Algebraic subgroups of the group of birational transformations of ruled surfaces
Pascal Fong
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
我们分类了当C是光滑的正亏曲线时，Bir(CxPP^1)的最大代数子群。
We classify the maximal algebraic subgroups of Bir(CxPP^1), when C is a smooth projective curve of positive genus.
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Let $S$ be a birationally ruled surface. We show that the moduli schemes $M_S(r,c_1,c_2)$ of semistable sheaves on $S$ of rank $r$ and Chern classes $c_1$ and $c_2$ are irreducible for all $(r,c_1,c_2)$ provided the polarization of $S$ used satisfies a simple numerical condition. This is accomplished by proving that the stacks of prioritary sheaves on $S$ of fixed rank and Chern classes are smooth and irreducible.
We call a group $G$ nilpotently Jordan of class at most $c$ $(c\in\mathbb{N})$ if there exists a constant $J\in\mathbb{Z}^+$ such that every finite subgroup $H\leqq G$ contains a nilpotent subgroup $K\leqq H$ of class at most $c$ and index at most $J$. We show that the birational automorphism group of a $d$ dimensional variety over a field of characteristic zero is nilpotently Jordan of class at most $d$.
We classify surfaces of general type whose bicanonical map is composed with a rational map of degree 2 onto a rational or ruled surface.
We call a group $G$ nilpotently Jordan of class at most $c$ $(c\in\mathbb{N})$ if there exists a constant $J\in\mathbb{Z}^+$ such that every finite subgroup $H\leqq G$ contains a nilpotent subgroup $K\leqq H$ of class at most $c$ and index at most $J$. We show that the birational automorphism group of a variety over a field of characteristic zero is nilpotently Jordan of class at most two.
Finite abelian subgroups in the groups of birational and bimeromorphic selfmaps
该研究针对复射影簇的双有理自同构群中含任意大阶的固定秩有限阿贝尔子群的问题，证明秩不超过簇维数的两倍，等号成立等价于簇双有理等价于阿贝尔簇，并将结果推广到紧Kähler空间的双亚纯自同构群。
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 finite $p$-subgroups of birational automorphism groups. By virtue of boundedness theorem of Fano varieties, we prove that there exists a constant $R(n)$ such that a rationally connected variety of dimension $n$ over an algebraically closed field is rational if its birational automorphism group contains a $p$-subgroups of maximal rank for $p > R(n)$. Some related applications on Jordan property are discussed.
We classify the singularities of a surface ruled by conics: they are rational double points of type $A_n$ or $D_n$. This is proved by showing that they arise from a precise series of blow-ups of a suitable surface geometrically ruled by conics. We determine also the family of such surfaces which are birational models of a given surface ruled by conics and obtained in a "minimal way" from it.
In this paper we classify nodal rational non-$\mathbb{Q}$-factorial del Pezzo threefolds of degree 2 which can be $G$-birationally rigid for some subgroup $G\subset \operatorname{Aut}(X)$.
Jordan property of birational automorphism groups of surfaces and birational permutations
本文证明了有限域上几何不可约代数曲面的双有理自同构群是Jordan群，并指出高维情形不成立，同时证明了有限域上双有理置换群的有限p'-子群有界，因此是p-Jordan群。
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。
Assuming a particular case of Borisov--Alexeev--Borisov conjecture, we prove that finite subgroups of the automorphism group of a finitely generated field over Q have bounded orders. Further, we investigate which algebraic varieties have groups of birational selfmaps satisfying the Jordan property.
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.
We prove that a finite group acting by birational automorphisms of a non-trivial Severi-Brauer surface over a field of characteristic zero contains a normal abelian subgroup of index at most 3. Also, we find an explicit bound for orders of such finite groups in the case when the base field contains all roots of 1.
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.
On G-birational rigidity of projective spaces
本文研究了有限子群G⊂Aut(ℙⁿ)，使得ℙⁿ在G下理性刚性。对于n≥3，证明了Aut(ℙⁿ)中至多存在有限多个此类子群。对于n=4，证明当G≈PSp₄(𝔽₃)时，ℙ⁴在G下理性超刚性。
Using a filtration on the Grothendieck ring of triangulated categories, we define the categorical dimension of a birational map between smooth projective varieties. We show that birational automorphisms of bounded categorical dimension form subgroups, which provide a nontrivial filtration of the Cremona group. We discuss some geometrical aspect and some explicit example. In the case of threefolds, we can moreover recover the genus of a birational automorphism, and the filtration defined by Frumkin.
For a geometrically rational surface X over an arbitrary field of characteristic different from 2 and 3 that contains all roots of 1, we show that either X is birational to a product of a projective line and a conic, or the group of birational automorphisms of X has bounded finite subgroups. As a key step in the proof, we show boundedness of finite subgroups in any anisotropic reductive algebraic group over a perfect field that contains all roots of 1. Also, we provide applications to Jordan property for groups of birational automorphisms.
We give a sharp bound for orders of elementary abelian 2-groups of birational automorphisms of rationally connected threefolds.
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 prove an existence result for stable vector bundles with arbitrary rank on an algebraic surface, and determine the birational structure of certain moduli space of stable bundles on a rational ruled surface.
Birational rigidity of quartic three-folds with a double point of rank 3
本文证明复射影空间中仅含秩3二重点的一般四次三重流形具有双有理刚性，其双有理自映射群为25个2阶循环群的自由积，并由此推出刚性阶乘四次流形补集的余维数至少为3。
We explicitly describe the $\mathbb A^1$-chain homotopy classes of morphisms from a smooth henselian local scheme into a smooth projective surface, which is birationally ruled over a curve of genus $> 0$. We consequently determine the sheaf of naive $\mathbb A^1$-connected components of such a surface and show that it does not agree with the sheaf of its genuine $\mathbb A^1$-connected components when the surface is not a minimal model. However, the sections of the sheaves of both naive and genuine $\mathbb A^1$-connected components over schemes of dimension $\leq 1$ agree. As a consequence, w…
We characterise smooth curves in a smooth cubic threefold whose blow-ups produce a weak-Fano threefold. These are curves $C$ of genus $g$ and degree $d$, such that (i) $2(d-5) \le g$ and $d\le 6$; (ii) $C$ does not admit a 3-secant line in the cubic threefold. Among the list of ten possible such types $(g,d)$, two were previously left as open numerical possibilities, namely $(g,d) = (0,5)$ and $(2,6)$. Using the Sarkisov link associated with a curve of type $(2,6)$, we are able to produce the first example of a pseudo-automorphism with dynamical degree greater than $1$ on a smooth threefold wi…
We discuss the problem of stable conjugacy of finite subgroups of Cremona groups. We show that the group $H^1(G,Pic(X))$ is a stable birational invariant and compute this group in some cases.
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…
On $G$-birational rigidity of del Pezzo surfaces
本文证明，若代数闭域上的光滑 del Pezzo 曲面是 $H$-双有理刚性的，则它也是 $G$-双有理刚性的，从而在二维情形下正面回答了 Kollár 问题的几何版本。
We improve a result of Prokhorov and Shramov on the rank of finite $p$-subgroups of the birational automorphism group of a rationally connected variety. Known examples show that they are sharp in many cases.
A remark on polycyclic groups of birational automorphisms
该研究针对代数簇双有理自同构群中的虚拟多循环子群，证明其虚拟导出长度的上界为2d+1，曲面情形可优化为3且该界最优。
We classify all finite subgroups of the plane Cremona group which have a fixed point. In other words, we determine all rational surfaces X with an action of a finite group G such that X is equivariantly birational to a surface which has a G-fixed point.
Linearization of finite subgroups of Cremona groups over non-closed fields
研究全局域上Cremona群有限子群的线性化性质，构造了在整体域上非k-线性化但在所有位点k_v-线性化的双有理对合，主要工具是推广Manin-Voskresenskiı和Bogomolov-Prokhorov的不变量。
We consider 3-subgroups in groups of birational automorphisms of rationally connected threefolds and show that any 3-subgroup can be generated by at most five elements. Moreover, we study groups of regular automorphisms of terminal Fano threefolds and prove that in all cases which are not among several explicitly described exceptions any 3-subgroup of such group can be generated by at most four elements.
Rank-two vector bundles on non-minimal ruled surfaces
本文通过引入两个数值不变量将稳定秩二向量丛表示为自然扩张，研究非极小直纹曲面上模空间的双有理几何，并证明有理曲面上模空间的不可约分支均为有理或稳定有理。
Smooth projective surfaces with pseudo-effective tangent bundles
该数学代数几何研究刻画了非单直纹光滑射影曲面切丛伪有效的等价条件，并探讨了非有理直纹曲面吹胀的相关性质。
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.
Modular Symbols and Equivariant Birational Invariants
本文研究经典模符号与有限阿贝尔群相关的Kontsevich-Pestun-Tschinkel群之间的关系，探讨其等变双有理不变量。
A conic bundle or quadric bundle in characteristic 2 can have generic fiber which is nowhere smooth over the function field of the base variety. In that case, the generic fiber is called a quasilinear quadric. We solve some of the main problems of birational geometry for quasilinear quadrics. First, if there is a rational map from one quasilinear quadric to another of the same dimension, and also a map back, then the two quadrics are birational. Next, we determine exactly which quasilinear quadrics are ruled, that is, birational over the base field to the product of some variety with the proje…
A determinant on birational maps of Severi-Brauer surfaces
本研究在完美域非平凡Severi-Brauer曲面自同构群上定义行列式并推广到双有理映射，结合已有群同态得到其双有理变换群的首个非平凡自同构几何有理曲面的阿贝尔化，进而找出该群的极大子群。
We give some properties of the subgroup $G_n(\mathbb{C})$ of the group of birational self-maps of $\mathbb{P}^n_\mathbb{C}$ generated by the standard involution and the group of automorphisms of $\mathbb{P}^n_\mathbb{C}$. We prove that there is no nontrivial finite-dimensional linear representation of $G_n(\mathbb{C})$. We also establish that $G_n(\mathbb{C})$ is perfect, and that $G_n(\mathbb{C})$ equipped with the Zariski topology is simple. Furthermore if $φ$ is an automorphism of $\mathrm{Bir}(\mathbb{P}^n_\mathbb{C})$, then up to birational conjugacy, and up to the action of a field autom…
We prove that for each $n \geq 2$, there exists a ruled variety X of dimension n such that Bir(X) contains connected algebraic subgroups which are not lying in a maximal one.
Algebraic subgroups of the plane Cremona group over a perfect field
该研究针对完美域上平面Cremona群的代数子群问题，证明了其任意无限代数子群都含于某个极大代数子群，并通过双有理映射共轭关系完成了极大群及有理点子群的分类。
Simple subgroups of the real space Cremona group
本文研究实三维射影空间双有理自映射群的有限非交换单子群，证明其中仅存在交错群$\mathfrak{A}_5$与$\mathfrak{A}_6$两类。
Birationally integrable vector fields on complex projective surfaces
本文证明复射影光滑曲面上的双有理可积向量场可正则化，并推广至有限维李代数，给出二维及半单情形的完全双有理分类及极大代数的刻画。
We show that the decomposition group of a line $L$ in the plane, i.e. the subgroup of plane birational transformations that send $L$ to itself birationally, is generated by its elements of degree 1 and one element of degree 2, and that it does not decompose as a non-trivial amalgamated product.
A conjecture of Morel asserts that the sheaf of $\mathbb A^1$-connected components of a space is $\mathbb A^1$-invariant. Using purely algebro-geometric methods, we determine the sheaf of $\mathbb A^1$-connected components of a smooth projective surface, which is birationally ruled over a curve of genus $>0$. As a consequence, we show that Morel's conjecture holds for all smooth projective surfaces over an algebraically closed field of characteristic $0$.
We propose a new method to study birational maps between Fano varieties based on multiplier ideal sheaves. Using this method, we prove equivariant birational rigidity of four Fano threefolds acted on by the group A6. As an application, we obtain that the Cremona group of rank 3 has at least five non-conjugate subgroups isomorphic to A6.
This article gives the proof of results announced in [J. Blanc, Finite Abelian subgroups of the Cremona group of the plane, C.R. Acad. Sci. Paris, Sér. I 344 (2007), 21-26.] and some description of automorphisms of rational surfaces. Given a finite Abelian subgroup of the Cremona group of the plane, we provide a way to decide whether it is birationally conjugate to a group of automorphisms of a minimal surface. In particular, we prove that a finite cyclic group of birational transformations of the plane is linearisable if and only if none of its non-trivial elements fix a curve of positive gen…
We show that hyperelliptic symplectic Lefschetz fibrations are symplectically birational to two-fold covers of rational ruled surfaces, branched in a symplectically embedded surface. This reduces the classification of genus 2 fibrations to the classification of certain symplectic submanifolds in rational ruled surfaces.
An equivariant stable birational invariant of an action of a finite group on a smooth projective variety is the first cohomology group of the Picard module. Bogomolov-Prokhorov and Shinder computed this for actions of cyclic groups on rational surfaces, with maximal stabilizers, in terms of the geometry of the fixed point locus. Using the Brauer group of the quotient stack, we extend the computation to more general actions and relate it to the equivariant Burnside group formalism.
The decomposition group of an irreducible plane curve $X\subset\mathbb P^2$ is the subgroup $\mathrm{Dec}(X)\subset\mathrm{Bir}(\mathbb P^2)$ of birational maps which restrict to a birational map of $X$. We show that $\mathrm{Dec}(X)$ is generated by its elements of degree $\leq2$ when $X$ is either a conic or rational cubic curve.
We prove that if $X$ is a rationally connected threefold and $G$ is a $p$-subgroup in the group of birational selfmaps of $X$, then $G$ is an abelian group generated by at most $3$ elements provided that $p\ge 17$. We also prove a similar result for $p\ge 11$ under an assumption that $G$ acts on a (Gorenstein) $G$-Fano threefold, and show that the same holds for $p\ge 5$ under an assumption that $G$ acts on a $G$-Mori fiber space.
We classify finite groups acting by birational transformations of a non-trivial Severi--Brauer surface over a field of characteristc zero that are not conjugate to subgroups of the automorphism group. Also, we show that the automorphism group of a smooth cubic surface over a field $K$ of characteristic zero that has no $K$-points is abelian, and find a sharp bound for the Jordan constants of birational automorphism groups of such cubic surfaces.
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…
This paper classifies surfaces of general type $S$ with $p_g=q=1$ having an involution $i$ such that $S/i$ has non-negative Kodaira dimension and that the bicanonical map of $S$ factors through the double cover induced by $i.$ It is shown that $S/i$ is regular and either: a) the Albanese fibration of $S$ is of genus 2 or b) $S$ has no genus 2 fibration and $S/i$ is birational to a $K3$ surface. For case a) a list of possibilities and examples are given. An example for case b) with $K^2=6$ is also constructed.
We prove that an analogue of Jordan's theorem on finite subgroups of general linear groups holds for the groups of biregular automorphisms of elliptic ruled surfaces. This gives a positive answer to a question of Vladimir L. Popov.
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.
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 describe conjugacy classes of finite subgroups of odd order in the group of birational automorphisms of the real projective plane.
Finite groups of symplectic birational transformations of IHS manifolds of $OG10$ type
本文分类了 $OG10$ 型 IHS 流形上忠实辛双有理变换的有限群，证明其作用共轭于 375 个等距群的子群，并给出由三次四重体自同构决定的准则，结果经计算机辅助验证并公开于 Zenodo。
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.
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