the classical theory of divisors on a Riemann surface.

From now on, let be any fixed irreducible smooth curve.

In this paper two different new methods that can be used to show “the pair can imposes independent conditions on . ” will be presented. The first one, that we call Method I, is a sort of generalization of Knutsen’s method in [5] (cf. [5, Lemma 6.3]). The second one, Method II, is completely new. For application purposes, in all cases in Theorem 2.9, Theorem 2.12 and Theorem 3.1 of this paper we apply Method I to show without using Method II at all. Some cases in Theorem 2.9, Theorem 2.12 and Theorem 3.1 also follow from Method II but the others do not. However, theoretically it is possible that in certain situations we can not apply Method I (for example, the condition i) of Corollary 1.8 is not satisfied.) but we can still apply Method II. It is hoped that Method II can find applications somewhere else. Because we only use Method I in §2 and §3, Method II is put in the Appendix A.

1.1. Method I

Proposition 1.5.

Define , where is the n-th symmetric product of . We assume . Then can impose independent conditions on if the following two conditions are satisfied:

i) is irreducible;

ii) the following restriction map is injective

Proof.

Suppose conditions i) and ii) are satisfied.

Define the restriction map is not injective. Let be the natural projection map. since is a finite surjective map.

Define the restriction map is injective By condition ii), is open dense in So without loss of generality, we can assume the in ii) consists of distinct points.

Now because the restriction map is injective, by linear algebra subset , is a set of distinct points and the restriction map is injective too. Therefore, no member of can contain , which implies but By i) So cannot dominate via the map .

On the other hand, all divisors in the complementary of in impose independent conditions on . By the assumption , the natural restriction map is surjective. So can impose independent conditions on

The following general result gives us a nice criterion for the irreducibility of .

Lemma 1.6.

Let be any smooth projective curve and is a very ample line bundle on . , . Define the incidence variety , where . Then is irreducible if .

Proof.

If or , obviously is irreducible.

Suppose . The complete linear system induces an embedding with and is an isomorphism. Then is a nondegenerate algebraic curve of degree . Let be the set of hyperplanes of . By abusing notation, we can identify with , and use them interchangeably later on.

Define distinct points, and any set of imposes independent conditions on . By Castelnuovo’s general position theorem, is open dense in .

Let . Fix any . Because is dense in , we can find a one-parameter family of hyperplanes such that: , and . Clearly then we can find , specialize to . Therefore, , which implies is open dense in .

In order to show is irreducible, we only need to show is irreducible. To this end, consider the projection map , where . Let , i.e. imposes independent conditions on . Clearly is open dense in . Furthermore, is irreducible since the fibers are irreducible and of the same dimension. Because open subset, is irreducible too.

Next, suppose . Define . By the argument above, we know that is irreducible. However, obviously so is irreducible too.

The lemma is proved.

Remark 1.7.

All cases in Knutsen’s paper [5] satisfy in Lemma 1.6. Actually, in [5] the method used to prove requires (cf.[5, Lemma 6.3]). All cases in this paper can only satisfy .

Corollary 1.8.

We assume . Then can impose independent conditions on if the following two conditions are satisfied:

i) ;

ii) .

Furthermore, implies the condition ii).

Proof.

implies either or , so is irreducible by Lemma 1.6.

Clearly implies the condition ii) in Proposition 1.5.

Next, we need to show implies .To this end let’s consider the following exact sequence of sheaves,

Taking cohomology groups, we have exact. By assumption and hence since , so implies .

Remark 1.9.

In §2 and §3, we will just use Corollary 1.8. to show for all cases.

2. Curves on K3 surfaces in nodal Calabi-Yau threefolds

In the rest of this paper, we will use Theorem 0.1 to show the existence of smooth and isolated curves in general complete intersection Calabi-Yau (CICY) threefolds. In this section, these curves are obtained by deforming a careful chosen continuous family of curves on a complete intersection K3 surface in a nodal CICY threefold.

We first recall how we can embed a complete intersection K3 surface into a nodal CICY threefold. We will follow notations used in [5, §6].

It is well known that there are three types of complete intersection K3 surfaces in projective space, namely the intersection types (4) in , (2,3) in and in . Similarly, there are five types of CICY threefolds in projective space, namely the intersection types in , and in , in and in .

Table 1

4
4
5
5
5
5
6
6
7
Remark 2.1.

Table 1 here is a part of [5, Table 1 in §6]. Notice that in [5, Table 1 in §6] the complete intersection types of are denoted as .

Our goal is to embed a given smooth complete intersection K3 surface of type into a nodal CICY threefold of type . To this end, we first choose generators of degrees for the ideal of . So

Define

where are general in if and are general in .

Then define

If the coefficient forms are chosen in a sufficient general way, has only ordinary double points and they lie on . This can be checked using Bertini’s theorem. In fact, the nodes are the intersection points of two general elements of and (distinct, when ). As above, we denote the set of nodes by .

Moreover, for general , Bertini’s theorem yields that the fourfold

is smooth. (Note that if .)

We are therefore in the setting of Theorem 0.1 with ,

and .

Remark 2.2.

Actually, as mentioned in §1, the integers and correspond to the integers and used in §1.

As mentioned in the introduction, in this section we will prove the main theorems Theorem 2.9 and Theorem 2.12.

To apply Theorem 0.1 to prove Theorem 2.9 and Theorem 2.12, we first need a smooth regular surface and a line bundle on . All surfaces which will be used in the proofs of Theorem 2.9 and Theorem 2.12 are complete intersection K3 surfaces as in Table 1 with , where is the hyperplane section of and is a smooth irreducible curve on . Furthermore, the genus and degree of are exactly the genus and degree of the desired smooth and isolated curves in general CICY threefolds. Then we define to be . The existence of these K3 surfaces are guaranteed by the following theorem due to Knutsen.

Theorem 2.3.

([6, Theorem 1.1]). Let be integers. Then there exists a K3 surface of degree in containing a smooth curve of degree d and genus if and only if

(i) and there exist integers and such that and divides ,

(ii) except in the following cases

(a) ,

(b) and ,

(c) and ,

(d) and or divides ,

(iii) and is not divisible by ,

(iv) and .

Furthermore, in case (i) can be chosen such that and in cases (ii)-(iv) such that , where is the hyperplane section of .

If can be chosen to be scheme-theoretically an intersection of quadrics in cases (i), (iii) and (iv), and also in case (ii), except when and or and , in which case has to be an intersection of both quadrics and cubics.

In order to apply Theorem 0.1, we need to show that can be embedded into a nodal CICY threefold such that the conditions (A1)-(A7) mentioned in the introduction are all satisfied. As explained in the proofs of Theorem 2.9 and Theorem 2.12, all of them except the condition can be easily verified by using results in [5]. We will use Corollary 1.8 to check .

Definition 2.4.

Let X be a K3 surface. A divisor is called -2 divisor if the self-intersection .

Remark 2.5.

It is easy to see that a K3 surface has a -2 divisor if and only if it contains a smooth rational curve.

In order to use Corollary 1.8, we need to show , where is any smooth irreducible member in . As explained in the Corollary 1.8, it suffices to show . Then there are two different situations: 1) the K3 surfaces do not have -2 divisors; 2) the K3 surfaces have -2 divisors. In the first situation, it is very easy to check . (Cf. Lemma 2.8) In the second situation, we explicitly compute the closed cone of curves and the nef cone and then use the information about these cones to check .

2.1. doesn’t have -2 divisors

2.1.1. Vanishing of the first cohomology group of line bundles on a K3 surface without -2 divisors

Proposition 2.6.

([10, Cor. 3.2]). Let be a complete linear system on a K3 surface. Then has no base points outside its fixed components.

Proposition 2.7.

([6, Prop. 2.3]) Let be a complete linear system without fixed components on a K3 surface such that . Then every member of can be written as a sum where for and is a smooth curve of genus 1.

In other words, is a multiple of an elliptic pencil.

In particular, if is part of a basis of , then the generic member of is smooth and irreducible.

Lemma 2.8.

Let be a K3 surface without -2 divisors. Let be a divisor on . Then if and only if the following two conditions are satisfied

(i)

(ii) a smooth elliptic curve on and an integer , and .

In particular, if is part of a basis of , then (ii) is automatically true.

Proof.

By Riemann-Roch, easily implies . also implies (ii). Otherwise, a smooth elliptic curve on and an integer , and . We may assume is positive, then we have an exact sequence of cohomology groups . But and , so and hence by Serre duality, contradiction.

On the other hand, suppose both (i) and (ii) are satisfied. Firstly, if , by R-R, either or is non-empty. We may assume is effective. Every irreducible curve on has non-negative self-intersection, so the linear system has no fixed components, and hence is base point free by Proposition 2.6. Therefore, every irreducible curve on is a nef divisor, and hence every effective divisor on is nef. So is nef and big, then by Kawamata-Viehweg vanishing Theorem. Secondly, if , again we may assume is effective. Clearly doesn’t have fixed components, then by Proposition 2.7 and (ii) is actually an elliptic pencil. Therefore, contains an irreducible member, and hence . Lastly, if , then both and are empty, by R-R.

2.1.2. Curves on deformed with CICY threefolds

Theorem 2.9.

Let and be integers. Then in any of the following cases the general Calabi-Yau complete intersection threefold of a particular type contains an isolated, smooth curve of degree and genus :

(a) and ; and ; and ; and ; and .

(b) and ; and ; and ; and ; and ; and ; and ; and ; and ; and ; and ; and .

(c) and .

(d) and .

(e) and ; and .

Proof.

Case (a) g=23 and d=18:

By Theorem 2.3, there exists a surface of degree 6 in with , where is the hyperplane section of and is a smooth irreducible curve of degree 18 and genus 23. Clearly, is actually a complete intersection K3 surface of type (2,3) in . is defined to be the line bundle .

Using notations introduced above (also the same notations as in [5, Table 1 in §6] except that [5] doesn’t use and ). : (cf. [5, Table 1 in §6]), and .

There are two conditions in [5, Prop 7.2]:

(7.1)

or

(7.2)

The condition (7.1) is satisfied since . The trouble is that condition (7.2) is not satisfied (notice that in the language of Lemma 1.6, condition (7.2) is exactly ). However, by looking closely at the proof of [5, Prop 7.2], the condition (7.2) is only used to prove the following two statements:

Statement 1). , where is the number of nodes on a general quintic threefold containing as before.

Statement 2). For general , the set of nodes imposes independent conditions on , .

Therefore, in order to get the conclusion of [5, Prop. 7.2] we only need to show statements 1) and 2).

. Statement 1) is proved.

As in §1, we let . Next we are going to use Corollary 1.8 to show .

Notice that in §1, we assume throughout . So we need to show that in current situation we do have . (Actually, the following proof for can be found in the proof of [5, Prop. 7.2]. For the reader’s convenience, we repeat it here.) By [6, Prop. 1.3] for all if and only if

which is condition (7.1). Next We note from the cohomology of

twisted by , Kodaira vanishing and Serre duality, that

so that also if condition (7.1) holds, as we have just seen.

Now as in §1 we fix any smooth irreducible . First of all, condition i) is satisfied since and . In order to show condition ii), we only need to prove . However, using some softwares (e.g. Mathematica) it is very easy to check that has no -2 divisors. For example, the following picture shows us how to do this by using Mathematica. We use to represent a divisor .

By Lemma 2.8, since . By Serre duality, . Therefore, condition ii) in Corollary 1.8 is also true. So can impose independent conditions on . Then by Bertini’s Theorem, for general , the set of nodes imposes independent conditions on . Therefore, we have the conclusion of [5, Prop7.2], which says the conditions (A1)-(A7)are satisfied. Then by Theorem 0.1, the general quintic threefold in contains an isolated, smooth curve of degree 18 and genus 23.

The proofs for all the other cases are similar to that for the case (a) and . However, we need to specify what the complete intersection type of K3 surface is for each case. The information needed is listed in Table 2 in the Appendix B. (All K3 surfaces in this table have no -2 divisors.)

2.2. X has -2 divisors

2.2.1. Nef cone of K3 surfaces with -2 divisors

If a K3 surface does not have -2 divisors, it is very easy to compute the closed cone of curves and hence the nef cone . (Cf. [8, Corollary 2.3]). If there are -2 divisors in , it could be difficult to compute the nef cone of in general. However, if the Picard number of is 2, it is not hard to do so. In this subsection we will assume throughout that the K3 surface has a -2 divisor and its Picard number is 2.

Lemma 2.10.

Let X be a smooth projective K3 surface with Picard number 2. Then , is an ample divisor, and .

Proof.

Choose any basis for , say . Because is projective, suppose is an ample class, and we may assume integers a and b are coprime. Then