Rational curves on genus-one fibrations
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© The Author(s) 2020
Fabrizio Anella
Rational curves on genus-one fibrations Received: 18 July 2019 / Accepted: 30 October 2020 Abstract. 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 → B does contain vertical rational curves if and only if it not isomorphic to a finite étale quotient of a product B˜ × 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. Conjecture 0.1. Every (possibly singular) Calabi–Yau variety does contain rational curves. This conjecture is unsolved even for smooth Calabi–Yau manifolds in dimension three. We started studying Calabi–Yau varieties that admit a genus-one fibration and we got a positive answer in [2, Theorem 1]. At this point it is natural to ask the following question. Question 0.2. Under which conditions a genus-one fibration does contain rational curves? Without asking anything on the base of the fibration, we can say something only on the rational curves that are vertical for the fibration. The main purpose of this article is to prove the following answer to Question 0.2 that gives a complete characterization in the case of relatively minimal genus-one fibration. π
→ B be genus-one fibration such that K X ∼Q π ∗ L for some Theorem 0.3. Let X − Q-Cartier Q-divisor on B. Then X does not contain vertical rational curves if and only if there exists a finite cover B˜ of B and a genus-one curve E such that X is a finite étale quotient of B˜ × E over B. F. Anella (B): Dipartimento di Matematica e Fisica, Università Roma 3, Largo San Leonardo Murialdo 1, 00146 Rome, Italy. e-mail: [email protected]; [email protected] Present Address: F. Anella: Mathematisches institut, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany Mathematics Subject Classification: Primary: 14J32 · Secondary: 14E30, 14D06
https://doi.org/10.1007/s00229-020-01259-2
F. Anella
A key ingredient for the proof of this result is the proof of the following theorem, that is a generalization of [2, Theorem 1] and of [7, Theorem 1.1]. Theorem 0.4. Let (X, ) be a klt pair such that there exists a surjective morphism φ : X → B to a variety of dimension n − 1. Suppose moreover K X + ∼Q φ ∗ L for some Q-Cartier Q-divisor L on B and the augmented irregularity of X is zero, then there exists a subvariety of X of dimension n − 1 covered by rational curves contracted by φ. This theorem has some interesting consequences, like the following. Corollary 0.5. Let X be a projective variety of dimension n with log terminal singularities, κ(X ) = n − 1 and q(X ˜ ) = 0. The variety X does contain rational curves. Using a very nice result of Lazi´c and Peternell [20, Theorem 6.12] we can do slightly better in the smooth case. Corollary 0.6. L
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