Related papers: Rationally cubic connected manifolds II
In this paper 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…
We study a particular class of rationally connected manifolds, $X\subset \p^N$, such that two general points $x,x' \in X$ may be joined by a conic contained in $X$. We prove that these manifolds are Fano, with $b_2\leq 2$. Moreover, a…
In this paper we investigate Fano manifolds $X$ whose Chern characters $ch_k(X)$ satisfy some positivity conditions. Our approach is via the study of polarized minimal families of rational curves $(H_x,L_x)$ through a general point $x\in…
The Picard number of a Fano manifold X obtained by blowing up a curve in a smooth projective variety is known to be at most 5, in any dimension greater than or equal to 4. We show that the Picard number attains to the maximal if and only if…
Let $X$ be a smooth complex projective variety and let $H \in \pic(X)$ be an ample line bundle. Assume that $X$ is covered by rational curves with degree one with respect to $H$ and with anticanonical degree greater than or equal to $(\dim…
As an application of a recent characterization of complete flag manifolds as Fano manifolds having only ${\mathbb P}^1$-bundles as elementary contractions, we consider here the case of a Fano manifold $X$ of Picard number one supporting an…
Let $X$ be a projective variety and let $C$ be a rational normal curve on $X$. We compute the normal bundle of $C$ in a general complete intersection of hypersurfaces of sufficiently large degree in $X$. As a result, we establish the…
Let X be a Fano manifold with Picard number one such that the tangent bundle T_X is big. If X admits a rational curve with trivial normal bundle, we show that X is isomorphic to the del Pezzo threefold of degree five.
For a generic anti-canonical hypersurface in each smooth toric Fano 4-fold with rank 2 Picard group, we prove there exist three isolated rational curves in it. Moreover, for all these 4-folds except one, the contractions of generic…
In this paper we address Fano manifolds X with a locally unsplit dominating family of rational curves of anticanonical degree equal to the dimension of X. We first observe that their Picard number is at most 3, and then we provide a…
We study smooth, complex Fano 4-folds X with a rational contraction onto a 3-fold, namely a rational map X-->Y that factors as a sequence of flips X-->X' followed by a surjective morphism X'->Y with connected fibers, where Y is normal,…
We classify smooth Fano manifolds X with the Picard number $\rho_X \geq 3$ such that there exists an extremal ray which has a birational contraction that maps a divisor to a point.
We investigate the possible homological classes of rational curves on the moduli space $X_n=\bar{\mathcal{M}_{0,n}}$ of rational nodal curves with $n$ marked points. In the case of $X_5$ and $X_6$ the relevant homology classes belong to…
We continue the study, begun by the second author in math.AG/0701889, of secant defective manifolds having "simple entry loci". We prove that such manifolds are rational and describe them in terms of tangential projections. Using also our…
Let (X,D) be a dlt pair, where X is a normal projective variety. Let S denote the support of the rounddown of D, and K the canonical divisor of X. We show that any smooth family of canonically polarized varieties over X\S is isotrivial if…
In this paper we address Fano manifolds with positive higher Chern characters. They are expected to enjoy stronger versions of several of the nice properties of Fano manifolds. For instance, they should be covered by higher dimensional…
Let $X$ be an $n$-dimensional complex Fano manifolds $(n\geq 3)$. Assume that $X$ contains a divisor $A$, which is isomorphic to a rational homogeneous space with Picard number one, such that the conormal bundle $\mathscr{N}^*_{A/X}$ is…
Given a morphism between smooth projective varieties $f: W \to X$, we study whether $f$-relatively free rational curves imply the existence of $f$-relatively very free rational curves. The answer is shown to be positive when the fibers of…
For $X$ a smooth cubic threefold we study the Pl\"ucker embedding of the Fano surface of lines $S$ of $X$. We prove that if $X$ is general then the minimal gonality of a covering family of curves of $S$ is four and that this happens for a…
Pasquier and Perrin discovered that the ${\rm G}_2$-horospherical manifold ${\bf X}$ of Picard number 1 can be realized as a smooth specialization of the rational homogeneous space parameterizing the lines on the 5-dimensional hyperquadric,…