Related papers: Explicit Birational Geometry of Fano threefold com…

We prove birational superrigidity of Fano double hypersurfaces of index one with quadratic and multi-quadratic singularities, satisfying certain regularity conditions, and give an effective explicit lower bound for the codimension of the…

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 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…

For a general Fano $3$-fold of index $1$ in the weighted projective space $\mathbb{P}(1,1,1,1,2,2,3)$ we construct $2$ new birational models that are Mori fibre spaces, in the framework of the so-called Sarkisov program. We highlight a…

This is a continuation of a series of papers studying the birational Mori fiber structures of anticanonically embedded $\mathbb{Q}$-Fano $3$-fold weighted complete intersections of codimension $2$. We have proved that $19$ families consists…

We give a brief survey of the concept of birational rigidity, from its origins in the two-dimensional birational geometry, to its current state. The main ingredients of the method of maximal singularities are discussed. The principal…

In this paper we prove birational rigidity of large classes of Fano-Mori fibre spaces over a base of arbitrary dimension, bounded from above by a constant that depends on the dimension of the fibre only. In order to do that, we first show…

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

We prove that every non-trivial structure of a rationally connected fibre space (and so every structure of a Mori-Fano fibre space) on a general (in the sense of Zariski topology) hypersurface of degree $M$ in the $(M+1)$-dimensional…

We establish birational superrigidity for a large class of singular projective Fano hypersurfaces of index one. In the special case of isolated singularities, our result applies for instance to: (1) hypersurfaces with semi-homogeneous…

We study birational transformations into elliptic fibrations and birational automorphisms of quasismooth anticanonically embedded weighted Fano 3-fold hypersurfaces having terminal singularities classified by A.R. Iano-Fletcher, J. Johnson,…

For a Zariski general (regular) hypersurface $V$ of degree $M$ in the $(M+1)$-dimensional projective space, where $M$ is at least 16, with at most quadratic singularities of rank at least 13, we give a complete description of the structures…

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…

Among smooth non-rational Fano 3-folds, the non-hyperelliptic Fano 3-fold X(10) of genus 6 has the unique property to admit a non-trivial orbit of birationally isomorphic 3-folds, inside its moduli space. Here we prove that these orbits…

We prove that in the parameter space of $M$-dimensional Fano complete intersections of index one and codimension two the locus of varieties that are not birationally superrigid has codimension at least $\frac12 (M-9)(M-10)-1$.

We prove that every quasi-smooth hypersurface in the 95 families of weighted Fano threefold hypersurfaces is birationally rigid.

We prove that normal projective stable families of maximal variation, of fixed dimension, and with bounded adjoint volume are birationally bounded. This is a consequence of a substantially stronger statement, formulated a priori…

We prove that a general Fano fibration $\pi\colon V\to {\mathbb P}^1$, the fiber of which is a double Fano hypersurface of index 1, is birationally superrigid provided it is sufficiently twisted over the base. In particular, on $V$ there…

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…

We determine birational superrigidity for a quasi-smooth prime Fano 3-fold of codimension 4 with no projection centers. In particular we prove birational superrigidity for Fano 3-folds of codimension 4 with no projection centers which are…