相关论文: Birational geometry of algebraic varieties with a …

We show that any quasismooth Fano threefold weighted complete intersections of type $(12, 14)$ in $\mathbb{P} (1, 2, 3, 4, 7, 11)$ is birationally solid.

We study rationality questions for Fano schemes of linear spaces on smooth complete intersections of two quadrics, especially over non-closed fields. Our approach is to study hyperbolic reductions of the pencil of quadrics associated to…

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…

Let $f\colon X\to Z$ be a Mori fibre space. McKernan conjectured that the singularities of $Z$ are bounded in terms of the singularities of $X$. Shokurov generalised this to pairs: let $(X,B)$ be a klt pair and $f\colon X\to Z$ a…

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.

We characterize the birational geometry of some hyperk\"ahler fourfolds of Picard rank $3$ obtained as the Fano varieties of lines on cubic fourfolds containing pairs of cubic scrolls. In each of the two cases considered, we identify all of…

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

In this paper, we prove various results on boundedness and singularities of Fano fibrations and of Fano type fibrations. A Fano fibration is a projective morphism $X\to Z$ of algebraic varieties with connected fibres such that $X$ is Fano…

We give a sufficient condition for birational superrigidity of del Pezzo fibrations of degree $1$ with only $\frac{1}{2} (1,1,1)$ singular points, generalizing the so called $K^2$-condition. As an application, we also prove that a del Pezzo…

In this paper, we investigate singularities on fibrations and related topics. We prove conjectures of McKernan and Shokurov on singularities on Fano type fibrations and a conjecture of the author on singularities on log Calabi-Yau…

Let $X$ be an analytic subset of an open neighbourhood $U$ of the origin $\underline{0}$ in $\mathbb{C}^n$. Let $f\colon (X,\underline{0}) \to (\mathbb{C},0)$ be holomorphic and set $V =f^{-1}(0)$. Let $\B_\epsilon$ be a ball in $U$ of…

We prove that a quasi-smooth Fano threefold hypersurface is birationally rigid if and only if it has Fano index one.

Determining when the birational automorphism group of a Fano variety is finite is an interesting and difficult problem. The main technique for studying this problem is by the Noether-Fano method. This method has been effective in studying…

We prove that every smooth complete intersection X defined by s hypersurfaces of degree d_1, ... , d_s in a projective space of dimension d_1 + ... + d_s is birationally superrigid if 5s +1 is at most 2(d_1 + ... + d_s + 1)/sqrt{d_1...d_s}.…

Koll\'ar proved that a very general $n$-dimensional complex hypersurface of degree at least $3\lceil (n+3)/4\rceil$ is not birational to a fibration in rational curves. This is most interesting when the hypersurface is Fano, in which case…

In this paper we study the geometry of mildly singular Fano varieties on which there is an effective prime divisor of Picard number one. Afterwards, we address the case of toric varieties. Finally, we treat the lifting of extremal…

A consistent exposition of the arguments and constructions of the method of maximal singularities, the aim of which is to describe birational iso/automorphisms of Fano varieties and Fano fibrations. The principal elements of the method are…

We prove divisorial canonicity of Fano double hypersurfaces of general position.

This is the unabridged web version of the paper that will be published on the American Journal of Mathematics. In this paper, we study the birational geometry of certain examples of mildly singular quartic 3-folds. A quartic 3-fold is an…

We highlight a relation between the existence of Sarkisov links and the finite generation of (certain) Cox rings. We introduce explicit methods to use this relation in order to prove birational rigidity statements. To illustrate, we…