相关论文: Birationally rigid Fano-Mori fibre spaces

In this paper we prove the birational superrigidity of Fano-Mori fibre spaces $\pi\colon V\to S$, every fibre of which is a complete intersection of type $d_1\cdot d_2$ in the projective space ${\mathbb P}^{d_1+d_2}$, satisfying certain…

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 develop the quadratic technique of proving birational rigidity of Fano-Mori fibre spaces over a higher-dimensional base. As an application, we prove birational rigidity of generic fibrations into Fano double spaces of dimension…

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 birational superrigidity of generic Fano fiber spaces $V/{\mathbb P}^1$, the fibers of which are Fano complete intersections of index 1 and dimension $M$ in ${\mathbb P}^{M+k}$, provided that $M\geq 2k+1$. The proof combines the…

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…

We continue to study birational geometry of Fano fibrations $\pi\colon V\to {\mathbb P}^1$ the fibers of which are Fano double hypersurfaces of index 1. For a majority of families of this type, which do not satisfy the condition of…

In this paper we describe the birational geometry of Fano double spaces $V\stackrel{\sigma}{\to}{\mathbb P}^{M+1}$ of index 2 and dimension $\geqslant 8$ with at mostquadratic singularities of rank $\geqslant 8$, satisfying certain…

We prove birational superrigidity of direct products $V=F_1\times...\times F_K$ of primitive Fano varieties of the following two types: either $F_i\subset{\mathbb P}^M$ is a general hypersurface of degree $M$, $M\geq 6$, or…

We prove birational superrigidity of every hypersurface of degree N in P^N with singular locus of dimension s, under the assumption that N is at least 2s+8 and it has only quadratic singularities of rank at least N-s. Combined with the…

We prove that a generic (in the sense of Zariski topology) Fano complete intersection $V$ of the type $(d_1,...,d_k)$ in ${\mathbb P}^{M+k}$, where $d_1+...+d_k=M+k$, is birationally superrigid if $M\geq 7$, $M\geq k+3$ and $\mathop{\rm…

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 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…

We study birational geometry of Fano varieties, realized as double covers $\sigma\colon V\to {\mathbb P}^M$, $M\geq 5$, branched over generic hypersurfaces $W=W_{2(M-1)}$ of degree $2(M-1)$. We prove that the only structures of a rationally…

We complete the analysis on the birational rigidity of quasismooth Fano 3-fold deformation families appearing in the Graded Ring Database as a complete intersection. When such a deformation family $X$ has Fano index at least 2 and is…

We prove birational superrigidity of generic Fano complete intersections $V$ of type $2^{k_1}\cdot 3^{k_2}$ in the projective space ${\mathbb P}^{2k_1+3k_2}$, under the condition that $k_2\geq 2$ and $k_1+2k_2=\mathop{\rm dim} V\geq 12$,…

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…

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 prove that a Fano complete intersection of codimension $k$ and index 1 in the complex projective space ${\mathbb P}^{M+k}$ for $k\geqslant 20$ and $M\geqslant 8k\log k$ with at most multi-quadratic singularities is birationally…

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 $\frac12 (M-9)(M-10)-1$.