Related papers: Rational endomorphisms of Fano hypersurfaces

We describe two ways to construct finite rational morphisms between fiber products of rational elliptic surfaces with section and some Calabi--Yau manifolds. We use them to construct correspondences between such fiber products that admit at…

We study the spaces of rational curves on Fano threefolds with Gorenstein terminal singularities. We generalize the results regarding Geometric Manin's Conjecture for smooth Fano threefolds, including the classification of subvarieties with…

In this paper a large class of Fano double quadrics and cubics are shown to be factorial and birationally superrigid, in particular they admit no non-trivial structure of a fibration with rationally connected fibres and are therefore…

We prove analogues of several well-known results concerning rational morphisms between quadrics for the class of so-called quasilinear $p$-hypersurfaces. These hypersurfaces are nowhere smooth over the base field, so many of the geometric…

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…

A classical result of Bondal-Orlov states that a standard flip in birational geometry gives rise to a fully faithful functor between derived categories of coherent sheaves. We complete their embedding into a semiorthogonal decomposition by…

We prove a structural result for geometrically non-reduced varieties and give applications to Fano varieties. For example, we show that if $X$ is the generic fibre of a Mori fibre space of relative dimension $n$, and the characteristic is…

Given a minimal surface equipped with a generically finite map to an Abelian variety, we give an optimal bound on the canonical degree of a rational or an elliptic curve. As a corollary, we obtain the finiteness of rational and elliptic…

It is shown that a formal mapping between two real-analytic hypersurfaces in complex space is convergent provided that neither hypersurface contains a nontrivial holomorphic variety. For higher codimensional generic submanifolds,…

For a one-parameter family of general type hypersurfaces with bases of holomorphic n-forms, we construct open covers using tropical geometry. We show that after normalization, each holomorphic n-form is approximately supported on a unique…

We complete the study of birational geometry of Fano fiber spaces $\pi\colon V\to {\mathbb P}^1$, the fiber of which is a Fano double hypersurface of index 1. For each family of these varieties we either prove birational rigidity or produce…

We prove divisorial canonicity of Fano double hypersurfaces of general position.

A Fano-Enriques threefold is a three-dimensional non-Gorenstein Fano variety of index 1 with at most canonical singularities. We study the birational geometry of Fano-Enriques threefolds with terminal cyclic quotient singularities. We…

We show that being a general fibre of a Mori fibre space is a rather restrictive condition for a Fano variety. More specifically, we obtain two criteria (one sufficient and one necessary) for a Q-factorial Fano variety with terminal…

We consider a rational surface with a relatively minimal fibration. Picard number of a such fibred surface is bounded in terms of the genus of a general fibre. When Picard number is the maximum for any given genus, we characterize a such…

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 singularities on fibrations and related topics. We prove conjectures of McKernan and Shokurov on singularities on Fano type fibrations and a conjecture of the author on singularities on log Calabi-Yau…

In this paper we study boundedness properties and singularities of log Calabi-Yau fibrations, particularly those admitting Fano type structures. A log Calabi-Yau fibration roughly consists of a pair $(X,B)$ with good singularities and a…

We present a variant of the classical Darboux-Jouanolou Theorem. Our main result provides a characterization of foliations which are pull-backs of foliations on surfaces by rational maps. As an application, we provide a structure theorem…

For a Fano manifold of pseudo-index at least 3 and $c_1^2-2c_2$ nef, we show irreducibility of certain spaces of curves on the Fano manifold implies the manifold is a union of rational surfaces.