Revista Matemática Iberoamericana

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Volume 28, Issue 3, 2012, pp. 815–838
DOI: 10.4171/RMI/692

Published online: 2012-07-16

Rationally cubic connected manifolds II

Gianluca Occhetta[1] and Valentina Paterno[2]

(1) Università di Trento, Povo (Trento), Italy
(2) Università di Trento, Povo (Trento), Italy

We study smooth complex projective polarized varieties $(X,\!H)$ of dimension $ n \ge 2$ which admit a dominating family $V$ of rational curves of $H$-degree $3$, such that two general points of $X$ may be joined by a curve parametrized by $V$ and which do not admit a covering family of lines (i.e., rational curves of $H$-degree one). We prove that such manifolds are obtained from RCC manifolds of Picard number one by blow-ups along smooth centers. If we further assume that $X$ is a Fano manifold, we obtain a stronger result, classifying all Fano RCC manifolds of Picard number $\rho_X \ge 3$.

Keywords: Rationally connected manifolds, rational curves, Fano manifolds

Occhetta Gianluca, Paterno Valentina: Rationally cubic connected manifolds II. Rev. Mat. Iberoam. 28 (2012), 815-838. doi: 10.4171/RMI/692