Related papers: Fano manifolds having (n-1,0)-type extremal rays w…

Let $X$ be a $\mathbb Q$-factorial canonical weak Fano variety of dimension $n\geq 2$. We show that if the $\mathbb Q$-Fano index $q_{\mathbb Q}(X)\geq 3$, then $X$ satisfies a Kawamata--Miyaoka type inequality: \[c_1(X)^n\leq 4\,\hat…

As a special case of a conjecture by Schwede and Smith, we prove that a smooth complex projective threefold with nef anti-canonical divisor is weak Fano if it is of globally $F$-regular type.

We consider some families of smooth Fano hypersurfaces $X_{n+2}$ in ${\bf P}^{n+2} \times {\bf P}^3$ given by a homogeneous polynomial of bidegree $(1,3)$. For these varieties we obtain lower bounds for the number of $F$-rational points of…

For a complex connected semisimple linear algebraic group $G$ of adjoint type and of rank $n$, De Concini and Procesi constructed its wonderful compactification $\bar{G}$, which is a smooth Fano $G \times G$-variety of Picard number $n$…

We prove the birational rigidity of Fano complete intersections of index 1 with a singular point of high multiplicity, which can be close to the degree of the variety. In particular, the groups of birational and biregular automorphisms of…

We describe the closed cone of moving curves of smooth Fano three- and fourfolds by giving finitely many equations that cut out the cone. The equations are induced by the exceptional divisors of divisorial contractions and by nef divisors…

We prove that a smooth Fano hypersurface $V=V_M\subset{\Bbb P}^M$, $M\geq 6$, is birationally superrigid. In particular, it cannot be fibered into uniruled varieties by a non-trivial rational map and each birational map onto a minimal Fano…

In this short note we give a characterization of smooth projective varieties of Picard number one that are separably uniruled but not separably rationally connected. We also give a sufficient condition involving the torsion order and the…

In the present paper we discuss coherent sheaves of rank > 1 whose projectivization gives rise to smooth varieties - varieties of this type are also called smooth scrolls. We prove some basic properties of these varieties and we give some…

We consider weak Fano manifolds with small contractions obtained by blowing up successively curves and subvarieties of codimension 2 in products of projective spaces. We give a classification result for a special case. In the process of…

We prove the boundedness theorem for Fano threefolds with log-terminal singularities of any fixed index. This is an improvement of our earlier result, where we required additionally that the variety is Q-factorial, with Picard number 1. The…

A V_{12} Fano threefold is a smooth Fano threefold X of index 1 with Pic X = Z and (-K_X)^3=12. We show that the bounded derived category of coherent sheaves on any V_{12} threefold X admits a semiorthogonal decomposition consisting of two…

We construct prime Fano manifolds from spin representations of $Spin_n$ for $n\le 14$. In this range, and if $n\ne 13$, the projectivizations of these representations are prehomogeneous, and we deduce that our Fano manifolds are locally…

T.Kishimoto raised the problem to classify all compactifications of contractible affine 3-folds into smooth Fano 3-folds with second Betti number two and classified such compactifications whose log canonical divisors are not nef. In this…

We show that the optimal upper bound for the anticanonical degrees of non-Gorenstein $\mathbb{Q}$-factorial canonical Fano threefolds with Picard number one is 200/3.

In this paper, we classify smooth toric Fano 5-folds of index 2. There exist exactly 10 smooth toric Fano 5-folds of index 2 up to isomorphisms.

We study the question whether rational homogeneous spaces are rigid under Fano deformation. In other words, given any smooth connected family f:X -> Zof Fano manifolds, if one fiber is biholomorphic to a rational homogeneous space S,…

A complete classification is presented of elliptic and K3 fibrations birational to certain mildly singular complex Fano 3-folds. Detailed proofs are given for one example case, namely that of a general hypersurface X of degree 30 in…

We consider the structure of the derived categories of coherent sheaves on Fano threefolds with Picard number 1 and describe a strange relation between derived categories of different threefolds. In the Appendix we discuss how the ring of…

For a toric Fano manifold $X$ denote by $Crit(X) \subset (\mathbb{C}^{\ast})^n$ the solution scheme of the Landau-Ginzburg system of equations of $X$. Examples of toric Fano manifolds with $rk(Pic(X)) \leq 3$ which admit full strongly…