Birational geometry of singular Fano varieties

Birational Unboundedness of $\Bbb Q$-Fano threefolds

In this paper, we prove that the family of $\Bbb Q$-Fano threefolds with Picard number one is birationally unbounded.

Birationally rigid varieties with a pencil of Fano double covers. III

We complete the study of birational geometry of Fano fiber spaces $π\colon V\to {\mathbb P}^1$, 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 explicitly non-trivial structures of Fano fiber spaces. A new linear method of studying movable systems on Fano fiber spaces $V/{\mathbb P}^1$ is developed.

关于大Fano指数的Q-Fano三维流形的理性质，I

On the birational geometry of $\mathbf{Q}$-Fano threefolds of large Fano index, I

本文研究Fano指数≥2的Q-Fano三维流形的理性问题，探讨其几何性质和分类方法。

Birational rigidity of Fano varieties and field extensions

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

法诺三维加权完全交的K-稳定性与双有理刚性

K-stability and birational rigidity of Fano 3-fold weighted complete intersections

本研究证明了双有理刚性的良构拟光滑法诺三维加权完全交具有K-稳定性。

$\mathbb{Q}$-Fano threefolds with three birational Mori fiber structures

In this paper we give first examples of $\mathbb{Q}$-Fano threefolds whose birational Mori fiber structures consist of exactly three $\mathbb{Q}$-Fano threefolds. These examples are constructed as weighted hypersurfaces in a specific weighted projective space. We also observe that the number of birational Mori fiber structures does not behave upper semi-continuously in a family of $\mathbb{Q}$-Fano threefolds.

On the second and third exterior power of tangent bundles of Fano manifolds with birational contractions

In this paper, we classify Fano manifolds with elementary contractions of birational type such that the second or third exterior power of tangent bundles are numerically effective.

关于大Fano指数的Q-Fano三维流形的理性几何

On the birational geometry of $\mathbf{Q}$-Fano threefolds of large Fano index, II

本文研究Fano指数≥3的Q-Fano三维流形的理性问题，延续前文研究。

Birationally rigid Fano varieties

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 results of the theory of birational rigidity of higher-dimensional Fano varieties and fibrations are given and certain natural conjectures are formulated.

Birationally rigid Fano hypersurfaces with isolated singularities

It is proved that a general Fano hypersurface of index 1 (in the projective space) with isolated singularities of general position is birationally rigid. Therefore it cannot be fibered into uniruled varieties of a smaller dimension by a rational map. The group of birational self-maps is either trivial or cyclic of order two.

Birationally rigid Fano fibre spaces. II

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 that if every fibre of a Fano-Mori fibre space satisfies certain natural conditions, then every birational map onto another Fano-Mori fibre space is fibre-wise. After that we construct large classes of fibre spaces (into Fano double spaces of index one and into Fano hypersurfaces of index one) which satisfy those conditions. The paper is a follow up of my pre…

Fano hypersurfaces and their birational geometry

We survey some results on the nonrationality and birational rigidity of certain hypersurfaces of Fano type. The focus is on hypersurfaces of Fano index one, but hypersurfaces of higher index are also discussed.

Birationally rigid Fano hypersurfaces

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 variety of the same dimension is a biregular isomorphism. The proof is based on the method of maximal singularities combined with the connectedness principle of Shokurov and Koll\' ar.

On the birational geometry of toric Fano 4-folds

We announce a factorization result for equivariant birational morphisms between toric 4-folds whose source is Fano: such a morphism is always a composite of blow-ups along smooth invariant centers. Moreover, we show with a counterexample that, differently from the 3-dimensional case, even if both source and target are Fano, the intermediate varieties can not be chosen Fano.

On Fano manifolds with a birational contraction sending a divisor to a curve

Let X be a smooth Fano variety of dimension at least 4. We show that if X has an elementary birational contraction sending a divisor to a curve, then the Picard number of X is smaller or equal to 5.

On the birational geometry for irreducible symplectic 4-folds related to the Fano schemes of lines

There is two group actions on the Fano scheme of lines such that the quotient becomes an irreducible symplectic manifold. We showed that both quotients are birational to the generalized Kummer variety or the 2-points Hilbert scheme of a K3 surface.

On birational superrigidity and conditional birational superrigidity of certain Fano hypersurfaces

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 results of I. A. Chel'tsov and T. de Fernex, this completes the list of birationally superrigid singular hypersurfaces with only ordinary double points except in dimension 4 and 6. Further we impose an additional condition on the base locus of a birational map to a Mori fiber space. Then we prove conditional birational superrigidity of certain smooth Fano hypers…

Birationally rigid varieties with a pencil of Fano double covers. I

We prove that a general Fano fibration $π\colon V\to {\mathbb P}^1$, 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 are no other structures of a rationally connected fibration. The proof is based on the method of maximal singularities.

双有理刚性的Fano 3-重超曲面

Birationally solid Fano 3-fold hypersurfaces

本文基于Picard数为1的Fano簇的双有理刚性定义，完成了拟光滑双有理刚性Fano 3-重加权超曲面的分类。

Higher index Fano varieties with finitely many birational automorphisms

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 this problem for Fano varieties of index one and two. The purpose of this paper is to give a new approach to this problem, and to show that in every positive characteristic there are Fano varieties of arbitrarily large index with finite (or even trivial) birational automorphisms. To do this we prove that these varieties admit ample and birationally equivarian…

Birational Mori fiber structures of $\mathbb{Q}$-Fano $3$-fold weighted complete intersections, III

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 of birationally rigid varieties and $14$ families consists of birationally birigid varieties. The aim of this paper is to continue the work systematically and prove that, among the remaining $52$ families, $21$ families consist of birationally birigid varieties.

Birationally rigid varieties with a pencil of Fano double covers. II

We continue to study birational geometry of Fano fibrations $π\colon V\to {\mathbb P}^1$ 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 sufficient twistedness over the base, we prove birational rigidity (in particular, it means that there are no other structures of a fibration into rationally connected varieties) and compute their groups of birational self-maps. We considerably improve the principal components of the method of maximal singularities, in the first place, the technique of counting multip…

Fano hypersurfaces with no finite order birational automorphisms

We use the specialization homomorphism for the birational automorphism group to study finite order birational automorphisms. For a family of varieties over a DVR, we prove that a birational automorphism of order coprime to the residue characteristic cannot specialize to the identity. As an application, we show that very general $n$-dimensional hypersurfaces of degree $d \geq 5 \lceil (n+3)/6 \rceil$ have no finite order birational automorphisms.

Birational geometry of Fano double spaces of index two

We study birational geometry of Fano varieties, realized as double covers $σ\colon V\to {\mathbb P}^M$, $M\geq 5$, branched over generic hypersurfaces $W=W_{2(M-1)}$ of degree $2(M-1)$. We prove that the only structures of a rationally connected fiber space on $V$ are the pencils-subsystems of the free linear system $|-\frac12 K_V|$. The groups of birational and biregular self-maps of the variety $V$ coincide.

双有理超刚性Fano三维加权超曲面的K-稳定性

K-stability of birationally superrigid Fano 3-fold weighted hypersurfaces

该研究证明指数为1的拟光滑Fano三维加权超曲面的α不变量≥1/2，结合已有双有理超刚性与K-稳定性的关系结论，推导出这类双有理超刚性曲面的K-稳定性。

关于特征 $p>0$ 下 Fano 簇的典范丛公式与有效双有理性

On the canonical bundle formula and effective birationality for Fano varieties in char $p>0$

本文研究了正特征下 Fano 型簇的双有理几何与有界性问题，证明了有界性在正规化下不变，给出了更易用的典范丛公式，并证实了具有良好奇点的弱 Fano 簇的有效双有理性。

Birational geometry of algebraic varieties with a pencil of Fano complete intersections

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 traditional quadratic techniques of the method of maximal singularities with the linear techniques based on the connectedness principle of Shokurov and Koll\' ar. Certain related results are also considered.

Birational geometry of algebraic varieties, fibred into Fano double spaces

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 $M\geqslant 4$ and index one over a rationally connected base of dimension at most $\frac12 (M-2)(M-1)$. An estimate for the codimension of the subset of hypersurfaces of a given degree in the projective space with a positive-dimensional singular set is obtained, which is close to the optimal one.

Mori dream spaces and birational rigidity of Fano 3-folds

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 complete the birational rigidity results of Okada for Fano complete intersection 3-folds in singular weighted projective spaces.

Examples of Fano varieties of index one that are not birationally rigid

A conjecture of Pukhlikov states that a smooth Fano variety of dimension at least four and index one is birationally rigid. We show that a general member of the linear system given by the ample generator of the Picard group of the moduli space of stable, rank two bundles with fixed determinant of odd degree on a curve of genus at least three, is not birationally rigid.

Birational Mori fiber structures of Q-Fano 3-fold weighted complete intersections

In this paper we consider Q-Fano 3-fold weighted complete intersections of codimension 2 in the 85 families listed in the Iano-Fletcher's list and determine which cycle is a maximal center or not. For each maximal center, we construct either a birational involution which untwists the maximal singularity or a Sarkisov link centered at the cycle to an another explicitly described Mori fiber space. As a consequence, 19 families are proved to be birationally rigid and the remaining 66 families are proved to be birationally nonrigid.

Exceptional collections on toric Fano threefolds and birational geometry

Bernardi and Tirabassi show the existence of full strong exceptional collections consisting of line bundles on smooth toric Fano $3$-folds under assuming Bondal's conjecture, which states that the Frobenius push-forward of the structure sheaf $\mc O_X$ generates the derived category $D^b(X)$ for smooth projective toric varieties $X$. In this article, we show Bondal's conjecture for smooth toric Fano $3$-folds and also improve their result, using birational geometry.

Birationally rigid iterated Fano double covers

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 projective spaces. A generic variety in these families is proved to be birationally superrigid; in particular, it admits no non-trivial structures of a fibration into rationally connected (or uniruled) varieties, it is non-rational and its groups of birational and biregular self-maps coincide.

具有有理等价 Fano 直线簇的三次四重体

Cubic fourfolds with birational Fano varieties of lines

本文给出了几对非同构三次四重体例子，证明其 Fano 直线簇是有理等价的，为 Fourier-Mukai 伙伴猜想提供新证据，并提出三次四重体与其 Fano 直线簇有理等价性相关的猜想。

Effective birational rigidity of Fano double hypersurfaces

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 set of non-rigid varieties in the natural parameter space of the family. The lower bound is quadratic in the dimension of the variety. The proof is based on the techniques of hypertangent divisors combined with the recently discovered $4n^2$-inequality for complete intersection singularities.

On explicit birational geometry for weak Fano varieties and polarised Calabi-Yau varieties

Given a natural number $l$ and a weak Fano $n$-fold $X$ with $\operatorname{dim}\overline{φ_{-lK_X}(X)}\geq n-1$, we study the lower bound of the anti-canonical volume and the upper bound of the anti-canonical stability index. The method can also be used to give similar bounds for polarised Calabi-Yau varieties.

包含三次滚动面对的三次四重线上Fano簇的双有理几何

Birational geometry of Fano varieties of lines on cubic fourfolds containing pairs of cubic scrolls

研究包含三次滚动面对的三次四重线上Fano簇（超Kähler四重）的双有理几何，识别所有双有理模型并给出显式映射，同时刻画双有理自同构群的生成元和关系。

Birational rigidity and K-stability of Fano hypersurfaces with ordinary double points

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 Kähler--Einstein metric.

The Curve Exclusion Theorem for elliptic and K3 fibrations birational to Fano 3-fold hypersurfaces

The theorem referred to in the title is a technical result that is needed for the classification of elliptic and K3 fibrations birational to Fano 3-fold hypersurfaces in weighted projective space. We present a complete proof of the Curve Exclusion Theorem, which first appeared in the author's unpublished PhD thesis [Ry02] and has since been relied upon in solutions to many cases of the fibration classification problem. We give examples of these solutions and discuss them briefly.

Classification of elliptic and K3 fibrations birational to some Q-Fano 3-folds

A complete classification is presented of elliptic and K3 fibrations birational to certain mildly singular complex Fano 3-folds. Detailed proofs are given for one example case, namely that of a general hypersurface X of degree 30 in weighted P^4 with weights 1,4,5,6,15; but our methods apply more generally. For constructing birational maps from X to elliptic and K3 fibrations we use Kawamata blowups and Mori theory to compute anticanonical rings; to exclude other possible fibrations we make a close examination of the strictly canonical singularities of (X,(1/n)H), where H is the linear system …

Toric Fano varieties and birational morphisms

In this paper we study smooth toric Fano varieties using primitive relations and toric Mori theory. We show that for any irreducible invariant divisor D in a toric Fano variety X, we have $0\leqρ_X-ρ_D\leq 3$, for the difference of the Picard numbers of X and D. Moreover, if $ρ_X-ρ_D>0$ (with some additional hypotheses if $ρ_X-ρ_D=1$), we give an explicit birational description of X. Using this result, we show that when dim X=5, we have $ρ_X\leq 9$. In the second part of the paper, we study equivariant birational morphisms f whose source is Fano. We give some general results, and in dimension …

Explicit Birational Geometry of Fano threefold complete intersections

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 minimally embedded in a weighted projective space in codimension 2, we determine which cyclic quotient singularity is a maximal centre. If a cyclic quotient singularity is a maximal centre, we construct a Sarkisov link to a non-isomorphic Mori fibre space or a birational involution. This allows, in particular, the construction of new examples of Fano 3-folds of co…

Birational rigidity of singular Fano hypersurfaces

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 singularities of multiplicity asymptotically bounded by twice the square root of the dimension of the hypersurface, (2) hypersurfaces with isolated singularities whose Tyurina numbers satisfy a similar bound, and (3) hypersurfaces with isolated singularities whose dual variety is a hypersurface of degree sufficiently close to the expected degree.

Birational geometry of Fano hypersurfaces of index two

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 projective space for $M\geq 14$ is given by a pencil of hyperplane sections. In particular, the variety $V$ is non-rational and its group of birational self-maps coincide with the group of biregular automorphisms and is therefore trivial. The proof is based on the techniques of the method of maximal singularities and the inversion of adjunction.

Birational geometry of singular Fano hypersurfaces of index two

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 of rationally connected (or Fano-Mori) fibre space: every such structure over a positive-dimensional base is a pencil of hyperplane sections. This implies, in particular, that $V$ is non-rational and its groups of birational and biregular automorphisms coincide. The set of non-regular hypersurfaces has codimension at least $\frac12(M-11)(M-10)-10$ in the na…

Fano 簇的双有理刚性与 alpha 不变量

Birational rigidity and alpha invariants of Fano varieties

本文构造双有理超刚性及带有限群作用的 Fano 簇，证明其 alpha 不变量可任意接近 $1/2$，群版本 alpha 不变量可任意小。

Birational geometry of Fano direct products

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 $F_i\stackrelσ{\to}{\mathbb P}^M$ is a general double space of index 1, $M\geq 3$. In particular, each structure of a rationally connected fiber space on $V$ is given by a projection onto a direct factor. The proof is based on the connectedness principle of Shokurov and Koll\' ar and the technique of hypertangent divisors.

Birationally rigid Fano complete intersections. II

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 $\mathop{\rm max} \{d_i\}\geq 4$. In particular, on the variety $V$ there is exactly one structure of a Mori fibre space (or a rationally connected fibre space), the groups of birational and biregular self-maps coincide, $\mathop{\rm Bir} V= \mathop{\rm Aut} V$, and the variety $V$ is non-rational. This fact covers a considerably larger range of complete intersections than …

Birational superrigidity and slope stability of Fano manifolds

We show a relation between the birational superrigidity of Fano manifold and its slope stability in the sense of Ross-Thomas.

K-stability of birationally superrigid Fano varieties

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 variety of dimension $n$ is at least $\frac{1}{n+1}$ (under mild assumptions) and that the moduli space (if exists) of birationally superrigid Fano varieties is separated.

Birational geometry of singular Fano double spaces of index two

In this paper we describe the birational geometry of Fano double spaces $V\stackrelσ{\to}{\mathbb P}^{M+1}$ of index 2 and dimension $\geqslant 8$ with at mostquadratic singularities of rank $\geqslant 8$, satisfying certain additional conditions of general position: we prove that these varieties have no structures of a rationally connected fibre space over a base of dimension $\geqslant 2$, that every birational map $χ\colon V\dashrightarrow V'$ onto the total space of a Mori fibre space $V'/{\mathbb P}^1$ induces an isomorphism $V^+\cong V'$ of the blow up $V^+$ of the variety $V$ along $σ^{…

双有理刚性 Fano–Mori 纤维空间

Birationally rigid Fano-Mori fibre spaces

本文在纤维空间充分扭曲的条件下，证明了一类由高余维指数1 Fano完全交构成的Fano–Mori纤维空间的双有理刚性，并给出基维数仅依赖于纤维维数的上界估计。

Birational geometry of defective varieties

Here we investigate the birational geometry of projective varieties of arbitrary dimension having defective higher secant varieties. We apply the classical tool of tangential projections and we determine natural conditions for uniruledness, rational connectivity, and rationality.

Exercises in the birational geometry of algebraic varieties

This a collection of about 100 exercises. It could be used as a supplement to the book Kollár--Mori: Birational geometry of algebraic varieties.