Jérémy Blanc, Ivan Pan, Thierry Vust
This article deals with the study of the birational transformations of the projective complex plane which leave invariant an irreducible algebraic curve. We try to describe the state of art and provide some new results on this subject.
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We survey new results on finite groups of birational transformations of algebraic varieties.
We show that any birational map between projective hyperKähler manifolds of dimension 4 is composed of a sequence of simple flops or elementary Mukai transformations under the assumption that each irreducible component of the indeterminacy of the birational map is normal.
A priori, the set of birational transformations of an algebraic variety is just a group. We survey the possible algebraic structures that we may add to it, using in particular parametrised family of birational transformations.
We study birational transformations P^n--->S \subseteq P^N defined by linear systems of quadrics whose base locus is smooth and irreducible of dimension \leq3 and whose image S is sufficiently regular.
In this article, we study birational transformations belonging to Galois points for certain plane quartic curve. In fact, we see that they can be extended to Cremona transformations. In particular, we determine the conjugacy class of them. We have that they are all conjugate to linear transformations.
We give a corrected version of Corollary 3.33 in: H. Flenner, S. Kaliman, and M. Zaidenberg, Birational transformations of weighted graphs. Affine algebraic geometry. Osaka Univ. Press, 2007, 107-147.
In a previous paper (arXiv:1410.5207) certain birational transformations were constructed between the noncommutative schemes associated to quadratic and cubic three dimensional Sklyanin algebras. In the current paper we consider the inverse birational transformations and show that they are of the same type. Moreover we extend everything to the $\mathbb{Z}$-algebras context, which allows us to incorporate the noncommutative quadrics introduced by Van den Bergh.
Characterizing Varieties Using Birational Transformations
该研究通过birational变换和基域自同构来特征化变种，揭示了被规则变种与非被规则变种在birational变换群性质上的差异。
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 introduce the notion of induced birational transformations of irreducible holomorphic symplectic sixfolds of the sporadic deformation type discovered by O'Grady. We give a criterion to determine when a manifold of $OG_6$ type is birational to a moduli space of sheaves on an abelian surface. Then we determine when a birational transformation of the moduli space is induced by an automorphism of the abelian surface. Referring to the Mongardi--Rapagnetta--Saccá birational model of manifolds of $OG_6$ type, we give a result to determine when a birational transformation is induced at the quotient…
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.
We study and classify up to a linear conjugation germs of flows in the space of quadratic birational transformations of the complex projective space of dimension 3. As a consequence we show that every quadratic flow preserves a pencil of planes through a line.
Geometric realizations of birational transformations via $\mathbb{C}^*$-actions
本文研究通过环面作用实现双有理变换的几何构造，对一类特定双有理变换给出显式实现，并刻画了其 Mori、Nef 与 Movable 锥等几何性质。
Extending some results of Crauder and Katz, and Ein and Shepherd-Barron on special Cremona transformations, we study birational transformations of the complex projective spaces onto prime Fano manifolds such that the base locus X of the transformations is smooth, irreducible and reduced. We get a complete classification when X has dimension 1, or dimension 2, or codimension 2. We also get partial results when X has dimension 3.
Let ϕ: \mathbb{P}^{r}\dashrightarrow Z be a birational transformation with a smooth connected base locus scheme, where Z\subseteq\mathbb{P}^{r+c} is a nondegenerate prime Fano manifold. We call ϕa quadro-quadric special briational transformation if ϕand ϕ^{-1} are defined by linear subsystems of |\mathcal{O}_{\mathbb{P}^{r}}(2)| and |\mathcal{O}_{Z}(2)| respectively. In this paper we classify quadro-quadric special birational transformations in the cases where either (i) Z is a complete intersection and the base locus scheme of ϕ^{-1} is smooth, or (ii) Z is a hypersurface.
In this paper we generalize some classical birational transformations to the non-commutative case. In particular we show that 3-dimensional quadratic Sklyanin algebras (non-commutative projective planes) and 3-dimensional cubic Sklyanin algebras (non-commutative quadrics) have the same function field. In the same vein we construct and analogue of the Cremona transform for non-commutative projective planes.
We study transformations as in the title with emphasis on those having smooth connected base locus, called "special". In particular, we classify all special quadratic birational maps into a quadric hypersurface whose inverse is given by quadratic forms by showing that there are only four examples having general hyperplane sections of Severi varieties as base loci.
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。
A birational transformation f: P^n --> Z, where Z is a nonsingular variety of Picard number 1, is called a special birational transformation of type (a, b) if f is given by a linear system of degree a, its inverse is given by a linear system of degree b and the base locus S \subset P^n of f is irreducible and nonsingular. In this paper, we classify special birational transformations of type (2,1). In addition to previous works Alzati-Sierra and Russo on this topic, our proof employs natural C^*-actions on Z in a crucial way. These C^*-actions also relate our result to the problem studied in ou…
We give a necessary and sufficient condition for an automorphism of the Hilbert scheme of points on a K3 surface (non necessarily algebraic) to be induced by an automorphism of the surface. We prove furthermore that the group of birational transformations of a projective irreducible holomorphic symplectic manifold is finitely generated.
A birational map from a projective space onto a not too much singular projective variety with a single irreducible non-singular base locus scheme (special birational transformation) is a rare enough phenomenon to allow meaningful and concise classification results. We shall concentrate on transformations defined by quadratic equations onto some varieties (especially projective hypersurfaces of small degree), where quite surprisingly the base loci are interesting projective manifolds appearing in other contexts; for example, exceptions for adjunction theory, small degree or small codimensional …
We study how relative quantum cohomology, defined by Tseng--You and Fan--Wu--You, varies under birational transformations. For toric complete intersections with simple normal crossings divisors that contain the loci of indeterminacy, we prove that their respective relative $I$-functions can be directly identified. For toric complete intersections with smooth divisors, we prove that their respective relative $I$-functions are related by analytic continuation. We also study some connections with extremal transitions and FJRW theory.
Symplectic birational transformations of finite order on O'Grady's sixfolds
本文证明了OG6型流形上的有限阶辛自同构和双有理变换分别在BBF格与判别群上作用平凡，并分类了所有可能的不变与余不变子格。
We classify the group of birational automorphisms of Hilbert schemes of points on algebraic K3 surfaces of Picard rank one. We study whether these automorphisms are symplectic or non-symplectic and if there exists a hyperkähler birational model on which they become biregular. We also present new geometrical constructions of these automorphisms.
Algebraic subgroups of the group of birational transformations of ruled surfaces
研究光滑正亏曲线C的 ruled 曲面的理性的变换群Bir(CxPP^1)的最大代数子群分类
The Structure of Algebraic Families of Birational Transformations
本文刻画代数簇上双有理变换的代数族结构，证明到 Bir(X) 的态射的 Chevalley 型结果、纤维维数公式及代数子群刻画，并研究保持纤维化的代数族。
Special birational transformations $Φ:\p^r\da Z$ defined by quadric hypersurfaces are studied by means of the variety of lines $\mathcal L_z\subset\p^{r-1}$ passing through a general point $z\in Z$. Classification results are obtained when $Z$ is either a Grassmannian of lines, or the 10-dimensional spinor variety, or the $E_6$-variety. In the particular case of quadro-quadric transformations, we extend the well-known classification of Ein and Shepherd-Barron coming from Zak's classification of Severi varieties to a wider class of prime Fano manifolds $Z$. Combining both results, we get a clas…
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。
In our previous works (2012, 2013), we provided a finite list of properties characterizing all potential types of quadratic birational transformations of a projective space into a factorial variety, whose base locus is smooth and irreducible. However, some existence problems remained open. Among them one had to prove that the image of a given transformation was factorial, but in two particular situations, even the mere existence of the transformation was left as an open problem. In this paper, we use computer algebra methods to construct explicitly four examples of such transformations; two of…
An application of $p$-adic integration to the dynamics of a birational transformation preserving a fibration
针对Kodaira维数非负的射影流形上保纤维化的双有理变换，通过p进积分方法证明诱导的基伪自同构阶有限，并推论K3ⁿ型等辛流形的首个动力学次数可刻画具Zariski稠轨道的双有理变换。
We classify simple groups that act by birational transformations on compact complex Kähler surfaces. Moreover, we show that every finitely generated simple group that acts non-trivially by birational transformations on a projective surface over an arbitrary field is finite.
A criterion for the existence of a birational embedding of an algebraic curve into a projective plane with two Galois points is presented. Several novel examples of plane curves with two inner Galois points as an application are described.
Birational transformations on irreducible compact Hermitian symmetric spaces
本文研究了不可约紧致赫尔米特对称空间上的双有理变换，通过构造吹胀和吹降操作将其转化为射影空间，并解决了Landsberg和Manivel提出的双有理映射问题。
Let A^2 denote the affine plane over an algebraically closed field of arbitrary characteristic. Besides contributing several new results in the general theory of birational endomorphisms of A^2, this article describes certain classes of birational endomorphisms f defined by requiring that the missing curves or contracting curves of f are lines. The last part of the article is concerned with the monoid structure of the set of birational endomorphisms of A^2.
Given a surface $S$ and a finite group $G$ of automorphisms of $S$, consider the birational maps $S\dashrightarrow S'$ that commute with the action of $G$. This leads to the notion of a $G$-minimal variety. A natural question arises: for a fixed group $G$, is there a birational $G$-map between two different $G$-minimal surfaces? If no such map exists, the surface is said to be $G$-birationally rigid. This paper determines the $G$-rigidity of the projective plane for every finite subgroup $G\subset\mbox{PGL}_3\left(\mathbb{C}\right)$.
We describe a close relation between wall crossings in the birational geometry of moduli space of Gieseker stable sheaves $M_H(v)$ on $\bb{P}^2$ and mini-wall crossings in the stability manifold $Stab(D^b(\bb{P}^2))$.
We classify surfaces of general type whose bicanonical map is composed with a rational map of degree 2 onto a rational or ruled surface.
Birational involutions of the real projective plane fixing an irrational curve
本文详细阐述了关于实射影平面上固定一条无理曲线的双有理对合的研究，基于2022年克拉科夫会议报告。
The dynamical degree $λ(f)$ of a birational transformation $f$ measures the exponential growth rate of the degree of the formulae that define the $n$-th iterate of $f$. We study the set of all dynamical degrees of all birational transformations of projective surfaces, and the relationship between the value of $λ(f)$ and the structure of the conjugacy class of $f$. For instance, the set of all dynamical degrees of birational transformations of the complex projective plane is a closed and well ordered set of algebraic numbers.
We define a new notion of "b-stability" for a polarised algebraic variety, adapted to the existence problem for Kahler-Einstein metrics on Fano manifolds.
We study the birational geometry of moduli spaces of semistable sheaves on the projective plane via Bridgeland stability conditions. We show that the entire MMP of their moduli spaces can be run via wall-crossing. Via a description of the walls, we give a numerical description of their movable cones, along with its chamber decomposition corresponding to minimal models. As an application, we show that for primitive vectors, all birational models corresponding to open chambers in the movable cone are smooth and irreducible.
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。
Birational transformations of threefold $\mathbf{Q}$-conic bundles
本文针对三维$\mathbf{Q}$-圆锥丛，提出一种算法将其变换为标准形式，该丛满足终端$\mathbf{Q}$-阶乘奇点、反典范除子$f$-丰沛及相对皮卡数1等条件。
We prove that the group of birational transformations of a Del Pezzo fibration of degree 3 over a curve is not simple, by giving a surjective group homomorphism to a free product of infinitely many groups of order 2. As a consequence we also obtain that the Cremona group of rank 3 is not generated by birational maps preserving a rational fibration. Besides, its subgroup generated by all connected algebraic subgroups is a proper normal subgroup.
Let $X$ be a projective variety of dimension $r$ over an algebraically closed field. It is proven that two birational embeddings of $X$ in $¶^n$, with $n\geq r+2$ are equivalent up to Cremona transformations of $¶^n$.
In this paper we study the birational geometry of HyperKaehler manifolds by combining the method of minimal model program and the traditional approach of symplectic geometry.
We introduce the notion of a standard weighted graph and show that every weighted graph has an essentially unique standard model. Moreover we classify birational transformations between such models. Our central result shows that these are composed of elementary transformations. The latter ones are defined similarly to the well known elementary transformations of ruled surfaces. In a forthcoming paper, we apply these results in the geometric setup to obtain standard equivariant completions of affine surfaces with an action of certain algebraic groups. We show that these completions are unique u…
Birational involutions of the real projective plane
研究实射影平面的双射反演分类，发现12种不同类别的反演，与复数域上的4种类别不同，且固定曲线不决定共轭类。
Addendum/Erratum to the paper "Weyl groups and Birational transformations among minimal models"
本文为1995年关于Fano三维簇与Weyl群论文的补遗和勘误，补入Mori–Mukai分类中遗漏的第88种形变类型，并修正相关错误与计算。
Small bandwidth ${\mathbb C}^*$-actions and birational geometry
该研究针对带一维复环面作用的光滑射影簇与配极配对，定义了作用的双有理几何对应物以实现重构，并分析了低复杂度作用与Atiyah翻转、Cremona变换的对应关系。
Let $f\colon X\dashrightarrow X$ be a bimeromorphic transformation of a complex irreducible symplectic manifold $X$. Some important dynamical properties of $f$ are encoded by the induced linear automorphism $f^*$ of $H^2(X,\mathbb Z)$. Our main result is that a bimeromorphic transformation such that $f^*$ has at least one eigenvalue with modulus $>1$ doesn't admit any invariant fibration (in particular its generic orbit is Zariski-dense).
We study the group of symplectic birational transformations of the plane. It is proved that this group is generated by $\mathrm{SL}(2,\mathbb{Z})$, the torus and a special map of order $5$, as it was conjectured by A. Usnich. Then we consider a special subgroup $H$, of finite type, defined over any field which admits a surjective morphism to the Thompson group of piecewise linear automorphisms of $\mathbb{Z}^2$. We prove that the presentation for this group conjectured by Usnich is correct.
In this paper we consider the birational classification of pairs (S,L), with S a rational surfaces and L a linear system on S. We give a classification theorem for such pairs and we determine, for each irreducible plane curve B, its "Cremona minimal" models, i.e. those plane curves which are equivalent to B via a Cremona transformation, and have minimal degree under this condition.
We prove the conjecture that two projective symplectic resolutions for a symplectic variety $W$ are related by Mukai's elementary transformations over $W$ in codimension 2 in the following cases: (i). nilpotent orbit closures in a classical simple complex Lie algebra; (ii). some quotient symplectic varieties.
We look at sequences of positive integers that can be realized as degree sequences of iterates of rational dominant maps of smooth projective varieties over arbitrary fields. New constraints on the degree growth of endomorphisms of the affine space and examples of degree sequences are displayed. We also show that the set of all degree sequences of rational maps is countable; this generalizes a result of Bonifant and Fornaess.
Two divisors in $\mathbb P^n$ are said to be Cremona equivalent if there is a Cremona modification sending one to the other. In this paper I study irreducible cones in $\mathbb P^n$ and prove that two cones are Cremona equivalent if their general hyperplane sections are birational. In particular I produce examples of cones in $\mathbb P^3$ Cremona equivalent to a plane whose plane section is not Cremona equivalent to a line in $\mathbb P^2$.
We construct a determinantal family of quarto-quartic transformations of a complex projective space of dimension $3$ from trigonal curves of degree $8$ and genus $5$. Moreover we show that the variety of $(4,4)$-birational maps of $\mathbb{P}_3(\mathbb{C})$ has at least four irreducible components and describe three of them.
We show that the ordinal of the dynamical degrees of all complex birational maps of the projective plane is $ω^ω$.
Birational conjugacies between endomorphisms on the projective plane
本文证明了特征0下射影平面上次数至少为2的支配自同态若双有理共轭则必自同构共轭，并给出正特征反例。
Complex projective algebraic varieties with $\mathbb{C}^*$-actions can be thought of as geometric counterparts of birational transformations. In this paper we describe geometrically the birational transformations associated to rational homogeneous varieties endowed with a $\mathbb{C}^*$-action with no proper isotropy subgroups.
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