Rational curves on fibered varieties
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Mathematische Zeitschrift
Rational curves on fibered varieties Fabrizio Anella1,2 Received: 18 July 2019 / Accepted: 28 August 2020 © The Author(s) 2020
Abstract Let X be a complex projective variety with log terminal singularities and vanishing augmented irregularity. In this paper, we prove that if X admits a relatively minimal genus-one fibration, then it contains a subvariety of codimension one covered by rational curves contracted by the fibration. We then focus on the case of varieties with numerically trivial canonical bundle and we discuss several consequences of this result. Keywords Genus-one fibrations · Elliptic fiber spaces · Calabi–Yau varieties · Rational curves Mathematics Subject Classification 14J32 · 14E30 · 14D06
Introduction The existence of rational curves on algebraic varieties is an interesting and often challenging problem. In the case of Calabi–Yau varieties, for instance, their existence is fully proved only in dimension two by Bogomolov–Mumford [28, Appendix], while in higher dimensions we have only partial results. On K3 surfaces, any ample linear series contains rational curves; this result justifies the definition of the Beauville–Voisin class as the zero-cycle class of a point on a rational curve [3]. Defining this class in higher dimensions is much more difficult, because it is hard to find i
an ample divisor H − → X with i ∗ (CH0 (H )) = Z, and we do not expect in general that there exists a divisor such that CH0 (H ) = Z, e.g. H rational. The experience with the minimal model program suggests that, even when one is mainly interested in smooth varieties, the natural setting is to allow, at least, log terminal singularities. The aim of this paper is to extend the results proven in [5] to the singular setting typical of the
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Fabrizio Anella [email protected]; [email protected]
1
Department of Mathematics and Physics, Roma Tre University, Largo San Leonardo Murialdo 1, 00146 Rome, Italy
2
Present Address: Mathematisches institut, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany
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F. Anella
minimal model program. Some of the techniques we use lead us to prove some new results also in the smooth case. The main result of this paper is the following. Theorem 1 Let X be a complex normal projective variety of dimension n with log terminal singularities and vanishing augmented irregularity, i.e. the irregularity of any quasi-étale cover of X is zero. Suppose that there exists a surjective morphism φ from X to a variety B of dimension n − 1. If there exists a Cartier divisor L on B such that φ ∗ L ∼ K X , then there exists a subvariety of codimension one in X that is covered by rational curves contracted by φ. If the variety X has numerically trivial canonical bundle, as a corollary of this theorem we have the following result. Corollary 1 Let X be a Calabi–Yau variety of dimension n as in Definition 5. Suppose that there exists a surjective morphism φ from X to a variety B of dimension n − 1. Then there exists a subvariety of codimension one in X t
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