Related papers: Higher Fano manifolds and rational surfaces

The minimum of intersection numbers of the anti-canonical divisor with rational curves on a Fano manifold is called pseudo-index. It is expected that the intersection number of anti-canonical divisor attains to the minimum on an extremal…

We prove that very general non-rational Fano threefolds which are not birational to cubic threefolds are not stably rational.

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…

Chains of minimal degree rational curves have been used as an important tool in the study of Fano manifolds. Their own geometric properties, however, have not been studied much. The goal of the paper is to introduce an infinitesimal method…

The main result is that a quasi-projective surface has negative log Kodaira dimension (i.e. no log pluricanonical sections) iff it is dominated by images of the affine line. This follows from our main intermediate result, that the smooth…

We study the space of rational curves on del Pezzo surfaces in positive characteristic. For most primes p we prove the irreducibility of the moduli space of rational curves of a given nef class, extending results of Testa in characteristic…

Let F be a polarized irreducible holomorphic symplectic fourfold, deformation equivalent to the Hilbert scheme parametrizing length-two zero-dimensional subschemes of a K3 surface. The homology group H^2(F,Z) is equipped with an integral…

We exhibit several families of Fano threefolds with a multiplicative Chow-K\"unneth decomposition, in the sense of Shen-Vial. As a consequence, a certain tautological subring of the Chow ring of powers of these threefolds injects into…

Let X be a compact K\"ahler manifold such that the anticanonical bundle $-K_X$ is nef. A classical conjecture claims that the Albanese map is submersive. We prove this conjecture if the general fibre is a weak Fano manifold. If X is…

It is proved that a general Fano hypersurface of index 1 (in the projective space) with isolated singularities of general position is birationally rigid. Therefore it cannot be fibered into uniruled varieties of a smaller dimension by a…

We prove the irreducibility of the spaces of rational curves on del Pezzo manifolds of Picard rank 1 and dimension at least 4 by analyzing the fibers of evaluation maps. As a corollary, we prove Geometric Manin's Conjecture in these cases.

We show that the degrees of rational endomorphisms of very general complex Fano and Calabi-Yau hypersurfaces satisfy certain congruence conditions by specializing to characteristic p. As a corollary we show that very general n-dimensional…

The purpose of this note is to show that the minimal $e$ for which every smooth Fano hypersurface of dimension $n$ contains a free rational curve of degree at most $e$ cannot be bounded by a linear function in $n$ when the base field has…

In this paper, we investigate Fano manifolds whose Chern characters satisfy some positivity conditions. We prove that such manifolds admit long chains of higher order minimal families of rational curves and are covered by higher rational…

We prove several classification results for the components of the moduli space of rational curves on a smooth Fano threefold. In particular, we prove a conjecture of Batyrev on the growth of the number of components as the degree increases.…

This paper explores the Fano variety of lines in hypersurfaces, particularly focusing on those with mild singularities. Our first result explores the irreducibility of the variety $\Sigma$ of lines passing through a singular point $y$ on a…

In this note we prove that any toric Fano manifold with nef tangent bundle is a product of projective spaces. In particular, it implies that Campana-Peternell conjecture hold for toric manifolds.

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…

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.

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…