Related papers: Fano 4-folds with a small contraction

We study Fano manifolds of pseudoindex greater than one and dimension greater than five, which are blow-ups of smooth varieties along smooth centers of dimension equal to the pseudoindex of the manifold. We obtain a classification of the…

We prove that a smooth projective variety $X$ of dimension $n$ with strictly nef third, fourth or $(n-1)$-th exterior power of the tangent bundle is a Fano variety. Moreover, in the first two cases, we provide a classification for $X$ under…

We show that deformations of a surjective morphism onto a Fano manifold of Picard number 1 are unobstructed and rigid modulo the automorphisms of the target, if the variety of minimal rational tangents of the Fano manifold is non-linear or…

Generalising toric geometry we study compact varieties admitting lower dimensional torus actions. In particular we describe divisors on them in terms of convex geometry and give a criterion for their ampleness. These results may be used to…

We prove that a fibration X \to \Bbb P_1, the general fiber of which is a smooth Fano threefold, is rationally connected. The proof is based on a generalization of Tsen's classical theorem: a fibration X/C over a curve the general fiber of…

In this paper, we introduce the notion of toric special weak Fano manifolds, which have only special primitive crepant contractions. We study the structure of them, and in particular completely classify smooth toric special weak Fano…

In this paper, we prove a special case of Campana--Peternell's conjecture in dimension 4. Specifically, we show that a projective smooth fourfold $X$ with $c^2_1(X)\cdot c_2(X)\neq 0$ and strictly nef anti-canonical divisor $-K_X$ is a Fano…

We prove that a general rational smooth Fano threefold admits a toric model. More precisely, for a general rational smooth Fano threefold $X$, we show the existence of a boundary divisor $D$ for which $(X,D)\simeq_{\rm cbir}…

Let $X$ be a smooth complex prime Fano fourfold having a semi-free action of $\mathbb{C}^*$, then $X$ is contained in one of the families $\mathbb{P}^4,Q^4,W_5, X^{m}_{8}$. All of the families contain members that have a semi-free…

We exploit an elementary specialization technique to study some properties of rational curves on index $n-1$ Fano $n$-folds. We prove a simple formula for counting rational curves passing through a suitable number of points in the case…

We prove rationality criteria over algebraically non-closed fields of characteristic $0$ for five out of six types of geometrically rational Fano threefolds of Picard number $1$ and geometric Picard number bigger than $1$. For the last type…

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…

We give some bounds on the anticanonical degrees of Fano varieties with Picard number 1 and mild singularities, extending results of Koll\'ar et al. from the early 90's and improving them even in the smooth case. The proof is based on a…

Let $X$ be a cubic fourfold that has only simple singularities and does not contain a plane. We prove that the Fano variety of lines on $X$ has the same analytic type of singularity as the Hilbert scheme of two points on a surface with only…

A Fano variety of Picard number $1$ is said to be \textit{birationally solid} if it is not birational to a Mori fiber space over a positive dimensional base. In this paper we complete the classification of quasi-smooth birationally solid…

The present paper provides a geometric characterization of complete flag varieties for semisimple algebraic groups. Namely, if $X$ is a Fano manifold whose all elementary contractions are $\mathbb P^1$-fibrations then $X$ is isomorphic to…

We determine the Cox rings of the minimal resolutions of cubic surfaces with at most rational double points, of blow ups of the projective plane at non-general configurations of six points and of three dimensional smooth Fano varieties of…

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…

Small codimensional embedded manifolds defined by equations of small degree are Fano and covered by lines. They are complete intersections exactly when the variety of lines through a general point is so and has the right codimension. This…

This paper considers Q-Fano 3-folds X with \rho=1. The aim is to determine the maximal Fano index f of X. We prove that f<= 19, and that in case of equality, the Hilbert series of X equals that of weighted projective space PP(3,4,5,7). From…