Related papers: Birationally rigid singular double quadrics and do…

We study (smooth, complex) Fano 4-folds X having a rational contraction of fiber type, that is, a rational map X-->Y that factors as a sequence of flips followed by a contraction of fiber type. The existence of such a map is equivalent to…

Building on work of Segre and Koll'ar on cubic hypersurfaces, we construct over imperfect fields of characteristic p\geq 3 particular hypersurfaces of degree p, which show that geometrically rational schemes that are regular and whose…

It is proved that the group of birational automorphisms of a three-dimensional double quadric with a singular point arising from a double point on the branch divisor is a semidirect product of the free group generated by birational…

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 some conditions for the existence of higher dimensional algebraic fibering of group extensions. This leads to various corollaries on incoherence of groups and some geometric examples of algebraic fibers of type $F_n$ but not…

Following an idea of Ciliberto we show that double covers of projective r-space branched over an hypersurface of degree 2d are unirational provided r is sufficiently big with respect to d.

In this paper we give a criterion for birational rigidity of del Pezzo fibrations of degree 1 and 2 with only quotient singularities. As an application, we prove birational rigidity of suitable del Pezzo fibrations admitting an action of…

We prove that $n$-dimensional smooth hypersurfaces of degree $n+1$ are superrigid. Starting with the work of Fano in 1915, the proof of this Theorem took 100 years and a dozen researchers to construct. Here I give complete proofs, aiming to…

We give an algorithm to classify singular fibers of finite cyclic covering fibrations of a ruled surface by using singularity diagrams. As the first application, we classify all fibers of 3-cyclic covering fibrations of genus 4 of a ruled…

An affine algebraic variety $X$ is rigid if the algebra of regular functions ${\mathbb K}[X]$ admits no nonzero locally nilpotent derivation. We prove that a factorial trinomial hypersurface is rigid if and only if every exponent in the…

It is proved that the special fibre of the Nash transform of an irreducible isolated singularity has maximal dimension.

We construct singular quartic double fivefolds whose Kuznetsov component admits a crepant categorical resolution of singularities by a twisted Calabi--Yau threefold. We also construct rational specializations of these fivefolds where such a…

We prove a criterion of nonsingularity of a complete intersection of two fiberwise quadrics in a scroll over $P^1$. As a corollary we derive the following addition to the Alexeev theorem on rationality of standard Del Pezzo fibrations of…

We classify birational maps into elliptic fibrations of a general quasismooth hypersurface in $\mathbb{P}(1,a_{1},a_{2},a_{3},a_{4})$ of degree $\sum_{i=1}^{4}a_{i}$ that has terminal singularities.

We survey what is known about Fano threefold weighted complete intersections from the point of view of birational rigidity.

We describe the singular fibers of a parabolic fibration $f:X\to Y$ whose moduli divisor $M_Y$ is numerically trivial and discriminant divisor $B_Y$ is zero.

We study the polynomial fibration induced by the equation of the Klein surfaces obtained as quotient of finite linear groups of automorphisms of the plane; this surfaces are of type A, D, E, corresponding to their singularities. The generic…

We study the equivariant geometry of special quadric fibrations, called Umemura quadric fibrations, as well as the maximality of their automorphism groups inside $Cr_n(\mathbb{C})$. We produce infinite families of pairwise non-conjugate…

A rational elliptic surface with section is a smooth, rational, complex, projective surface $\mathcal{X}$ that admits a relatively minimal fibration $f: \mathcal{X}\longrightarrow \bbP^1$ such that its general fibre is a smooth irreducible…

We prove the non-rationality of a double cover of $\mathbb{P}^{n}$ branched over a hypersurface $F\subset\mathbb{P}^{n}$ of degree $2n$ having isolated singularities such that $n\ge 4$ and every singular points of the hypersurface $F$ is…