Related papers: Birational rigidity and alpha invariants of Fano v…

We prove that every birationally superrigid Fano variety whose alpha invariant is greater than (resp. no smaller than) $\frac{1}{2}$ is K-stable (resp. K-semistable). We also prove that the alpha invariant of a birationally superrigid Fano…

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…

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 prove the birational rigidity of Fano complete intersections of index 1 with a singular point of high multiplicity, which can be close to the degree of the variety. In particular, the groups of birational and biregular automorphisms of…

Iterating the procedure of making a double cover over a given variety, we construct large families of smooth higher-dimensional Fano varieties of index 1. These varieties can be realized as complete intersections in various weighted…

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…

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 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 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 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 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 the alpha invariant of a quasi-smooth Fano 3-fold weighted hypersurface of index $1$ is greater than or equal to $1/2$. Combining this with the result of Stibitz and Zhuang \cite{SZ19} on a relation between birational…

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 show that fixed dimensional klt weak Fano pairs with alpha-invariants and volumes bounded away from $0$ and the coefficients of the boundaries belong to the set of hyperstandard multiplicities $\Phi(\mathscr{R})$ associated to a fixed…

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 prove birational superrigidity of Fano cyclic covers of index 1 over hypersurfaces in the projective space.

It is shown that hypersurfaces of degree $M$ in ${\mathbb P}^M$, $M\geqslant 5$, with at most quadratic singularities of rank at least 3, satisfying certain conditions of general position, are birationally superrigid Fano varieties and the…

We introduce an inductive argument for proving birational superrigidity and K-stability of singular Fano complete intersections of index one, using the same types of information from lower dimensions. In particular, we prove that a…

The aim of this note is to settle some foundational questions about the behavior of birational rigidity in extensions of algebraically closed fields.

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