Rational curves on genus one fibrations

In this paper we look for necessary and sufficient conditions for a genus one fibration to have rational curves. We show that a projective variety with log terminal singularities that admits a relatively minimal genus one fibration $X\rightarrow B$ does contain vertical rational curves if and only if it not isomorphic to a finite \'etale quotient of a product $\tilde{B}\times E$ over $B$. Many sufficient conditions for the existence of rational curves in a variety that admits a genus one fibration are proved in this paper.


Introduction
The starting point of this work has been the following folklore conjecture.
This conjecture is unsolved even in dimension three. We started studying Calabi-Yau varieties that admit an elliptic fibration and we got a positive answer in [Ane18]. At this point it is natural to ask the following question. The main purpose of this article is to prove the following partial answer to Question 0.2.
Theorem 0.5. Let X be a smooth projective variety of dimension n ≥ 2 andq(X) = 0. If X is covered by elliptic curves, then it contains a rational curve.
In Section 1 we fix some notations and definitions used in the subsequent parts. In Section 3 we prove Theorem 0.3. The proof is organized in several lemmas which are of some interest. In Section 4 we explain some generalizations and consequences of Theorem 0.3 and we give other partial answers to Question 0.2. An interesting generalization of Theorem 0.3 is Theorem 4.1. There will be a more accurate analysis of Question 0.2 in the forthcoming PhD thesis of the author.
Acknowledgements. The author would like to thank Andreas Höring for important comments and discussions during his visit in Rome. The author would also like to thank his advisor, Simone Diverio, for the continuous help provided during this work.

Preliminaries
In this paper every variety will be an irreducible projective variety over the complex numbers. The variety X will be always normal and of dimension n ≥ 2. The notations and standard properties about singularities that is used in this article can be found for example in [KM98]. For the reader's convenience we recall some definitions.
Definition 1.1. Let φ : X → B be a surjective morphism of normal projective varieties an D ∈ WDiv(X) be a prime Weil divisor. We say that D is φ-exceptional if cod B (φ(D)) ≥ 2. We say that D is of insufficient fiber type if cod B (φ(D)) = 1 and there exists another prime Weil divisor D ′ = D such that φ(D ′ ) = φ(D). In either of the above cases, we say that D is degenerate.
Definition 1.2. A morphism f : Z → Y between normal varieties is called quasi-étale if f is quasi-finite andétale in codimension one. The augmented irregularity of a variety Y is the following, not necessarily finite, positive integerq Remark 1.3. The above definition of quasi-étale morphism is not the same of [Cat07].
Remark 1.5. The image of the singular locus of f i.e. f (sing(f )) 1 , is the singular values of f .
One can associate to any elliptic curve a complex number called its jinvariant. This association is modular, which means that an elliptic family f : Y → B comes with a rational map j : B C called j-function that is at least defined over the smooth values of f . For some standard facts about the j-function of an elliptic family the reference can be found in [Kod63].
In some recent works on Beauville-Bogomolov decomposition (see for example [GKP16], [HP17], [Dru18] ) appeared several definitions of Calabi-Yau varieties. In our Definition 1.6 we include also products of Calabi-Yau and irreducible holomorphic symplectic varieties in the sense of [HP17].
Definition 1.7. A klt variety Y (resp. a pair (Y, ∆)) with a fibration Y f → B is a Calabi-Yau fiber space (resp. a log Calabi-Yau fiber space) if the generic fiber Y t of f (resp. the pairs (Y t , ∆ t )) has numerically trivial canonical bundle (resp. K Yt + ∆ t ≡ num 0).
If the general fiber is a curve we are mainly interested in the case the boundary ∆ does not intersect the general fiber. Indeed if the intersection is non-trivial then Y is uniruled.
Definition 1.8. An elliptic fiber space is a morphism π : Y → B between normal projective varieties with connected fibers and such that the general fiber is a smooth genus one curve.
A particular case of Calabi-Yau fiber space is the following.
An important class of examples of Calabi-Yau fiber spaces is given by orbibundles. We just recall the construction of orbibundles because they are a key point in the proof of Theorem 0.3. More properties and details can be found in the article of Kollár [Kol15].
Definition 1.10. LetB be a normal variety, F a klt variety with K F ≡ num 0 andỸ :=B×F their product. Let G be a finite group and ρ B :

Some remarks on the augmented irregularity
The augmented irregularity is not a birational invariant for projective varieties with canonical singularities. Indeed the standard construction of a Kummer surface is a counterexample: take an elliptic curve E and consider the quotient X := E×E ± of the product of two copies of E by the involution.
The quotient map E × E → X is a quasi-étale cover soq(X) ≥ 2, moreover by [Dru18,Remark 4.3] it holds the equality. However a minimal resolutioñ X of X is a K3 surface. In particularX is simply connected, henceq(X) = 0.
The variety X is an example of a regular variety with non trivial augmented irregularity.
It is natural to ask whether there exists some manageable conditions for the vanishing of the augmented irregularity of a variety. It is easy to check [Ane18, Remark 1.14 ] that a variety X, say smooth for simplicity, with finite fundamental group hasq(X) = 0. For varieties with numerically trivial canonical divisor an interesting characterization is given in [GGK17, Theorem 11.1]. We prove that an implication still holds without the assumption on the canonical bundle: the following is a sufficient condition for the vanishing of the augmented irregularity which does not rely on computations of invariants on quasi-étale covers, but only on invariants of the variety under investigation.
Proposition 2.1. Let X be a projective variety with klt singularities. If [KM98,Proposition 5.20] and by [GGK17, Proposition 6.9] there is a non-zero reflexive form ω ∈ H 0 (X, Ω By definition the sections of a reflexive sheaf are exactly the sections on the regular part of X. So to construct a non-zero global section of Sym [k] Ω 1 X we construct an element in H 0 (X reg , Sym k Ω 1 Xreg ). Now we consider just the restriction to the regular of X: This is anétale finite cover, so we can find a furtherétale finite cover over the regular partỸ → X reg that is Galois. Let G be the group of deck trasmormations ofỸ over X reg . By abuse of notations we call again ω the pullback toỸ of ω. Now consider the sectionα : . This section is invariant under the action of the deck trasformations, so it descends to a section α of H 0 (X reg , (Ω 1 Xreg ) ⊗N ). By construction it is easy to check that this section is symmetric, i.e. ω belongs to H 0 (X reg , Sym N (Ω 1 Xreg )). It is harder to prove thatα, and hence α, is non-zero.
For any non-zero element γ ∈ H 0 (Ỹ , Ω 1 Y ) and a generic point p ∈Ỹ the space Ker(γ) ⊂ TỸ ,p is a proper subspace. Since ω = 0 and the elements ρ ∈ G are automorphisms (and we are working over C), also ρ * ω are nonzero elements in H 0 (Ỹ , Ω 1 Y ). So for generic p ∈Ỹ we can choose a tangent vector Now we can evaluate our sectionα at the vector v ⊕N . The computations are the following: So we have constructed a non-zero section of H 0 (X reg , Sym N Ω 1 Xreg )) that corresponds to a non-zero section of H 0 (X, Sym [N ] Ω 1 X )

Proof of the main theorem
We start with a lemma that is already stated in [DFM16].
Lemma 3.1. Let X π → B be an elliptic fiber space. If we suppose the subvariety of the singular values of π, i.e. Z := π(sing(π)), has codimension at least two then the family π is isotrivial.
Proof. Since B is normal it is smooth in codimension one and also the subvariety Z ∪ B sing has codimension at least two. We denote B 0 := Z c ∩ B reg . It is well defined the regular function that sends a point t ∈ B 0 the j-invariant of the elliptic curve over t [Kod63]. Since (B 0 ) c has codimension at least two and B is normal, the function j 0 can be extended to a function j : B → C. Since B is projective this function must be constant. This means that the fibers over B 0 are isomorphic. Now we study the general fibers over the singular locus of an elliptic fibration.
Lemma 3.2. Let φ : X → B be an elliptic fibration and Z := φ(sing(φ)). Suppose that cod B (Z) = 1, then a general fiber over Z is mE + m i R i where E is an elliptic curve and R i are rational curves.
Proof. We can study the restriction of φ to a surface as follows. Let H be a very ample divisor on B such that (n − 2)H + L is globally generated. The pullback φ * H is a globally generated Cartier divisor. Moreover there is an isomorphism because φ has connected fibers. This implies that general elements in |H| are general also in |φ * (H)|. So we choose n − 2 general divisors D 1 , . . . , D n−2 ∈ |H| such that C := D 1 ∩ . . . ∩ D n−2 is a smooth irreducible curve in B reg not contained in the locus of singularities of φ and S := φ −1 (D 1 )∩. . .∩φ −1 (D n−2 ) is a normal surface. Looking at the Kodaira's table [BHPVdV04, Section V.7] it is easy to check that the singular fibers of φ| S are mE + m i R i where E is an elliptic curve and R i are rational curves. The condition on the dimension of Z insures that a general point in Z lies on a curve obtained as general intersection of hyperplane sections.
Remark 3.3. If X contains no uniruled codimension one subvarieties but Z has codimension one in B, then the fibers over any general point of Z of dimension n − 2 is a multiple elliptic curve.
Remark 3.4. It follows from Lemma 3.2 and from a result of Kawamata [Kaw91] that an elliptic fiber space with no uniruled codimension one subvarieties has no degenerate divisors. Lemma 3.2 can be seen as a soft version of Kodaira's table in higher dimension. With the same strategy of the proof of this lemma one can certainly do a better classification of singular fibers. Now we can merge together these lemmas and prove the following result. Proof of Lemma 3.5. We can assume by Lemma 3.1 that Z := φ(sing(φ)) has codimension one in B. The general fiber over Z is classified by Lemma 3.2. Since there are no uniruled codimension one subvarieties, in the general fiber over Z there are only multiple elliptic curves. Now we can proceed cutting with hyperplane sections as in the proof of Lemma 3.2. In this way we get many curves in B with only elliptic fibers (possibly multiple) over them. The j-invariant for multiple elliptic curves is well-defined as one can easily check with a semistable reduction.
For each curve C obtained in this way we get an holomorphic function C j → C that by compactness must be constant. From this fact it follows that the j-invariant is constant on B. We prove this fact by induction on the dimension of B. There is nothing to prove if the dimension is one. By induction we can suppose that the j-invariant is constant on an ample subvariety H ⊂ B. Since this function is constant along curves that are general complete intersections and these curves must intersect H, this constant must be the same. The union of these curves dominates B, so the j-invariant is constant on B. This means that the family X φ → B is isotrivial.
Finally we can proceed with the proof of our main result.
Proof of Theorem 0.3. We can suppose, eventually passing to Stein factorization, that φ has connected fibers and B is normal.
Since K X + ∆ ≡ num φ * L, the restriction (K X + ∆)| Xt ∼ O Xt to a general fiber of φ is trivial. It follows from standard arguments that a general fiber of φ is a smooth curve contained in the smooth locus of X, so by adjunction formula K Xt ∼ K X | Xt . This implies K Xt ≡ num −∆| Xt and hence the general fiber has genus at most one. If the genus is zero the variety X is uniruled, so we can suppose φ is an elliptic fibration. Note that even if the ∆ and K X are not Q-Cartier Q-divisor their restriction on a neighborhood of a general fiber X t is Q-Cartier.
If the fibration is non-isotrivial then there exists an uniruled divisor in X by Lemma 3.5. It remains to study the case φ is an elliptic fibration without exceptional divisors. Under these conditions by a result of Kóllar [Kol15, Theorem 44] X is birational over B to an orbibundle X orb . By construction the variety X orb has no φ-exceptional divisors. Since X and X orb are birational over B, both have no degenerate divisors (see Remark 3.4)and they are relatively minimal over B, they are isomorphic in codimension one. Indeed under these conditions the birational morphism can't contract any divisor over B. SinceX :=B × F → X orb is quasi-étale [Kol15, see Lemma 38] there exists an open subsetX • ⊂X such that the induced morphism X • → X isétale and codX(X • ) ≥ 2. By Zariski Main Theorem a quasi-finite morphism factorizes through an open immersion and a finite morphism, so there exists a commutative diagram with Y projective and the morphism ψ isétale over the image ofX • . Since ψ is finite and the complementary of the image ofX • in X has codimension at least two, the morphism ψ is a quasi-étale cover of X. By some basic properties of the reflexive sheaves and by [GKP16, Proposition 6.9] the following chain of isomorphisms holds So Y is a quasi-étale cover of X that has positive irregularity and this is a contradiction.

Applications and remarks
In this section we talk about some applications of Theorem 0.3. The following theorem is a generalization of Theorem 0.3.
Theorem 4.1. Let (X, ∆) be a klt pair withq(X) = 0 such that there exists a surjective morphism φ : X → B to a variety of dimension n − 1. Suppose moreover K X +∆ ≡ num φ * L+ a i E i for some Q-Cartier Q-divisor L on B, some exceptional divisor E i and not all the coefficients are strictly negative. Then X does contain rational curves.
Proof. We can write K X + ∆ ≡ num φ * L + D − D ′ with D and D ′ exceptional effective divisors with no common components. Since the pair (X, ∆) is klt D and D ′ are Q-Cartier Q-divisors. If there are no exceptional divisors this is Theorem 0.3. It follows from Hodge index Theorem that the divisor D is covered by curves that intersect negatively K X + ∆ [Lai11, Lemma 2.9]. In the part we need of [Lai11, Lemma 2.9] it is unnecessary the Q-factoriality condition, but it is sufficient that the exceptional divisor is a Q-Cartier Qdivisor. It follows that K X + ∆ is not nef and by cone theorem there are rational curves in X.
Remark 4.2. Up to consider the new pair K X + ∆ + εE i one can prove that there exists rational curves provided that not every −a i is bigger than the log-canonical threshold of E i .
Returning to Question 0.2, an elliptic fiber space X φ → B withq(X) = 0 with no rational curves (if it exists) must be isotrivial by Lemma 3.5; if X is klt then there are some exceptional divisors E i such that the canonical bundle is K X ∼ Q φ * L + a i E i (if B is Q-factorial), and every a i is strictly negative. Moreover we can't expect that these coefficients are too small or the canonical bundle can't be nef or too big by Remark 4.2.
We stop trying to generalize Theorem 4.1 and we mention some particular cases and prove Corollary 0.4.
Proof of Corollary 0.4. If the canonical bundle of X is not nef, then there are rational curves in X by the cone theorem. The numerical dimension of K X is greater than the Kodaira dimension n − 1. If ν(K X ) = n then the canonical bundle is big and X is of general type, that is a contradiction. So ν(K X ) = n − 1 = κ(K X ) and the canonical bundle is semi-ample. In particular the Iitaka fibration of the canonical bundle gives an elliptic fibration ϕ K X : In particular we can apply Theorem 0.3 and get the thesis.
We state Theorem 0.3 without boundary. Corollary 4.4. Let X φ → B be an elliptic fiber space withq(X) = 0 and K X ≡ num φ * L + a i E i . If we suppose that some a i is non-negative, then X does contain rational curves. This is exactly Theorem 4.1 without boundary. For smooth varieties we can do slightly better. A very recent and interesting work on varieties covered by elliptic curves can be found in [LP18]. For smooth varieties we can prove using their results Theorem 0.5.
Proof of Theorem 0.5. Suppose by contradiction that X does not contain rational curves. We can apply [LP18, Theorem 6.12] and find an equidimensional fiber space X → W . This fibration is relatively minimal and we can proceed as in the proof of Theorem 0.3 and find an irregular quasi-étale cover of X.
In particular this proves the following result.
Corollary 4.5. Let X be a smooth projective variety covered by elliptic curves but with no rational curves. Then the fundamental group of X is infinite.
We conclude with an useful criterion to find elliptic fibration due to Kóllar.
Theorem 4.6. Let X be a log terminal variety of dimension n with nef canonical bundle and L a Cartier divisor on X. Assume moreover 1) L n−2 · Td 2 (X) > 0.
This result is [Kol15, Theorem 10]. In the same article there is also a log version of this theorem.
The divisors L and K X are nef, hence L n−1 · K X ≥ 0. It follows that L n−1 · K X = 0.
It follows from Theorem 4.6 the following result.
Theorem 4.8. Let X be a variety with log terminal singularities andq(X) = 0. If there exists a line bundle L on X such that the conditions from 1 to 5 of Theorem 4.6 are satisfied, then X does contain rational curves.