A Characterization of Semisimple Plane Polynomial Automorphisms

Locally tame plane polynomial automorphisms

For automorphisms of a polynomial ring in two variables over a domain R, we show that local tameness implies global tameness provided that every 2-generated invertible R-module is free. We give many examples illustrating this property.

The automorphism theorem and additive group actions on the affine plane

Due to Rentschler, Miyanishi and Kojima, the invariant ring for a ${\bf G}_a$-action on the affine plane over an arbitrary field is generated by one coordinate. In this note, we give a new short proof for this result using the automorphism theorem of Jung and van der Kulk.

Normal subgroup generated by a plane polynomial automorphism

We study the normal subgroup <f> generated by a non trivial element f in the group G of complex plane polynomial automorphisms having Jacobian determinant 1. On one hand if f has length at most 8 relatively to the classical amalgamated product structure of G, we prove that <f> = G. On the other hand if f is a sufficiently generic element of even length at least 14, we prove that <f> is a proper subgroup of G.

Automorphism groups of smooth plane curves

The author determines the structure of automorphism groups of smooth plane curves of degree at least four. Furthermore, he gives some upper bounds for the order of automorphism groups of smooth plane curves and classifies the cases with large automorphism groups. This paper also contains a simple proof of the uniqueness of smooth plane curves with the full automorphism group of maximal order for each degree.

Automorphism Groups of Affine Varieties and a Characterization of Affine n-Space

We show that the automorphism group of affine n-space $A^n$ determines $A^n$ up to isomorphism: If $X$ is a connected affine variety such that $Aut(X)$ is isomorphic to $Aut(A^n)$ as ind-groups, then $X$ is isomorphic to $A^n$ as a variety. We also show that every finite group and every torus appears as $Aut(X)$ for a suitable affine variety $X$, but that $Aut(X)$ cannot be isomorphic to a semisimple group. In fact, if $Aut(X)$ is finite dimensional and $X$ not isomorphic to the affine line $A^1$, then the connected component $Aut(X)^0$ is a torus. Concerning the structure of $Aut(A^n)$ we pro…

Infinite transitivity for automorphism groups of the affine plane

This paper is dedicated to the problem of infinite transitivity for algebraically generated automorphism groups of the affine plane. We provide a necessary and sufficient condition of infinite transitivity for a large family of subgroups generated by one-parameter additive groups.

Donaldson-Thomas Transformation of Double Bruhat Cells in Semisimple Lie Groups

Double Bruhat cells $G^{u,v}$ were studied by Fomin and Zelevinsky. They provide important examples of cluster algebras and cluster Poisson varieties. Cluster varieties produce examples of 3d Calabi-Yau categories with stability conditions, and their Donaldson-Thomas invariants, defined by Kontsevich and Soibelman, are encoded by a formal automorphism on the cluster variety known as the Donaldson-Thomas transformation. Goncharov and Shen conjectured in that for any semisimple Lie group $G$, the Donaldson-Thomas transformation of the cluster Poisson variety $H\backslash G^{u,v}/H$ is a slight m…

A vanishing theorem on fake projective planes with enough automorphisms

For every fake projective plane $X$ with automorphism group of order 21, we prove that $H^i(X, 2L)=0$ for all $i$ and for every ample line bundle $L$ with $L^2=1$. For every fake projective plane with automorphism group of order 9, we prove the same vanishing for every cubic root (and its twist by a 2-torsion) of the canonical bundle $K$. As an immediate consequence, there are exceptional sequences of length 3 on such fake projective planes.

Automorphism group of plane curve computed by Galois points, II

Recently, the first author classified finite groups obtained as automorphism groups of smooth plane curves of degree $d \ge 4$ into five types. He gave an upper bound of the order of the automorphism group for each types. For one of them, the type (a-ii), that is given by $\max \left\{ 2 d (d - 2), 60 d \right\}$. In this article, we shall construct typical examples of smooth plane curve $C$ by applying the method of Galois points, whose automorphism group has order $60 d$. In fact, we determine the structure of the automorphism group of those curves.

关于具有21阶自同构群的假射影平面的方程

On equations of fake projective planes with automorphism group of order $21$

研究Dolgachev椭圆面的双重和三重纤维，找到两个新的具有21个自同构的假射影平面方程，完成该自同构群的假射影平面方程研究任务，包括Keum发现的假射影平面。

Automorphism groups of smooth plane curves with many Galois points

We settle the automorphism groups of curves appearing in a classification list of smooth plane curves with at least two Galois points. One of them is an ordinary curve whose automorphism group exceeds the Hurwitz bound.

Invariants, Bitangents and Matrix Representations of Plane Quartics with 3-Cyclic Automorphisms

In this work we compute the Dixmier invariants and bitangents of the plane quartics with 3,6 or 9-cyclic automorphisms, we find that a quartic curve with 6-cyclic automorphism will have 3 horizontal bitangents which form an asysgetic triple. We also discuss the linear matrix representation problem of such curves, and find a degree 6 equation of 1 variable which solves the symbolic solution of the linear matrix representation problem for the curve with 6-cyclic automorphism.

On the automorphism group of non-singular plane curves fixing the degree

This note is devoted, after the result of Harui, arXiv:1306.5842, to solve some natural questions for non-singular plane curves of degree $d$ over an algebraically closed field $K$ of zero characteristic.

On the orbits of plane automorphisms and their stabilizers

Let $\Bbbk$ be a perfect field with algebraic closure $\overline{\Bbbk}$. If $H$ is a subgroup of plane automorphisms over $\Bbbk$ and $p\in\overline{\Bbbk}^2$ is a point, we describe the subgroup consisting of plane automorphisms which stabilize the orbit of $p$ under $H$, when this orbit has irreducible closure in $\overline{\Bbbk}^2$. As an application, we treat the case where $H$ is cyclic and the closure of the orbit of $p$ is an arbitrary (non-necessarily irreducible) curve.

Projective plane curves whose automorphism groups are simple and primitive

We study complex projective plane curves with a given group of automorphisms. Let $G$ be a simple primitive subgroup of $PGL(3, \mathbb{C})$, which is isomorphic to $A_{6}$, $A_{5}$ or $PSL(2, \mathbb{F}_{7})$. We obtain a necessary and sufficient condition on $d$ for the existence of a nonsingular projective plane curve of degree $d$ invariant under $G$. We also study an analogous problem on integral curves.

Hyperbolicity of cycle spaces and automorphism groups of flag domains

If G_0 is a real form of a complex semisimple Lie group G and Z is compact G-homogeneous projective algebraic manifold, then G_0 has only finitely many orbits on Z. Complex analytic properties of open G_0-orbits D (flag domains) are studied. Schubert incidence-geometry is used to prove the Kobayashi hyperbolicity of certain cycle space components C_q(D). Using the hyperbolicity of C_q(D) and analyzing the action of Aut(D) on it, an exact description of Aut(D) is given. It is shown that, except in the easily understood case where D is holomorphically convex with a nontrivial Remmert reduction, …

On the automorphisms of some $K3$ surface double cover of the plane

In this paper we study the automorphisms group of some $K3$ surfaces which are double covers of the projective plane ramified over a smooth sextic plane curve. More precisely, we study the case of a $K3$ surface of Picard rank two such that there is a rational curve of degree $d$ which is tangent to the sextic in $d$ points.

The stratification by automorphism groups of smooth plane sextic curves

We obtain the list of automorphism groups for smooth plane sextic curves over an algebraically closed field K of characteristic p=0 or p>21. Moreover, we assign to each group a geometrically complete family over K describing its corresponding stratum, that is, a generic defining polynomial equation with parameters such that any curve in the stratum is K-isomorphic to a non-singular plane model obtained by specializing the values of those parameters over K.

Automorphisms of Products of Drinfeld Half Planes

The Drinfeld upper half-planes play the role of symmetric spaces in the $p$-adic analytic world. We find the automorphism group of a product of such spaces, where each may be defined over a different field. We deduce a rigidity theorem for quotients of such products by discrete and torsion free groups.

The Bicanonical map of fake projective planes with an automorphism

We show, for several fake projective planes with nontrivial automorphism group, that the bicanonical map is an embedding.

关于$G$-诱导簇的自同构群

On Automorphism group of a $G$-induced variety

本文研究复数域上伴随型连通半单代数群$G$对应的$G$-诱导簇，计算了若干这类簇的代数自同构群中含恒等元的连通分支。

Automorphisms of the plane preserving a curve

We study the group of automorphisms of the affine plane preserving some given curve, over any field. The group is proven to be algebraic, except in the case where the curve is a bunch of parallel lines. Moreover, a classification of the groups of positive dimension occuring is also given in the case where the curve is geometrically irreducible and the field is perfect.

光滑平面曲线与唯一的外伽罗华点及其自同构群

Smooth plane curves with a unique outer Galois point and their automorphism groups

研究光滑平面曲线的外伽罗华点条件，揭示其自同构群结构及方程形式，分类非循环的自同构群情况。

An algorithmic approach to the Polydegree Conjecture for plane polynomial automorphisms

We study the interaction between two structures on the group of polynomial automorphisms of the affine plane: its structure as an amalgamated free product and as an infinite-dimensional algebraic variety. We introduce a new conjecture, and show how it implies the Polydegree Conjecture. As the new conjecture is an ideal membership question, this shows that the Polydegree Conjecture is algorithmically decidable. We further describe how this approach provides a unified and shorter method of recovering existing results of Edo and Furter.

具有大自同构群的超特殊平面五次曲线

Superspecial plane quintics with large automorphism groups

本文研究自同构群阶大于10以及循环群阶为8和10的平面五次曲线，利用高斯超几何级数的截断描述其超特殊性，并确定了自同构群同构于Z/10Z的超特殊平面五次曲线的同构类数目，同时给出了自同构群同构于Z/8Z的曲线的有效枚举算法及13<p<10000的计算结果。

A Fake Projective Plane with an order 7 automorphism

A fake projective plane is a compact complex manifold of dimension 2 which has the same Betti numbers as the complex projective plane, but not isomorphic to the complex projective plane. As was shown by D. Mumford, there exists at least one such surface. In this paper we prove the existence of a fake projective plane which is birational to a cyclic cover of degree 7 of a Dolgachev surface.

自由群自同构群到仿射代数簇自同构群的嵌入

Embeddings of automorphism groups of free groups into automorphism groups of affine algebraic varieties

构造了自同构群包含 Aut(F_n) 的有理仿射代数簇新无限系列，并据此改进了 Cremona 群包含 Aut(F_n) 与辫子群 B_n 的秩界。

仿射平面自同构群的拓扑简单性

Topological simplicity of the group of automorphisms of the affine plane

本文证明任意无限域k上无限维代数群SAut_k(𝔸²)是拓扑简单的，还指出高维闭正规子群含所有tame自同构，并讨论了差异显著的有限域情形。

On automorphisms group of some K3 surfaces

In this paper we study the automorphisms group of some K3 surfaces which are double covers of the projective plane ramified over a smooth sextic plane curve. More precisely, we study some particlar case of a K3 surface of Picard rank two.

Nonsingular plane filling curves of minimum degree over a finite field and their automorphism groups: Supplements to a work of Tallini

Our concern is a nonsingular plane curve defined over a finite field of q elements which includes all the rational points of the projective plane over the field. The possible degree of such a curve is at least q+2. We prove that nonsingular plane curves of degree q+2 having the property actually exist. More precisely, we write down explicitly all of those curves. Actually, Giuseppe Tallini studied such curves in his old paper in 1961. We explain the connection between his work and ours. Moreover we give another proof of his result on the automorphism group of such a curve, from the viewpoint o…

On plane curves given by separated polynomials and their automorphisms

Let $\mathcal{C}$ be a plane curve defined over the algebraic closure $K$ of a prime finite field $\mathbb{F}_p$ by a separated polynomial, that is $\mathcal{C}: A(y)=B(x)$, where $A(y)$ is an additive polynomial of degree $p^n$ and the degree $m$ of $B(X)$ is coprime with $p$. Plane curves given by separated polynomials are well-known and studied in the literature. However just few informations are known on their automorphism groups. In this paper we compute the full automorphism group of $\mathcal{C}$ when $m \not\equiv 1 \pmod {p^n}$ and $B(X)$ has just one root in $K$, that is $B(X)=b_m(X+…

Automorphism groups of non-singular plane curves of degree 5

Henn and Komiya-Kuribayashi listed, independently, the groups $G$ for which $\widetilde{M_3^{Pl}(G)}$ is non-empty. In this paper, we determine the loci $\widetilde{M_6^{Pl}(G)}$, corresponding to non-singular degree $5$ projective plane curves, which are non-empty. Also, we present the analogy of Henn's results for quartic curves concerning non-singular plane model equations associated to these loci.

局部有限导数的 tame 等距群研究

On the Tame Isotropy Group of Locally Finite Derivations of K[X,Y]

本文研究了K[X,Y]局部有限导数的tame等距群，通过Van den Essen的分类确定各规范形式对应的tame等距群，并证明其与关联的指数自同构的tame等距群一致，与全自同构群行为不同。

Linearization of holomorphic families of algebraic automorphisms of the affine plane

Let $G$ be a reductive group. We prove that a family of polynomial actions of $G$ on $\mathbb{C}^2$, holomorphically parametrized by an open Riemann surface, is linearizable. As an application, we show that a particular class of reductive group actions on $\mathbb{C}^3$ is linearizable. The main step of our proof is to establish a certain restrictive Oka property for groups of equivariant algebraic automorphisms of $\mathbb{C}^2$.

Cyclic Automorphisms groups of genus 10 non-hyperelliptic curves

In this paper, we list all cyclic automorphisms subgroups $H$ for which there exists a smooth projective non-hyperelliptic sextic curve $C$ with $H\preceq Aut(C)$. Furthermore, we attach to each group a defining equation of a plane sextic curve having exactly this group as cyclic automorphisms subgroup. These equations cover up to isomorphism all plane non-singular sextic curves having some non-trivial automorphism.

Sur des invariants géométriques associés aux automorphismes du plan affine

We associate to each automorphism of the plane, a geometric construction with some properties, it is the {\it{canonical resolution}}. We study the geometry of the canonical resolution, we deduce from it an upper bound for a geometric invariant (the ample index) linked to an automorphism of the affine plane and we link the structure of the effective cone of the surface we get to values of another geometric invariant (the effective index).

微分自同构与 $L$-valued $G$-Higgs 包的障碍理论

Infinitesimal automorphisms and obstruction theory on the moduli of $L$-valued $G$-Higgs bundles

研究 $L$-valued $G$-Higgs 包的微分自同构及稳定性模堆结构，构造对称完美障碍理论以定义 Vafa-Witten 不变量。

Cremona transformations, surface automorphisms and plane cubics

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.

平面曲线的自同构

Automorphisms of Plane Curves defined from Chebychev polynomials

本文研究了由切比雪夫多项式定义的平面曲线的自同构群，确定了d=4及一般情况下2d=p^r+1或4d=p^r+1时的自同构群结构，并应用结果证明某些同 genus 的极大曲线不相等。

$BG$的Brauer群与模空间的gerbe结构

The Brauer group of $BG$ and gerbe structures of moduli spaces

本文研究了具有阶$N$自同构的曲线的$μ_N$-gerbe结构，并确定了其Brauer类，关键在于计算$BG$的Brauer群。

Canonical height functions on the affine plane associated with polynomial automorphisms

Let $f: \mathbb{A}^2 \to \mathbb{A}^2$ be a polynomial automorphism of dynamical degree $δ\geq 2$ over a number field $K$. (This is equivalent to say that $f$ is a polynomial automorphism that is not triangularizable.) Then we construct canonical height functions defined on $\mathbb{A}^2(\bar{K})$ associated with $f$. These functions satisfy the Northcott finiteness property, and an $\bar{K}$-valued point on $\mathbb{A}^2(\bar{K})$ is $f$-periodic if and only if its height is zero. As an application of canonical height functions, we give an estimate on the number of points with bounded height …

Bloch猜想在具有3阶自同构的伪射影平面上的证明

Bloch's conjecture for fake projective planes with an automorphism of order 3

本文证明了对某些一般型曲面上的对合在零维Chow群上作用为恒等，从而验证了Bloch猜想，并推广到某些伪射影平面。

A class of pseudoreal Riemann surfaces with diagonal automorphism group

A Riemann surface $\mathcal{S}$ having field of moduli $\mathbb{R}$, but not a field of definition, is called \emph{pseudoreal}. This means that $\mathcal{S}$ has anticonformal automorphisms, but non of them is an involution. We call a Riemann surface $\mathcal{S}$ \emph{plane} if it can be described by a smooth plane model of some degree $d\geq4$ in $\mathbb{P}^2_{\mathbb{C}}$. We characterize pseudoreal-plane Riemann surfaces $\mathcal{S}$, whose conformal automorphism group $\operatorname{Aut}_+(\mathcal{S})$ is $\operatorname{PGL}_3(\mathbb{C})$-conjugate to a finite non-trivial group $G$ …

Rationality of the quotient of $\mathbb{P}^2$ by finite group of automorphisms over arbitrary field of characteristic zero

Let $\Bbbk$ be a field of characteristic zero and $G$ be a finite group of automorphisms of projective plane over $\Bbbk$. Castelnuovo's criterion implies that the quotient of projective plane by $G$ is rational if the field $\Bbbk$ is algebraically closed. In this paper we prove that $\mathbb{P}^2_{\Bbbk} / G$ is rational for an arbitrary field $\Bbbk$ of characteristic zero.

Plane curves with a large linear automorphism group in characteristic $p$

Let $G$ be a subgroup of the three dimensional projective group $\mathrm{PGL}(3,q)$ defined over a finite field $\mathbb{F}_q$ of order $q$, viewed as a subgroup of $\mathrm{PGL}(3,K)$ where $K$ is an algebraic closure of $\mathbb{F}_q$. For the seven nonsporadic, maximal subgroups $G$ of $\mathrm{PGL}(3,q)$, we investigate the (projective, irreducible) plane curves defined over $K$ that are left invariant by $G$. For each, we compute the minimum degree $d(G)$ of $G$-invariant curves, provide a classification of all $G$-invariant curves of degree $d(G)$, and determine the first gap $\varepsilo…

平面帕斯卡有限自同构的阶与帕斯卡深度

Order and Pascal depth of Pascal finite automorphisms of the plane

研究特征 \(p>0\) 域上平面帕斯卡有限自同构，证明其阶由帕斯卡深度决定，阶谱为 \(\{1,p,p^2\}\) 且深度 \(\leq p^2\)。对多项式群给出阶为 \(p^2\) 上限的另一证明，还证明界限精确，对比了平面与高维情况。

A note on the stratification by automorphisms of smooth plane curves of genus 6

In this note, we give a so-called representative classification for the strata by automorphism group of smooth $\bar{k}$-plane curves of genus $6$, where $\bar{k}$ is a fixed separable closure of a field $k$ of characteristic $p = 0$ or $p > 13$. We start with a classification already obtained by the first author and we use standard techniques. Interestingly, in the way to get these families for the different strata, we find two remarkable phenomenons that did not appear before. One is the existence of a non $0$-dimensional final stratum of plane curves. At a first sight it may sound odd, but …

特征标栈的 motivic 镜像对称

Motivic Mirror Symmetry for Character Stacks

本文提出任意半单群特征标栈的motivic镜像对称版本，推广胞腔分解至连通约化群，用其描述特征标栈自同构并通过Weil配对建立胞腔对偶性，还在秩二特殊线性群上验证了猜想。

关于对称平面四次曲线

On symmetric plane quartic curves

本文研究具有大自同构群的平面四次曲线的几何，重点分析了在$[2,2,2]$型初等阿贝尔群作用下不变的光滑平面四次曲线以及自同构群阶为$48$的光滑平面四次曲线的几何性质。

Dynamical degrees of (pseudo)-automorphisms fixing cubic hypersurfaces

We give a way to construct group of pseudo-automorphisms of rational varieties of any dimension that fix pointwise the image of a cubic hypersurface of $P^n. These group are free products of involutions, and most of their elements have dynamical degree >1. Moreover, the Picard group of the varieties obtained is not big, if the dimension is at least 3. We also answer a question of E. Bedford on the existence of birational maps of the plane that cannot be lifted to automorphisms of dynamical degree >1, even if we compose them with an automorphism of the plane.

关于常规平面多项式仿射变换族的公共周期点的统一界

Uniform bound on common periodic points for families of regular plane polynomial automorphisms

研究了常规平面多项式仿射变换族中公共周期点的统一界问题，证明了在特定条件下公共周期点数量被统一常数限制。

关于Severi-Brauer曲面的双有理自同构

On birational automorphisms of Severi-Brauer surfaces

本文显式描述了代数非闭域上非射影平面的Severi-Brauer曲面的双有理自同构群的生成元。

仿射平面上可交换多项式映射的自同构群

Automorphism Groups of Commuting Polynomial Maps of the Affine Plane

本文研究与秩为N的半单李代数相关联的多项式折叠映射，推导出A2、B2、G2对应的仿射平面上映射首项公式，并据此计算各折叠映射的仿射自同构群。

Involutions and higher order automorphisms of Higgs bundle moduli spaces

We consider the moduli space $\mathcal{M}(G)$ of $G$-Higgs bundles over a compact Riemann surface $X$, where $G$ is a complex semisimple Lie group. This is a hyperkähler manifold homeomorphic to the moduli space $\mathcal{R}(G)$ of representations of the fundamental group of $X$ in $G$. In this paper we study finite order automorphisms of $\mathcal{M}(G)$ obtained by combining the action of an element of order $n$ in $H^1(X,Z)\rtimes \mbox{Out}(G)$, where $Z$ is the centre of $G$ and $\mbox{Out}(G)$ is the group of outer automorphisms of $G$, with the multiplication of the Higgs field by an $n…