A characterization of projective spaces from the Mori theoretic viewpoint

Fano-Mori contractions of high length on projective varieties with terminal singularities

Let X be a projective variety with terminal singularities and let L be an ample Cartier divisor on X. We prove that if f is a birational contraction associated to an extremal ray $ R \subset \bar {NE(X)}$ such that R.(K_X+(n-2)L)<0, then f is a weighted blow-up of a smooth point. We then classify divisorial contractions associated to extremal rays R such that R.(K_X+rL)<0, where r is a non-negative rational number, and the fibres of f have dimension less or equal to r+1.

次极大长度的除子型Mori收缩

Divisorial Mori contractions of submaximal length

本文改变极大长度假设，证明次极大长度的除子型Mori收缩的例外轨迹双有理等价于射影丛或二次曲面丛，拓展了Mori收缩相关刻画研究。

Examples of Mori dream spaces with Picard number two

In this note, we give a sufficient condition such that a projective variety with Picard number two is a Mori dream space. Using this condition, we obtain examples of Mori dream spaces with Picard number two.

Birational geometry of hypersurfaces in products of projective spaces

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 results, we give new simple examples of non-Mori dream spaces and analogues of Mumford's example of a strictly nef line bundle which is not ample.

Remarks on bitrational relations of toric Mori fiber spaces

We can run the MMP for any divisor on any $\mathbb{Q}$-factorial projective toric variety. We show that two Mori fiber spaces, which are outputs of the above MMP, are connected by finitely many elementary transforms.

The Fulton-MacPherson compactification is not a Mori dream space

We show that the Fulton-MacPherson compactification of the configuration space of $n$ distinct labeled points in certain varieties of arbitrary dimension $d$, including projective space, is not a Mori dream space for $n$ larger than $d+8$.

Existence of Mori fibre spaces for 3-folds in char $p$

We prove the following results for projective klt pairs of dimension $3$ over an algebraically closed field of char $p>5$: the cone theorem, the base point free theorem, the contraction theorem, finiteness of minimal models, termination with scaling, existence of Mori fibre spaces, etc.

Morphisms of Projective Varieties from the viewpoint of Minimal Model Theory

In this Lecture Notes we present, in a sufficiently self contained way, our contributions and interests in the field of Minimal Model Theory. We study Fano-Mori spaces, both from the biregular and the birational point of view. For the former we recall and develop Kawamata's Base Point Free technique and some of Mori's deformation arguments. For the latter we lean on Sarkisov and #-Minimal Model Programs. In writing these notes we want to give our point of view on this area of research. We are not trying to give a treatment of the whole subject. These notes collect some topics we presented in t…

Completions of affine spaces into Mori fiber spaces with non-rational fibers

We describe a method to construct completions of affine spaces into total spaces of $\mathbb{Q}$-factorial terminal Mori fiber spaces over the projective line. As an application we provide families of examples with non-rational, birationally rigid and non-stably rational general fibers.

Explicit biregular/birational geometry of affine threefolds: completions of A^3 into del Pezzo fibrations and Mori conic bundles

We study certain pencils of del Pezzo surfaces generated by a smooth del Pezzo surface S of degree less or equal to 3 anti-canonically embedded into a weighted projective space P and an appropriate multiple of a hyperplane H. Our main observation is that every minimal model program relative to the morphism lifting such pencil on a suitable resolution of its indeterminacies preserves the open subset P H â A^3. As an application, we obtain projective completions of A^3 into del Pezzo fibrations over P^1 of every degree less or equal to 4. We also obtain completions of A^3 into Mori conic bundles…

Mori dream 纤维与几何一般纤维

Mori dream fibers and the geometric generic fiber

本文构造了有理曲面族，证明其纤维的 Mori dream 性质由法丛挠性决定，并给出纤维化成为 Mori dream 态射的充分条件。

构造Mori Dream Spaces之间双有理映射的几何实现

Constructing geometric realizations of birational maps between Mori Dream Spaces

本文为Mori Dream Spaces间的双有理映射构造几何实现，证明其仍为MDS且诱导墙穿越分解，并提供SageMath工具及环面情形下的矩多胞体构造。

Sur des variétés toriques non projectives

With the help of Mori theory for projective toric manifolds due to M. Reid, we study non projective toric manifolds which become projective after a single blow up along an invariant curve.

Conifold transitions and Mori theory

We show there is a symplectic conifold transition of a projective 3-fold which is not deformation equivalent to any Kaehler manifold. The key ingredient is Mori's classification of extremal rays on smooth projective 3-folds. It follows that there is a Lagrangian sphere in a projective variety which is not the vanishing cycle of any Kaehler degeneration, answering a question of Donaldson.

Projective surface automorphisms of positive topological entropy from an arithmetic viewpoint

Let X be a smooth projective surface over a number field, and $f: X \to X$ an automorphism of positive topological entropy. In this paper, we construct a height function on X that behaves well relative to f and deduce some arithmetic properties of (X; f).

Mori fibre spaces for Kähler threefolds

Let X be a compact Kaehler threefold such that the base of the MRC-fibration has dimension two. We prove that X is bimeromorphic to a Mori fibre space. Together with our earlier result arXiv:1304.4013 this completes the MMP for compact Kaehler threefolds: let X be a non-projective compact Kaehler threefold. Then X has a minimal model or X is bimeromorphic to a Mori fibre space over a non-projective Kaehler surface.

Mori dream spaces 的 Gotzmann 持续性定理

Gotzmann's persistence theorem for Mori dream spaces

本文将 Gotzmann 持续性定理从光滑射影环面簇推广到 Mori dream spaces，并给出射影空间乘积中点的更强持续性结果。

On Mori chamber and stable base locus decompositions

The effective cone of a Mori dream space admits two wall-and-chamber decompositions called Mori chamber and stable base locus decompositions. In general the former is a non trivial refinement of the latter. We investigate, from both the geometrical and the combinatorial viewpoints, the differences between these decompositions. Furthermore, we provide a criterion to establish whether the two decompositions coincide for a Mori dream space of Picard rank two, and we construct an explicit example of a Mori dream space of Picard rank two for which the decompositions are different, showing that our …

Cone theorem and Mori hyperbolicity

We discuss the cone theorem for quasi-log schemes and the Mori hyperbolicity. In particular, we establish that the log canonical divisor of a Mori hyperbolic projective normal pair is nef if it is nef when restricted to the non-lc locus. This answers Svaldi's question completely. We also treat the uniruledness of the degenerate locus of an extremal contraction morphism for quasi-log schemes. Furthermore, we prove that every fiber of a relative quasi-log Fano scheme is rationally chain connected modulo the non-qlc locus.

Mori Dream Spaces and Blow-Ups

The goal of the present article is to survey the general theory of Mori Dream Spaces, with special regards to the question: When is the blow-up of toric variety at a general point a Mori Dream Space? We translate the question for toric surfaces of Picard number one into an interpolation problem involving points in the projective plane. An instance of such an interpolation problem is the Gonzalez-Karu theorem that gives new examples of weighted projective planes whose blow-up at a general point is not a Mori Dream Space.

Characterization theorems for the projective space and vector bundle adjunction

We consider some conditions under which a smooth projective variety X is actually the projective space. We also extend to the case of positive characteristic some results in the theory of vector bundle adjunction. We use methods and techniques of the so called Mori theory, in particular the study of rational curves on projective manifolds.

Equivariant pliability of the projective space

We classify finite subgroups $G\subset\mathrm{PGL}_4(\mathbb{C})$ such that $\mathbb{P}^3$ is not $G$-birational to conic bundles and del Pezzo fibrations, and explicitly describe all $G$-Mori fibre spaces that are $G$-birational to $\mathbb{P}^3$ for these subgroups.

特征\(p>5\)的函数域上三维射影空间的森纤维空间

Mori fibre spaces for $3$-folds over imperfect fields

研究特征\(p>5\)的函数域上射影三维空间，证明典范除子非伪有效时LMMP有效，过程中表明此情形下三维空间对数典范阈值的ACC完全成立。

关于有限域代数闭包上对数典范三维簇的基点自由定理与Mori dream空间

On base point free theorem and Mori dream spaces for log canonical threefolds over the algebraic closure of a finite field

本文在特征大于5时去除了大性条件，将有限域代数闭包上三维对数典范射影对的基点自由定理推广至一般线丛，并讨论了此类域上的Mori dream空间。

关于Fujino和Sato猜想的再探讨

On a conjecture of Fujino and Sato

本文通过二次扇理论，证明非射影有理数极小弱Mori Dream空间可通过D-翻转转化为射影Mori Dream空间，并提供简短的证明方法。

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.

Mori's program for the moduli space of pointed stable rational curves

We prove that, assuming the F-conjecture, the log canonical model of the pair $(\bar{M}_{0,n}, \sum a_i ψ_i)$ is the Hassett's moduli space of weighted pointed stable rational curves without any modification of weight coefficients. For the boundary weight cases, we prove that the birational model is the GIT quotient of the product of the projective lines. This is a generalization of Simpson's theorem for symmetric weight cases.

Quotients of spectra of almost factorial domains and Mori dream spaces

We prove that a GIT chamber quotient of an affine variety $X=Spec(A)$ by a reductive group $G$, where $A$ is an almost factorial domain, is a Mori dream space if it is projective, regardless of the codimension of the unstable locus. This includes an explicit description of the Picard number, the pseudoeffective cone, and the Mori chambers in terms of GIT. We apply the results to quiver moduli, to show that they are Mori dream spaces if the quiver contains no oriented cycles, and if stability and semistability coincide. We give a formula for the Picard number in quiver terms. As a second applic…

关于投影三重曲面上的叶状结构的对数Sarkisov程序

On The Log Sarkisov Program For Foliations On Projective 3-Folds

本文证明了具有适当奇点的投影三重曲面上协秩为一的叶状结构的Sarkisov程序，并展示了两个具有秩一叶状结构的叶状Mori纤维空间之间的弱化理性刚性。

纤维 Mori 梦曲面的皮卡数

The Picard number of fibred Mori dream surfaces

研究光滑复射影曲面的皮卡数计算，提出利用纤维化可约纤维不可约分支计数的方法，证明特定几类曲面不是 Mori 梦空间。

Mori Dream Spaces and GIT

The main goal of this paper is to study varieties with the best possible Mori theoretic properties (measured by the existence of a certain decomposition of the cone of effective divisors). We call such a variety a Mori Dream Space. There turn out to be many examples, including quasi-smooth projective toric (or more generally, spherical) varieties, many GIT quotients, and log Fano 3-folds. We characterize Mori dream spaces as GIT quotients of affine varieties by a torus in a manner generalizing Cox's construction of toric varieties as quotients of affine space. Via the quotient description, the…

Toric varieties with ample tangent bundle

We give a simple combinatorial proof of the toric version of Mori's theorem that the only $n$-dimensional smooth projective varieties with ample tangent bundle are the projective spaces $\mathbb{P}^n$.

Varieties with Ample Tangent Sheaves

This paper generalises Mori's famous theorem about "Projective manifolds with ample tangent bundles" to normal projective varieties in the following way: A normal projective variety over $\mathbb{C}$ with ample tangent sheaf is isomorphic to the complex projective space.

Nef and Effective Cones on the Moduli Space of Torsion Sheaves on the Projective Plane

In this paper, we study the divisor theory of the Simpson moduli space of semistable sheaves of dimension 1 on the projective plane. We prove that these spaces are all Mori dream spaces, and calculate their nef cones. We also study the effective cones of these spaces for most choices of numerical invariants. As a consequence, we work out precisely when two such spaces are isomorphic.

Toric varieties whose canonical divisors are divisible by their dimensions

We totally classify the projective toric varieties whose canonical divisors are divisible by their dimensions. In Appendix, we show that Reid's toric Mori theory implies Mabuchi's characterization of the projective space for toric varieties.

从计算角度研究射影线上有理函数

Rational Functions on the Projective Line from a Computational Viewpoint

本文从计算角度研究了三次有理映射模空间的不变量描述，通过构造加权投影不变量和绝对不变量，实现了对模空间的嵌入和分类，为自动化分类自同构群提供了有效特征集。

Characterizing the projective space after Cho, Miyaoka and Shepherd-Barron

In this paper we give a short and elementary proof of a characterization of the projective space in terms of rational curves which was conjectured by Mori and Mukai. A proof was first given in a preprint by K. Cho, Y. Miyaoka and N. Shepherd-Barron. While our proof here is shorter and involves substantial technical simplifications, the essential ideas are taken from that preprint.

Maps of Mori Dream Spaces in Cox coordinates. Part I: existence of descriptions

Any rational map between affine spaces, projective spaces or toric varieties can be described in terms of their affine, homogeneous, or Cox coordinates. We show an analogous statement in the setting of Mori Dream Spaces. More precisely (in the case of regular maps) we show there exists a finite extension of the Cox ring of the source, such that the regular map lifts to a morphism from the Cox ring of the target to the finite extension. Moreover the extension only involves roots of homogeneous elements. Such a description of the map can be applied in practical computations.

Projectivized rank two toric vector bundles are Mori dream spaces

We prove that the Cox ring of the projectivization P(E) of a rank two toric vector bundle E, over a toric variety X, is a finitely generated k-algebra. As a consequence, P(E) is a Mori dream space if the toric variety X is projective and simplicial.

Non-commutative Mori contractions and $\PP^1$-bundles

We give a method for constructing maps from a non-commutative scheme to a commutative projective curve. With the aid of Artin-Zhang's abstract Hilbert schemes, this is used to construct analogues of the extremal contraction of a $K$-negative curve with self-intersection zero on a smooth projective surface. This result will hopefully be useful in studying Artin's conjecture on the birational classification of non-commutative surfaces. As a non-trivial example of the theory developed, we look at non-commutative ruled surfaces and, more generally, at non-commutative $\PP^1$-bundles. We show in pa…

有理连通光滑射影簇的动力视角刚性

Rigidity of rationally connected smooth projective varieties from dynamical viewpoints

本文从动力系统视角研究了有理连通光滑射影簇的刚性，证明了其成为环面簇或$(\mathbb{P}^1)^{\times n}$的充要条件分别取决于其自同态的分支除子不变性与特征值性质。

On manifolds with holomorphic normal projective connections

The authors give a complete classification of projective threefolds admitting a holomorphic normal projective connection. Moreover, they prove a general structure theorem on complex projective manifolds admitting a holomorphic normal projective connection, saying in particular, that any such manifold is either the projective space or minimal in the sense of Mori.

On covering and quasi-unsplit families of rational curves

We study extremality properties of covering families of rational curves on projective varieties. Among others, we show that on a normal and Q-factorial projective variety of dimension at most 4, every covering and quasi-unsplit family of rational curves generates a geometric extremal ray of the Mori cone.

光滑射影曲线上的Quot方案的birational几何学 via 稳定对模空间

Birational Geometry of Quot Schemes on smooth projective curves via Stable Pairs

研究光滑射影曲线上的Quot方案及其纤维的birational几何，利用稳定对模空间得到SQM，并证明其为Mori梦空间。

Birational geometry of blow-ups of projective spaces along points and lines

Consider the blow-up $X$ of $\mathbb{P}^3$ at 6 points in very general position and the 15 lines through the 6 points. We construct an infinite-order pseudo-automorphism $ϕ_X$ on $X$, induced by the complete linear system of a divisor of degree 13. The effective cone of $X$ has infinitely many extremal rays and hence, $X$ is not a Mori Dream Space. The threefold $X$ has a unique anticanonical section which is a Jacobian K3 Kummer surface $S$ of Picard number 17. The restriction of $ϕ_X$ on $S$ realizes one of Keum's 192 infinite-order automorphisms of Jacobian K3 Kummer surfaces. In general, w…

加权射影空间乘积中超曲面的双有理几何

Birational geometry of hypersurfaces in products of weighted projective spaces

本研究扩展了加权射影空间乘积中超曲面的双有理几何理论，确定了Mori梦想空间情形下的所有相关锥、双有理模型及Cox环，并证明了终端Calabi-Yau超曲面的Kawamata-Morrison锥猜想之双有理形式。

Canonical and log canonical thresholds of multiple projective spaces

In this paper we show that the global (log) canonical threshold of $d$-sheeted covers of the $M$-dimensional projective space of index 1, where $d\geqslant 4$, is equal to one for almost all families (except for a finite set). The varieties are assumed to have at most quadratic singularities, the rank of which is bounded from below, and to satisfy the regularity conditions. This implies birational rigidity of new large classes of Fano-Mori fibre spaces over a base, the dimension of which is bounded from above by a constant that depends (quadratically) on the dimension of the fibre only.

Characterizations of Projective Spaces and Hyperquadrics via Positivity Properties of the Tangent Bundle

Let $X$ be a smooth complex projective variety. A recent conjecture of S. Kovács states that if t\ he $p^{\text{th}}$-exterior power of the tangent bundle $T_X$ contains the $p^{\text{th}}$-exterior power of an ample vector bundle, then $X$ is either a projective space or a smooth quadric hypersurface. This conjecture is appealing since it is a common generalization of Mori's, Wahl's, Andreatt\ a-Wísniewski's, and Araujo-Druel-Kovács's characterizations of these spaces. In this paper I give a proof affirming this conjecture for varieties with Picard number 1.

Constructing non-Mori Dream Spaces from negative curves

We study blowups of weighted projective planes at a general point, and more generally blowups of toric surfaces of Picard number one. Based on the positive characteristic methods of Kurano and Nishida, we give a general method for constructing examples of Mori Dream Spaces and non-Mori Dream Spaces among such blowups. Compared to previous constructions, this method uses the geometric properties of the varieties and applies to a number of cases. We use it to fully classify the examples coming from two families of negative curves.

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。

Curves generating extremal rays in blowups of weighted projective planes

We consider blowups at a general point of weighted projective planes and, more generally, of toric surfaces with Picard number one. We give a unifying construction of negative curves on these blowups such that all previously known families appear as boundary cases of this. The classification consists of two classes of said curves, each depending on two parameters. Every curve in these two classes is algebraically related to other curves in both classes; this allows us to find their defining equations inductively. For each curve in our classification, we consider a family of blowups in which th…

Kovács 关于射影空间和超二次曲面特征化的猜想

Kovács' conjecture on characterization of projective space and hyperquadrics

本文证明了 Kovács 的猜想，指出在特定条件下，光滑复射影簇要么是射影空间，要么是二次超曲面，并推广了多个已有的特征化结果。

Log canonical pairs with boundaries containing ample divisors

Let $(X,Δ)$ be a projective log canonical pair such that $Δ\geq A$ where $A \geq 0$ is an ample $\mathbb{R}$-divisor. We prove that either $(X,Δ)$ has a good minimal model or a Mori fibre space. Moreover, if $X$ is $\mathbb{Q}$-factorial, then any Log Minimal Model Program on $K_X+Δ$ with scaling terminates. As an application we prove that a log Fano type variety $X$ with $\mathbb{Q}$-factorial log canonical singularities is a Mori dream space.

On the projective fourfolds with almost numerically positive canonical divisors

Let $X$ be a four-dimensional projective variety defined over the field of complex numbers with only terminal singularities. We prove that if the intersection number of the canonical divisor $K$ with every very general curve is positive ($K$ is almost numerically positive) then every very general proper subvariety of $X$ is of general type in the viewpoint of geometric Kodaira dimension. We note that the converse does not hold for simple abelian varieties.