János Kollár, Frédéric Mangolte
We show that the action of Cremona transformations on the real points of quadrics exhibits the full complexity of the diffeomorphisms of the sphere, the torus, and of all non-orientable surfaces. The main result says that if X is rational, then Aut(X), the group of algebraic automorphisms, is dense in Diff(X), the group of self-diffeomorphisms of X.
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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 …
We compute the multidegrees and the Segre numbers of general determinantal Cremona transformations, with generically reduced base scheme, by specializing to the standard Cremona transformation and computing its Segre class via mixed volumes of rational polytopes.
We construct Cremona transformations of P^3 with bidegrees (d,e), where d<e^2+1, e< d^2+1 and d,e> 0.
A Cremona transformation of five-dimensional projective space is constructed. The degree of the transformation is 7. The inequalities of Fano are not fulfilled for this transformation.
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.
This is a continuation of [Og12], concerning automorphisms of smooth quartic K3 surfaces and birational automorphisms of ambient projective three spaces.
We will show the rationality of the function field of two variables under certain Cremona transformation.
We apply the results of arXiv:1109.3573 to study quadro-quadric Cremona transformations in low-dimensional projective spaces. In particular we describe new very simple families of such birational maps and obtain complete and explicit classifications in dimension four and five.
Final version, appeared in C.R. Acad. Sci. Paris 339 (2004), no4, 257--259.
We prove that there is a smooth quartic K3 surface automorphism that is not derived from the Cremona transformation of the ambient three-dimensional projective space. This gives a negative answer to a question of Professor Marat Giz.atullin
We classify all monomial planar Cremona maps by multidegree using recent methods developed by Aluffi. Following the main result, we prove several more properties of the set of these maps, and also extend the results to the more general `r.c. monomial' maps.
We consider Cremona transformations of the complex projective space of dimension 3 which factorize as a product of at most two elementary links of type II, without small contractions, connecting two Fano 3-folds. We show that there are essentially eight classes of such transformations and we give a geometric description of elements in each of these classes.
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…
We prove a central limit theorem for the algebraic and dynamical degrees of a random composition of Cremona transformations.
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…
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.
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.
In this paper we address the following question arising from the work of P. Etingof, D. Kazhdan and A. Polishchuk (math.AG/0003009): given a homogeneous complex polynomial, when the rational map defined by its partials is of degree 1? We answer this question for reduced polynomials in three variables and also for reduced polynomials in four variables which factor into the product of linear polynomials.
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.
We study the quasi-projective variety Bir_d of plane Cremona transformations defined by three polynomials of fixed degree d and its subvariety Bir_d^o where the three polynomials have no common factor. We compute their dimension and the decomposition in irreducible components. We prove that Bir_d is connected for each d and Bir_d^o is connected when d < 7.
We give a method for constructing many examples of automorphisms with positive entropy on rational complex surfaces. The general idea is to begin with a quadratic Cremona transformation that fixes a reduced cubic curve and then use the group structure on the cubic to understand when the indeterminacy and exceptional behavior of the transformation may be eliminated by repeated blowing up.
A theoretical foundation for a generalization of the elliptic difference Painlevé equation to higher dimensions is provided in the framework of birational Weyl group action on the space of point configurations in general position in a projective space. By introducing an elliptic parametrization of point configurations, a realization of the Weyl group is proposed as a group of Cremona transformations containing elliptic functions in the coefficients. For this elliptic Cremona system, a theory of $τ$-functions is developed to translate it into a system of bilinear equations of Hirota-Miwa type f…
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。
Consider the gradient map associated to any non-constant homogeneous polynomial $f\in \C[x_0,...,x_n]$ of degree $d$, defined by \[ϕ_f=grad(f): D(f)\to \CP^n, (x_0:...:x_n)\to (f_0(x):...:f_n(x))\] where $D(f)=\{x\in \CP^n; f(x)\neq 0\}$ is the principal open set associated to $f$ and $f_i=\frac{\partial f}{\partial x_i}$. This map corresponds to polar Cremona transformations. In Proposition \ref{p1} we give a new lower bound for the degree $d(f)$ of $ϕ_f$ under the assumption that the projective hypersurface $V:f=0 $ has only isolated singularities. When $d(f)=1$, Theorem \ref{t4} yields very…
We present the abelianisation of the birational transformations of the real projective plane.
For any $n\geq 3$, we prove that there exist equivalences between these apparently unrelated objects: irreducible $n$-dimensional non degenerate projective varieties $X\subset \mathbb P^{2n+1}$ different from rational normal scrolls and 3-covered by twisted cubic curves, up to projective equivalence; quadro-quadric Cremona transformations of $ \mathbb P^{n-1}$, up to linear equivalence; $n$-dimensional complex Jordan algebras of rank three, up to isotopy. We also provide some applications to the classification of particular classes of varieties in the class defined above and of quadro-quadric …
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…
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。
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…
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.
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。
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.
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。
We complete the classical and modern work on the classification of conjugacy classes of finite subgroups of the group of birational transformations of the complex projective plane.
In this paper, we show that Cremona groups are sofic. We actually introduce a quantitative notion of soficity, called sofic profile, and show that the group of birational transformations of a d-dimensional variety has sofic profile at most polynomial of degree d. We also observe that finitely generated subgroups of the Cremona group have a solvable word problem. This provides examples of finitely generated groups with no embeddings into any Cremona group, answering a question of S. Cantat.
This article is motivated by the authors interest in the geometry of the Mori dream space $\mathbb{P}^4$ blown up in $8$ general points. In this article we develop the necessary technique for determining Weyl orbits of linear cycles for the four-dimensional case, by explicit computations in the Chow ring of the resolution of the standard Cremona transformation. In particular, we close this paper with applications to the question of the dimension of the space global sections of effective divisors having at most $8$ base points.
Let $\mathbf{K}$ be an algebraically closed field. The Cremona group $\operatorname{Cr}_2(\mathbf{K})$ is the group of birational transformations of the projective plane $\mathbb{P}^2_{\mathbf{K}}$. We carry out an overall study of centralizers of elements of infinite order in $\operatorname{Cr}_2(\mathbf{K})$ which leads to a classification of embeddings of $\mathbf{Z}^2$ into $\operatorname{Cr}_2(\mathbf{K})$, as well as a classification of maximal non-torsion abelian subgroups of $\operatorname{Cr}_2(\mathbf{K})$ when $\operatorname{char}(\mathbf{K})=0$.
We study the group of birational transformations of the plane that fix (each point of) a curve of geometric genus 1. A precise description of the finite elements is given; it is shown in particular that the order is at most 6, and that if the group contains a non-trivial torsion, the fixed curve is the image of a smooth cubic by a birational transformation of the plane. We show that for a smooth cubic, the group is generated by its elements of degree 3, and prove that it contains a free product of Z/2Z, indexed by the points of the curve.
The essential and Cremona dimensions of a group
本文通过实例支持猜想：有限群在复数域上的克雷莫纳维数不大于其本质维数。
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。
This paper contains a new proof of the classification of elements of prime order in the Cremona group Bir(P^2), up to conjugation. In addition, we give explicit geometric constructions of these Cremona transformations, and provide a parameterization of their conjugacy classes. Analogous constructions in higher dimensions are also discussed.
Let k be an algebraically closed field. We show that the Cremona group of all birational transformations of the projective plane P^2 over k is not a simple group. The strategy makes use of hyperbolic geometry, geometric group theory, and algebraic geometry to produce elements in the Cremona group that generate non trivial normal subgroups.
This work presents the conjugacy classes of finite abelian subgroups of the Cremona group of the plane. Using a well-known theory, this problem amounts to the study of automorphism groups of some Del Pezzo surfaces and conic bundles. We have thus to enumerate all the cases, which gives a long description, and then to show whether two cases are distinct or not, using some conjugacy invariants. For example, we use the non-rational curves fixed by one element of the group, and the action of the whole group on these curves. From this classification, we deduce a sequence of more general results on …
We present some (unfortunately not all) known properties on the Cremona group; when it's possible we mentioned links with the most known group of polynomial automorphisms of the affine plane. The mentioned properties are essentially algebraic properties: generators, relations, finite subgroups, subgroups of finite type, automorphisms of the Cremona group, Tits alternative... but also dynamical properties: classification of birational maps, centralizer, dynamic of an Heisenberg subgroup... We deal with the construction of entropic automorphisms on rational surfaces.
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 …
We look at algebraic embeddings of the Cremona group in $n$ variables $Cr_n(C)$ to the group of birational transformations $Bir(M)$ of an algebraic variety $M$. First we study geometrical properties of an example of an embedding of $Cr_2(C)$ into $Cr_5(C)$ that is due to Gizatullin. In a second part, we give a full classification of all algebraic embeddings of $Cr_2(C)$ into $Bir(M)$, where $dim(M)=3$, and generalize this result partially to algebraic embeddings of $Cr_n(C)$ into $Bir(M)$, where $dim(M)=n+1$, for arbitrary $n\geq 2$. In particular, this yields a classification of all algebraic…
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.
Homological stability fails for the Cremona groups, yet their stable homology is non-trivial
本文研究克雷莫纳群，发现其同调稳定性不成立，但证明其稳定同调由非平凡无限循环空间表示，为这类群的同调研究提供了新视角。
We study random walks on groups of isometries of non-proper delta-hyperbolic spaces under the assumption that at least one element in the group satisfies Bestvina-Fujiwara's WPD condition. We show that in this case typical elements are WPD, and the Poisson boundary coincides with the Gromov boundary. Moreover, we show that the random walk satisfies a form of asymptotic acylindricality, and we use this to show that the normal closure of random elements yields almost surely infinitely many different normal subgroups. Moreover, the probability that the normal closure is free tends to 1 if and onl…
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