Brendan Hassett, Kuan-Wen Lai
We exhibit a Cremona transformation of ${\bf P}^4$ such that the base loci of the map and its inverse are birational to K3 surfaces. The two K3 surfaces are derived equivalent but not isomorphic to each other. As an application, we show that the difference of the two K3 surfaces annihilates the class of the affine line in the Grothendieck ring of varieties.
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On Characteristic Classes Of Determinantal Cremona Transformations
我们计算的multidegrees和一般行列式克雷莫纳变换的Segre数，一般减少基计划，通过专门的标准克雷莫纳变换和计算其Segre类通过混合卷的合理多面体。
On Cremona Transformations of P^3 with all possible bidegrees
我们构造了P^3的具有二次（d，e）的Cremona变换，其中d<e^2+1，e< d^2+1且d，e> 0.
Fano's inequality is false for a simple Cremona transformation of five-dimensional projective space
构造了五维射影空间的一个Cremona变换。变换的次数为7。Fano不等式在这种变换中不成立。
A famous result of B. Crauder and S. Katz (1989) concerns the classification of special Cremona transformations whose base locus has dimension at most two. Furthermore, they also proved that a special Cremona transformation with base locus of dimension three has to be one of the following: 1) a quinto-quintic transformation of $\mathbb{P}^5$; 2) a cubo-quintic transformation of $\mathbb{P}^6$; or 3) a quadro-quintic transformation of $\mathbb{P}^8$. Special Cremona transformations as in case 1) have been classified by L. Ein and N. Shepherd-Barron (1989), while in our previous work (2013), we …
This article studies the possible degenerations of plane Cremona transformations of some degree into maps of smaller degree.
Cremona equivalence and log Kodaira dimension
本文研究了Cremona等价与log Kodaira维度的关系，发现一个有理曲面在项目空间中具有负log Kodaira维度但不与平面Cremona等价，展示了log版本的Castelnuovo理性准则不成立。
Smooth quartic K3 surfaces and Cremona transformations, II
这是[Og12]的继续，涉及光滑四次K3曲面的自同构和环境射影三空间的双有理自同构。
Quadro-quadric cremona transformations in low dimensions via the JC-correspondence
我们应用arXiv：1109.3573的结果研究低维射影空间中的二次-二次Cremona变换.特别是，我们描述了新的非常简单的家庭，这样的双有理映射，并获得完整和明确的分类在第四和第五维。
On Cremona transformations of prime order
最终版本，出现在C. R。Acad. Sci. Paris 339（2004），no4，257- 259.
Quartic K3 surfaces and Cremona transformations
我们证明了存在一个光滑的四次K3曲面自同构，它不是由周围三维射影空间的Cremona变换导出的。这就否定了马拉·吉兹·阿图林教授的一个问题
Classification of the monomial Cremona transformations of the plane
我们分类所有单项平面克雷莫纳地图的多度使用最近的方法开发的Brachffi。在主要结果的基础上，我们证明了这些映射的集合的几个性质，并将结果推广到更一般的r. c.单项映射
On Cremona Transformations of P^3 which factorize in a minimal form
我们考虑3维复射影空间的Cremona变换，它分解为至多两个II型基本链的乘积，没有小的收缩，连接两个Fano 3-folds。我们表明，基本上有八类这样的变换，我们给这些类中的每一个元素的几何描述。
Given a birational map in the three dimensional projective space defined by monomials of degree $d$, we prove that its inverse is defined by monomials of degree at most $d^2-d+1$.
We give a way of estimating quickly the length of the hyperbolic isometry associated to a plane Cremona transformation that is presented as the product of two involutions. We also show, using the ideas of Cantat and Lamy, that, provided that the ground field is either algebraic of characteristic not 2 or of characteristic zero, the normal closure of a high power of such a transformation is a proper subgroup of the Cremona group; this gives an effective instantiation of some of their results. Moreover, we show that any hyperbolic Cremona transformation is rigid and give a simple criterion for i…
Birational cubic fourfolds via an Enriques Cremona transformation
本文通过构造一个由Enriques曲面Fano模型上二重五次线性系统给出的Cremona变换，建立了判别式44的非常一般三次四fold与其唯一Fourier–Mukai伴侣之间的双有理等价。
We identify Feigin-Odesskii brackets $q_{n,1}(C)$, associated with a normal elliptic curve of degree $n$, $C\subset {\mathbb P}^{n-1}$, with the skew-symmetric $n\times n$ matrix of quadratic forms introduced by Fisher in arXiv:1510.04327 in connection with some minimal free resolutions related to the secant varieties of $C$. On the other hand, we show that for odd $n$, the generators of the ideal of the secant variety of $C$ of codimension $3$ give a Cremona transformation of ${\mathbb P}^{n-1}$, generalizing the quadro-cubic Cremona transformation of ${\mathbb P}^4$. We identify this transfo…
Quadratic Cremona transformations of $\mathbb{P}^4$
该代数几何研究证明$\text{Bir}_2(\boldsymbol{\text{P}}^4)$的不可约分支数量，并确定特定变换轨迹的不可约性。
New rational cubic fourfolds arising from Cremona transformations
本文证明判别式20的非常一般三次四维流形中Fourier-Mukai等价蕴含双有理等价，并借助Veronese曲面定义的Cremona变换发现新的有理三次四维流形。
Automorphisms of quartic surfaces and Cremona transformations
本文研究光滑四次曲面自同构是否由Cremona变换诱导，在ρ(S)=2时给出初步步骤，构造反例否定Oguiso问题。
A homogeneous polynomial of degree $d$ in $n+1$ variables is identifiable if it admits a unique additive decomposition in powers of linear forms. Identifiability is expected to be very rare. In this paper we conclude a work started more than a century ago and we describe all values of $d$ and $n$ for which a general polynomial of degree $d$ in $n+1$ variables is identifiable. This is done by classifying a special class of Cremona transformations of projective spaces.
We show that there is a pair of smooth complex quartic K3 surfaces $S_1$ and $S_2$ in ${\mathbf P}^3$ such that $S_1$ and $S_2$ are isomorphic as abstract varieties but not Cremona isomorphic. We also show, in a geometrically explicit way, that there is a pair of smooth complex quartic K3 surfaces $S_1$ and $S_2$ in ${\mathbf P}^3$ such that $S_1$ and $S_2$ are Cremona isomorphic, but not projectively isomorphic. This work is much motivated by several e-mails from Professors Tuyen Truong and János Kollár.
Polar Cremona transformations
在本文中，我们解决以下问题所产生的工作，P。Kazhdan和A. Polishchuk（数学。AG/0003009）：给定一个齐次复多项式，当有理映射定义的偏导数是1度？我们回答这个问题，减少多项式在三个变量，也减少多项式在四个变量的因素到产品的线性多项式。
A Rationality Problem of Some Cremona Transformation
我们将证明在一定的克雷莫纳变换下二元函数域的合理性。
This article studies the group generated by automorphisms of the projective space of dimension $n$ and by the standard birational involution of degree $n$. Every element of this group only contracts rational hypersurfaces, but in odd dimension, there are simple elements having this property which do not belong to the group. Geometric properties of the elements of the group are given, as well as a description of its intersection with monomial transformations.
Monoidal and submonoidal surfaces, and Cremona transformations
本文研究 P³ 中含高重数直线的 monoidal 与 submonoidal 曲面，将其联系到直线汇和 Cremona 变换，并以现代方法重证经典结果，进而推广到任意维超曲面。
Polar Cremona Transformations and Monodromy of Polynomials
考虑与由\[ϕ_f=grad(f): D(f)\to \CP^n, (x_0:...:x_n)\to (f_0(x):...:f_n(x))\]定义的次数为$d$的任何非常数齐次多项式$f\in \C[x_0,...,x_n]$相关联的梯度映射，其中$D(f)=\{x\in \CP^n; f(x)\neq 0\}$是与$f$和$f_i=\frac{\partial f}{\partial x_i}$相关联的主开集。这个映射对应于极Cremona变换。在命题\ref{p1}中，我们在射影超曲面$V:f=0 $只有孤立奇点的假设下，给出了$ϕ_f$的次数$d(f)$的一个新的下界.当$d(f)=1$时，定理\ref{t4}给出了关于$V$的奇性的很强的条件.
We present the abelianisation of the birational transformations of the real projective plane.
The Cremona group is the group of birational transformations of the plane. A birational transformation for which there exists a pencil of lines which is sent onto another pencil of lines is called a Jonquières transformation. By the famous Noether-Castelnuovo theorem, every birational transformation $f$ is a product of Jonquières transformations. The minimal number of factors in such a product will be called the length, and written $\mathrm{lgth}(f)$. Even if this length is rather unpredictable, we provide an explicit algorithm to compute it, which only depends on the multiplicities of the lin…
Two birational projective varieties in $P^n$ are Cremona Equivalent if there is a birational modification of $P^n$ mapping one onto the other. The minimal Cremona degree of $X\subset P^n$ is the minimal integer among all degrees of varieties that are Cremona Equivalent to $X$. The Cremona Equivalence and the minimal Cremona degree is well understood for subvarieties of codimension at least $2$ while both are in general very subtle questions for divisors. In this note I compute the minimal Cremona degree of quartic surfaces in $P^3$. This allows me to show that any quartic surface of elliptic r…
In this note we observe that the Cremona transformation in Oguiso's example of Cremona isomorphic but not projectively equivalent quartic K3 surfaces in three-dimensional projective space is the classical cubo-cubic transformation.
We study the birational self-maps of the projective plane over finite fields that induce permutations on the set of rational points. As a main result, we prove that no odd permutation arises over a non-prime finite field of characteristic two, which completes the investigation initiated by Cantat about which permutations can be realized this way. Main ingredients in our proof include the invariance of parity under groupoid conjugations by birational maps, and a list of generators for the group of such maps.
A Cremona transformation is a birational self-map of the projective space $ \mathbb{P}^{n} $. Cremona transformations of $ \mathbb{P}^{n} $ form a group and this group has a rational action on subvarieties of $ \mathbb{P}^{n} $ and hence on its Hilbert scheme. We study this action on the family of rational curves of $ \mathbb{P}^{3} $ and we prove the rectifiability of any one dimensional family. This shows that any uniruled surface is Cremona equivalent to a scroll and it answers a question of Bogomolov-Böhning related to the study of uniformly rational varieties. We provide examples of infin…
We prove a central limit theorem for the algebraic and dynamical degrees of a random composition of Cremona transformations.
Cremona transformations and diffeomorphisms of surfaces
我们表明，行动的克雷莫纳变换上的二次曲面的实点表现出完整的复杂性的球面，环面，和所有的非可定向曲面的同态。主要结果是，如果X是有理数，则代数自同构群Aut（X）在X的自同构群Diff（X）中稠密。
Extremal varieties 3-rationally connected by cubics, quadro-quadric Cremona transformations and rank 3 Jordan algebras
对于任何$n\geq 3$，我们证明，这些显然无关的对象之间存在等价：不可约$n$维非退化的投影品种$X\subset \mathbb P^{2n+1}$不同的合理正常涡卷和3-覆盖扭曲的三次曲线，射影等价;四次二次克雷莫纳变换的$ \mathbb P^{n-1}$，线性等价; $n$维复杂的约旦代数秩三，高达合痕。我们还提供了一些应用程序的分类的特定类别的品种在上述定义的类和quadro-quadric克雷莫纳变换，证明也是一个结构定理，这些双有理映射和品种3-覆盖扭曲的三次通过重新解释这些对象的可解性的根的约旦代数。
Plane Cremona maps preserving a Euclidean volume form
该研究给出了保持沿三重直线带极点微分形式的平面双有理变换群的生成元与关系表示，该群包含仿射平面自同构群。
On plane Cremona transformations of fixed degree
研究了由三个固定次数d的多项式定义的平面Cremona变换的拟射影簇Bir_d及其子簇Bir_d^o，其中三个多项式没有公共因子.我们计算它们的维数和不可约分量的分解。我们证明了对于每个d，Bir_d是连通的，并且当d < 7时，Bir_d^o是连通的。
Cremona transformations, surface automorphisms and plane cubics
给出了在有理复曲面上构造许多正熵自同构的方法。一般的想法是从一个二次克雷莫纳变换开始，它固定了一个简化的三次曲线，然后使用三次曲线上的群结构来理解何时可以通过重复爆破来消除变换的不确定性和异常行为。
Point configurations, Cremona transformations and the elliptic difference Painlevé equation
在射影空间中点位形空间上的双有理Weyl群作用的框架下，将椭圆型差分潘勒韦方程推广到高维，提供了理论基础.通过引入点构型的椭圆参数化，提出了Weyl群的一种实现，即一组系数中包含椭圆函数的Cremona变换。对于这个椭圆型Cremona系统，我们发展了一个$τ$ -函数理论，将其转化为格点上$τ$ -函数的Hirota-Miwa型双线性方程组.
In this work, we study a family of Cremona transformations of weighted projective planes which generalize the standard Cremona transformation of the projective plane. Starting from special plane projective curves we construct families of curves in weighted projective planes with special properties. We explain how to compute the fundamental groups of their complements, using the blow-up-down decompositions of the Cremona transformations, we find examples of Zariski pairs in weighted projective planes (distinguished by the Alexander polynomial). As another application of this machinery we study …
Galois Points and Cremona Transformations
本文研究平面曲线伽罗瓦群到$\text{Bir}(\text{ℙ}^2)$的扩张，证明阶≤3时可扩张为容基耶尔群子群、阶≥4时不成立，并给出两类扩张可能性不同的伽罗瓦扩张实例。
Two reduced projective schemes are said to be Cremona equivalent if there is a Cremona map that maps one in the other. In this paper I revise some of the known results about Cremona equivalence and extend the main result of [MP09] to reduced schemes. This allows to prove a very general contractibility result for union of rational subvarieties.
Birational geometry of Calabi-Yau pairs and 3-dimensional Cremona transformations
本文提出框架研究 Calabi-Yau 对的双有理几何，特别关注三维射影空间中四次曲面上的奇点如何影响其双有理性质。
On the inertia group of elliptic curves in the Cremona group of the plane
本文研究了平面上固定几何亏格为1的曲线的双有理变换群。一个精确的描述的有限元给出;它特别显示，该命令是最多6，并且，如果该组包含一个非平凡的扭转，固定曲线的图像是一个光滑的三次由双有理变换的平面。证明了光滑三次曲线的群是由其3次元素生成的，并证明了它包含Z/2 Z的自由积，由曲线的点来表示.
For each d we construct CAT(0) cube complexes on which Cremona groups rank d act by isometries. From these actions we deduce new and old group theoretical and dynamical results about Cremona groups. In particular, we study the dynamical behaviour of the irreducible components of exceptional loci, we prove regularization theorems, we find new constraints on the degree growth for non-regularizable birational transformations, and we show that the centralizer of certain birational transformations is small.
Quelques propriétés des transformations birationnelles du plan projectif complexe
我们目前的一些（不幸的是不是所有）已知的属性克雷莫纳组;当它是可能的，我们提到的链接与最知名的一组多项式自同构的仿射平面。上述性质基本上是代数性质：发电机，关系，有限子群，子群的有限型，自同构的克雷莫纳组，山雀替代.而且动力学性质：分类的双有理映射，中心化，动态的海森堡子群.本文讨论了有理曲面上熵自同构的构造。
We study birational transformations belonging to Galois points. Let $P$ be a Galois point for a plane curve $C$ and $G_P$ be a Galois group at $P$. Then an element of $G_P$ induces a birational transformation of $C$. In general, it is difficult to determine when this birational transformations can be extended to a Cremona (or projective) transformation. In this article, we shall prove that if the Galois group is isomorphic to the cyclic group of order three, then any element of the Galois group has an expression as a de Jonquières transformation. In particular, they can be extended to Cremona …
Consider an algebraically closed field k and the Cremona group of all birational transformations of the projective plane over k. We characterize infinite order elements of this group having a non-zero power generating a proper normal subgroup of the Cremona group.
In this paper we show how we can transform quadratic systems into new quadratic systems after some kind of birational transformations, the quadratic plane Cremona maps. We afterwards apply these transformations to the families of quadratic differential systems having an algebraic limit cycle. As a consequence, we provide a new family of quadratic systems having an algebraic limit cycle of degree 5. Moreover we show how the known families of quadratic differential systems having an algebraic limit cycle of degree greater than four are obtained using these transformations. We also provide the ph…
The Cremona group is the group of birational transformations of the complex projective plane. In this paper we classify its subgroups that consist only of elliptic elements using elementary model theory. This yields in particular a description of the structure of torsion subgroups. As an appliction, we prove the Tits alternative for arbitrary subgroups of the Cremona group, generalizing a result of Cantat. We also describe solvable subgroups of the Cremona group and their derived length, refining results from Déserti.
We extend our classification of special Cremona transformations whose base locus has dimension at most three to the case when the target space is replaced by a (locally) factorial complete intersection.
Finite abelian subgroups of the Cremona group of the plane
本文给出了平面Cremona群的有限交换子群的共轭类。使用一个著名的理论，这个问题相当于研究自同构群的一些德尔佩佐曲面和锥丛。因此，我们必须列举出所有的情况，这给出了一个很长的描述，然后使用一些共轭不变量来表明两种情况是否不同。例如，我们使用由群中的一个元素固定的非有理曲线，以及整个群对这些曲线的作用。从这种分类，我们推导出一系列更一般的结果双有理变换，例如存在无穷多共轭类的元素，以n，任何偶数n，结果假的奇数情况下。我们还证明了某些有穷级线性变换的根自身共轭于一个线性变换。
Lines, Quadrics, and Cremona Transformations in Two-View Geometry
本文针对两视图几何中由点对构成的矩阵$Z_k$（$7\le k\le9$）的秩亏缺问题，利用二次曲面、三次曲线与Cremona变换的代数几何工具给出几何刻画，并衔接已有$k\le6$的结果。
Fano's inequality is a mistake
给出了一个三维射影空间的Cremona变换的例子。对于这种变换，Fano不等式是不成立的.
Linearisation of finite abelian subgroups of the Cremona group of the plane
本文给出了[J. Blanc，Finite Abel subgroups of the Cremona group of the plane，C.R. Acad. Sci.巴黎先生I 344（2007），21-26.]以及有理曲面的自同构的一些描述。给出了平面Cremona群的有限Abel子群与极小曲面的自同构群群是否双有理共轭的一种判定方法。特别地，我们证明了平面的双有理变换的有限循环群是线性化的当且仅当它的非平凡元素都不固定正亏格的曲线。对于有限阿贝尔群，只有一个令人惊讶的例外，即同构于Z/2 ZxZ/4 Z的群，其非平凡元素不固定正亏格的曲线，但不共轭于极小有理曲面的自同构自同构群。我们也给出了del Pezzo曲面和锥丛的自同构（不一定是有穷级的）的一些描述。
Birational classification of curves on rational surfaces
本文研究了（S，L）对的双有理分类，其中S是有理曲面，L是S上的线性系。我们给出了一个分类定理，这样的对，我们确定，对于每个不可约的平面曲线B，它的“克雷莫纳最小”模型，即那些平面曲线是等价的B通过克雷莫纳变换，并在此条件下具有最小的程度。
Birational maps of Severi-Brauer surfaces, with applications to Cremona groups of higher rank
本研究通过刻画非平凡Severi-Brauer曲面的双有理变换群性质，推导得出秩≥4的Cremona群的商群结构、生成元性质、Hopfian性质及3-挠元不可数等关键群论结论。
2--elementary rational covers of the plane
本文研究了平面中Galois群形式为$\mathbb Z_2^r$的初等有理覆盖的分类问题。
Rational homogeneous spaces as geometric realizations of birational transformations
本文通过有理齐性空间上的 $\mathbb{C}^*$ 作用，实现了反演映射、特殊 Cremona 变换及类型 $(2,1)$ 的特殊双有理变换等经典双有理映射的几何实现。
One of the major open problems in noncommutative algebraic geometry is the classification of noncommutative projective surfaces (or, slightly more generally, of noetherian connected graded domains of Gelfand-Kirillov dimension 3). In a companion paper the authors described a noncommutative version of blowing down and, for example, gave a noncommutative analogue of Castelnuovo's classic theorem that lines of self-intersection (-1) on a smooth surface can be contracted. In this paper we will use these techniques to construct explicit birational transformations between various noncommutative surf…
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