Related papers: Birationally solid Fano 3-fold hypersurfaces

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…

This article settles the question of existence of smooth weak Fano threefolds of Picard number two with small anti-canonical map and previously classified numerical invariants obtained by blowing up certain curves on smooth Fano threefolds…

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.

Let X be a smooth complex Fano variety. We define and study 'quasi elementary' contractions of fiber type f: X -> Y. These have the property that rho(X) is at most rho(Y)+rho(F), where rho is the Picard number and F is a general fiber of f.…

We prove that every projectively normal Fano manifold in $\mathbb{P}^{n+r}$ of index $1$, codimension $r$ and dimension $n\geq 10r$ is birationally superrigid and K-stable. This result was previously proved by Zhuang under the complete…

We study the geography and birational geometry of 3-fold conic bundles over P^2 and cubic del Pezzo fibrations over P^1. We discuss many explicit examples and raise several open questions. This paper was submitted to the proceedings of the…

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…

We give a lower bound for the delta invariant of the fundamental divisor of a quasi-smooth weighted hypersurface. As a consequence, we prove K-stability of a large class of quasi-smooth Fano hypersurfaces of index 1 and of all smooth Fano…

We prove divisorial canonicity of Fano hypersurfaces and double spaces of general position with elementary singularities.

Extending previous results, we prove that for $n \ge 5$ all hypersurfaces of degree $n+1$ in ${\mathbb P}^{n+1}$ with isolated ordinary double points are birational superrigid and K-stable, hence admit a weak K\"ahler--Einstein metric.

Let $X$ be an $n$-dimensional complex Fano manifolds $(n\geq 3)$. Assume that $X$ contains a divisor $A$, which is isomorphic to a rational homogeneous space with Picard number one, such that the conormal bundle $\mathscr{N}^*_{A/X}$ is…

We study the birational properties of hypersurfaces in products of projective spaces. In the case of hypersurfaces in P^m x P^n, we describe their nef, movable and effective cones and determine when they are Mori dream spaces. Using these…

In this paper we classify mildly singular Fano varieties with maximal Picard number whose effective divisors are numerically eventually free.

We determine the complete list of anticanonically embedded quasi smooth log Fano 3-folds in weighted projective 4-spaces. This implies that the Reid-Fletcher list of 95 types of anticanonically embedded quasi smooth terminal Fano threefolds…

Let k be an uncountable algebraically closed field and let Y be a smooth projective k-variety which does not admit a decomposition of the diagonal. We prove that Y is not stably birational to a very general hypersurface of any given degree…

Under a hypothesis on $k$, $d$ and $n$ that is almost the best possible, we prove that for every smooth degree $d$ hypersurface in $P^n$, the $k$-plane sections dominate the moduli space of degree $d$ hypersurface in $P^k$. Using this we…

We study Fano threefolds with Picard number one equipped with a holomorphic section in $\Omega_V^1(1)$.

For $n\geq 4$, let $X$ be a complex smooth Fano $n$-fold whose minimal anticanonical degree of non-free rational curves on $X$ is at least $n-2$. We classify extremal contractions of such varieties. As an application, we obtain a…

The classification of toric Fano manifolds with large Picard number corresponds to the classification of smooth Fano polytopes with large number of vertices. A smooth Fano polytope is a polytope that contains the origin in its interior such…

We prove the following main result: Let X be a Fano 3-fold with terminal Q-factorial singularities and X does not have a small extremal ray and a face of Kodaira dimension 1 or 2 for Mori polyhedron of X. Then the Picard number \rho (X) <…