On group automorphisms in universal algebraic geometry

代数簇的自同构与无限传递性

Automorphisms of algebraic varieties and infinite transitivity

本文综述了仿射代数簇自同构群的多重与无限传递性，探讨了其与柔性的等价关系及代数几何推论，并分析了由有限单参数子群生成的情形及复分析与组合群论中的扩展应用。

Fixed points of automorphisms of real algebraic curves

We bound the number of fixed points of an automorphism of a real curve in terms of the genus and the number of connected components of the real part of the curve. Using this bound, we derive some consequences concerning the maximum order of an automorphism and the maximum order of an abelian group of automorphisms of a real curve. We also bound the full group of automorphisms of a real hyperelliptic curve.

On large automorphism groups of algebraic curves in positive characteristic

In his investigation on large $K$-automorphism groups of an algebraic curve, Stichtenoth obtained an upper bound on the order of the first ramification group of an algebraic curve $\cX$ defined over an algebraically closed field of characteristic $p$. Stichtenoth's bound has raised the problem of classifying all $\K$-automorphism groups $G$ of $\cX$ with the following property: There is a point $P\in \cX$ for which \begin{equation} |G_P^{(1)}|>\frac{p}{p-1}g. \end{equation} Such a classification is obtained here by proving Theorem 1.3

Algebraically hyperbolic manifolds have finite automorphism groups

A projective manifold $M$ is algebraically hyperbolic if there exists a positive constant $A$ such that the degree of any curve of genus $g$ on $M$ is bounded from above by $A(g-1)$. A classical result is that Kobayashi hyperbolicity implies algebraic hyperbolicity. It is known that Kobayashi hyperbolic manifolds have finite automorphism groups. Here we prove that, more generally, algebraically hyperbolic projective manifolds have finite automorphism groups.

Some automorphism groups are linear algebraic

Consider a normal projective variety $X$, a linear algebraic subgroup $G$ of Aut($X$), and the field $K$ of $G$-invariant rational functions on $X$. We show that the subgroup of Aut($X$) that fixes $K$ pointwise is linear algebraic. If $K$ has transcendence degree $1$ over $k$, then Aut($X$) is an algebraic group.

On algebraic automorphisms and their rational invariants

Let X be an affine irreducible variety over an algebraically closed field k of characteristic zero. Given an automorphism F, we denote by k(X)^F its field of invariants, i.e. the set of rational functions f on X such that f(F)=f. Let n(F) be the transcendence degree of k(X)^F over k. In this paper, we study the class of automorphisms F of X for which n(F)= dim X - 1. More precisely, we show that under some conditions on X, every such automorphism is of the form F=A_g, where A is an algebraic action of a linear algebraic group G of dimension 1 on X, and where g belongs to G. As an application, …

线性代数群作为自同构群的实现

Realisation of linear algebraic groups as automorphism groups

本文证明域 $F$ 上的任意线性代数群 $G$ 同构于某光滑射影 $F$-簇的自同构群概型，该簇由射影空间沿适当光滑子簇的爆破构造而成。

On automorphisms of algebraic curves

An irreducible, algebraic curve $\mathcal X_g$ of genus $g\geq 2$ defined over an algebraically closed field $k$ of characteristic $\mbox{char } \, k = p \geq 0$, has finite automorphism group $\mbox{Aut} (\mathcal X_g)$. In this paper we describe methods of determining the list of groups $\mbox{Aut} (\mathcal X_g)$ for a fixed $g\geq 2$. Moreover, equations of the corresponding families of curves are given when possible.

On topologically trivial automorphisms of compact Kähler manifolds and algebraic surfaces

In this paper, we investigate automorphisms of compact Kähler manifolds with different levels of topological triviality. In particular, we provide several examples of smooth complex projective surfaces X whose groups of $C^\infty$-isotopically trivial automorphisms, resp. cohomologically trivial automorphisms, have a number of connected components which can be arbitrarily large.

Algebraic integrable systems related to spectral curves with automorphisms

We apply a reduction to the Beauville systems to obtain a family of new algebraic completely integrable systems, related to curves with a cyclic automorphism.

关于Calabi-Yau流形的双有理自同构的代数动力学与泛Coxeter群

On algebraic dynamics of birational automorphisms of Calabi-Yau manifolds and the universal Coxeter groups

本文延续对复Calabi-Yau流形双有理自同构代数动力学的研究，聚焦原始性与Zariski稠密轨道的存在性，并利用泛Coxeter群及其Coxeter元素作为关键工具。

源于代数动力学的Calabi-Yau三维流形的自同构及第二陈类

Automorphisms of Calabi-Yau threefolds from algebraic dynamics and the second Chern class

本文是作者在MFO的报告记录，基于代数动力学视角，对严格Calabi-Yau三维流形的自同构及第二陈类问题进行了推测性探讨。

Algebraic curves with automorphism groups of large prime order

Let $\mathcal{X}$ be an algebraic curve of genus $g$ defined over an algebraically closed field $K$ of characteristic $p \geq 0$, and $q$ a prime dividing $|\mbox{Aut}(\mathcal{X})|$. We say that $\mathcal{X}$ is a $q$-curve. Homma proved that either $q \leq g+1$ or $q = 2g+1$, and classified $(2g+1)$-curves. In this note, we classify $(g+1)$-curves, and fully characterize the automorphism groups of $q$-curves for $q= 2g+1, g+1$. We also give some partial results on $q$-curves for $q = g, g-1$.

Algebraic groups as automorphism groups of algebras

We show that every algebraic group scheme over a field with at least 8 elements can be realized as the group of automorphisms of a nonassociative algebra. This is only a modest improvement of the theorem of Gordeev and Popov (2003), but it allows us to give a new characterization of algebraic Lie algebras and to simplify the standard descriptions of Mumford--Tate domains and Shimura varieties as moduli spaces. Once the original argument of Gordeev and Popov has been rewritten in the language of schemes, we find that it also applies to algebraic groups over discrete valuation rings.

Algebraic cycles on certain hyperkaehler fourfolds with an order $3$ non-symplectic automorphism II

Let $X$ be a hyperkähler variety, and assume that $X$ admits a non-symplectic automorphism $σ$ of order $k>{1\over 2}\dim X$. Bloch's conjecture predicts that the quotient $X/<σ>$ should have trivial Chow group of $0$-cycles. We verify this for Fano varieties of lines on certain cubic fourfolds having an order $3$ non-symplectic automorphism.

椭圆曲面的数值平凡自同构III：非代数情形

Numerically trivial automorphisms of elliptic surfaces III: the non-algebraic case

研究非代数恰当椭圆曲面\(S\)的数值平凡自同构群\(Aut_{\mathbb{Q}}(S)\)，完整处理其凯勒情形。

Algebraic cycles on certain hyperkaehler fourfolds with an order $3$ non-symplectic automorphism

Let $X$ be a hyperkähler variety, and assume $X$ has a non-symplectic automorphism $σ$ of order $>{1\over 2}\dim X$. Bloch's conjecture predicts that the quotient $X/<σ>$ should have trivial Chow group of $0$-cycles. We verify this for Fano varieties of lines on certain special cubic fourfolds having an order $3$ non--symplectic automorphism.

具有许多自同构的代数几何码的构造方法，源于代数曲线的Galois点

Algebraic-geometric codes with many automorphisms arising from Galois points

本文提出一种通过Galois点构造具有大量自同构的代数几何码的方法，展示了其在代数曲线中的应用。

On connected automorphism groups of algebraic varieties

Let $X$ be a normal projective algebraic variety, $G$ its largest connected automorphism group, and $A(G)$ the Albanese variety of $G$. We determine the isogeny class of $A(G)$ in terms of the geometry of $X$. In characteristic 0, we show that the dimension of $A(G)$ is the rank of every maximal trivial direct summand of the tangent sheaf of $X$. Also, we obtain an optimal bound for the dimension of the largest anti-affine closed subgroup of $G$ (which is the smallest closed subgroup that maps onto $A(G)$).

Some finiteness results on triangular automorphisms

In this note we prove that every finite collection of connected algebraic subgroups of the group of triangular automorphisms of the affine space generates a connected solvable algebraic subgroup.

具有大素数平方阶自同构群的代数曲线

Algebraic curves admitting automorphism groups of large prime square order

针对任意特征代数闭域上的曲线，本文给出了自同构群阶为素数平方的素数界、对应曲线族分类及最高界曲线的完全自同构群。

双有理自同构的正则化与代数稳定性

Regularization and algebraic stability of birational automorphisms

本文比较代数簇上五类双有理自同构，确定其包含关系，并证明在射影空间 $\mathbb{P}^k$（$k\geqslant 3$）中非平凡包含均为严格，通过显式例子验证。

由代数元组成的仿射簇的自同构群

Automorphism groups of affine varieties consisting of algebraic elements

本文证明若仿射代数簇自同构群的中性分支由代数元组成，则其为嵌套结构（代数子群的直极限），并利用连通 ind-group 的相关性质完成证明，改进了先前结论。

双Kähler流形族中双有理自同构群的代数熵

Algebraic entropy of birational automorphism groups in families of hyper-Kähler manifolds

本文研究射影双Kähler流形族中双有理自同构群的代数熵行为，探讨其性质与变化。

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 $\mathbb{C}^2$.

Abelian varieties as automorphism groups of smooth projective varieties in arbitrary characteristics

Let $A$ be an abelian variety over an algebraically closed field. We show that $A$ is the automorphism group scheme of some smooth projective variety if and only if $A$ has only finitely many automorphisms as an algebraic group. This generalizes a result of Lombardo and Maffei for complex abelian varieties.

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。

Algebraic varieties with automorphism groups of maximal rank

We confirm, to some extent, the belief that a projective variety X has the largest number (relative to the dimension of X) of independent commuting automorphisms of positive entropy only when X is birational to a complex torus or a quotient of a torus. We also include an addendum to an early paper though it is not used in the present paper.

Automorphisms of $\mathbb{C}^3$ Commuting with a $\mathbb{C}^+$-Action

Let $ρ$ be an algebraic action of the additive group $\mathbb{C}^+$ on the three-dimensional affine space $\mathbb{C}^3$. We describe the group $\textrm{Cent}(ρ)$ of polynomial automorphisms of $\mathbb{C}^3$ that commute with $ρ$. A particular emphasis lies in the description of the automorphisms in $\textrm{Cent}(ρ)$ coming from algebraic $\mathbb{C}^+$-actions. As an application we prove that the automorphisms in $\textrm{Cent}(ρ)$ that are the identity on the algebraic quotient of $ρ$ form a characteristic subgroup of $\textrm{Cent}(ρ)$.

Oka properties of groups of holomorphic and algebraic automorphisms of complex affine space

We show that the group of all holomorphic automorphisms of complex affine space $\mathbb C^n$, $n>1$, and several of its subgroups satisfy the parametric Oka property with approximation and with interpolation on discrete sets.

Birational automorphisms of algebraic varieties with a pencil of cubic surfaces

It is proved that on a smooth algebraic variety, fibered into cubic surfaces over the projective line and sufficiently ``twisted'' over the base, there is only one pencil of rational surfaces -- that is, this very pencil of cubics. In particular, this variety is non-rational; moreover, it can not be fibered into rational curves. The proof is obtained by means of the method of maximal singularities.

刚性仿射曲面的自同构群：单位元分支

Automorphism groups of rigid affine surfaces: the identity component

本文研究了仿射代数曲面自同构群的单位元分支成为代数群的条件，证明当且仅当曲面不允许加法群的有效作用时成立，此时该分支为秩不超过2的代数环面。

Large $3$-groups of automorphisms of algebraic curves in characteristic $3$

Let $S$ be a $p$-subgroup of the $\K$-automorphism group $\aut(\cX)$ of an algebraic curve $\cX$ of genus $\gg\ge 2$ and $p$-rank $γ$ defined over an algebraically closed field $\mathbb{K}$ of characteristic $p\geq 3$.In this paper we prove that if $|S|>2(\gg-1)$ then one of the following cases occurs. \begin{itemize} \item[(i)] $γ=0$ and the extension $\K(\cX)/\K(\cX)^S$ completely ramifies at a unique place, and does not ramify elsewhere. \item[(ii)] $γ>0$, $p=3$, $\cX$ is a general curve, $S$ attains the Nakajima's upper bound $3(γ-1)$ and $\K(\cX)$ is an unramified Galois extension of the …

Ordinary algebraic curves with many automorphisms in positive characteristic

Let $\mathcal{X}$ be an ordinary (projective, geometrically irreducible, nonsingular) algebraic curve of genus $\mathcal{g}(\mathcal{X}) \ge 2$ defined over an algebraically closed field $\mathbb{K}$ of odd characteristic $p$. Let $Aut(\mathcal{X})$ be the group of all automorphisms of $\mathcal{X}$ which fix $\mathbb{K}$ element-wise. For any solvable subgroup $G$ of $Aut(\mathcal{X})$ we prove that $|G|\leq 34 (\mathcal{g}(\mathcal{X})+1)^{3/2}$. There are known curves attaining this bound up to the constant $34$. For $p$ odd, our result improves the classical Nakajima bound $|G|\leq 84(\mat…

Embeddings of groups ${\rm Aut}(F_n)$ into automorphism groups of algebraic varieties

For every positive integer $n$, we construct, using algebraic groups, an infinite family of irreducible algebraic varieties $X$,whose automorphism group ${\rm Aut}(X)$ contains the automorphism group ${\rm Aut}(F_n)$ of a free group $F_n$ of rank $n$ as a subgroup. This property implies that, for $n \geqslant 2$, such groups ${\rm Aut}(X)$ are nonamenable, and, for $n \geqslant 3$, nonlinear and contain the braid group $B_n$ on $n$ strands. Some of these varieties $X$ are affine, and among affine, some are rational and some are not, some are smooth and some are singular. As an application, we …

Small bound for birational automorphism groups of algebraic varieties (with an Appendix by Yujiro Kawamata)

We give an effective upper bound of |Bir(X)| for the birational automorphism group of an irregular n-fold (with n = 3) of general type in terms of the volume V = V(X) under an ''albanese smoothness and simplicity'' condition. To be precise, |Bir(X)| < d_3 V^{10}. An optimum linear bound |Bir(X)|-1 < (1/3)(42)^3 V is obtained for those 3-folds with non-maximal albanese dimension. For all n > 2, a bound |Bir(X)| < d_n V^{10} is obtained when alb_X is generically finite, alb(X) is smooth and Alb(X) is simple.

Automorphism groups of rational elliptic and quasi-elliptic surfaces in all characteristics

We study the groups of automorphisms of rational algebraic surfaces that admit a relatively minimal pencil of curves of arithmetic genus one over an algebraically closed field of arbitrary characteristic. In particular, we classify such surfaces that admit non-trivial automorphisms that act trivially on the Picard group. As an application, we classify classical Enriques surfaces in characteristic $2$ that admit non-trivial numerically trivial automorphisms.

On algebraic curves with many automorphisms in characteristic p

Let $\mathcal{X}$ be an irreducible, non-singular, algebraic curve defined over a field of odd characteristic $p$. Let $g$ and $γ$ be the genus and $p$-rank of $\mathcal{X}$, respectively. The influence of $g$ and $γ$ on the automorphism group $Aut(\mathcal{X})$ of $\mathcal{X}$ is well-known in the literature. If $g \geq 2$ then $Aut(\mathcal{X})$ is a finite group, and unless $\mathcal{X}$ is the so-called Hermitian curve, its order is upper bounded by a polynomial in $g$ of degree four (Stichtenoth). In 1978 Henn proposed a refinement of Stichtenoth's bound of cube order in $g$ up to few ex…

实有理曲面的自同构群是n-传递的

The group of automorphisms of a real rational surface is n-transitive

该研究证明有理非奇异紧连通实代数曲面的实代数自同构群对所有自然数n均具有n-传递性，并以此给出此类曲面同构等价于拓扑同胚的更简洁新证明。

具有大循环自同构群的代数曲线

Algebraic curves with a large cyclic automorphism group

本文研究了正特征$p \neq 2$下具有阶至少为$2g(\mathcal{X}) + 1$的循环自同构群的代数曲线，成功给出了完整分类，克服了野分歧带来的挑战，填补了Irokawa-Sasaki结果在正特征下的对应。

Cox rings, semigroups and automorphisms of affine algebraic varieties

We study the Cox realization of an affine variety, i.e., a canonical representation of a normal affine variety with finitely generated divisor class group as a quotient of a factorially graded affine variety by an action of the Neron-Severi quasitorus. The realization is described explicitly for the quotient space of a linear action of a finite group. A universal property of this realization is proved, and some results on the divisor theory of an abstract semigroup emerging in this context are given. We show that each automorphism of an affine variety can be lifted to an automorphism of the Co…

On automorphisms and endomorphisms of projective varieties

We first show that any connected algebraic group over a perfect field is the neutral component of the automorphism group scheme of some normal projective variety. Then we show that very few connected algebraic semigroups can be realized as endomorphisms of some projective variety X, by describing the structure of all connected subsemigroup schemes of End(X).

Large $p$-groups of automorphisms of algebraic curves in characteristic $p$

Let $S$ be a $p$-subgroup of the $\mathbb K$-automorphism group $Aut(\mathcal X)$ of an algebraic curve $\mathcal X$ of genus $g\ge 2$ and $p$-rank $γ$ defined over an algebraically closed field $\mathbb{K}$ of characteristic $p\geq 3$. Nakajima proved that if $γ\ge 2$ then $|S|\leq \textstyle\frac{p}{p-2}(g-1)$. If equality holds, $\mathcal X$ is a Nakajima extremal curve. We prove that if $$|S|>\textstyle\frac{p^2}{p^2-p-1}(g-1)$$ then one of the following cases occurs: (i) $γ=0$ and the extension $\mathbb K(\mathcal X)|\mathbb K(\mathcal X)^S$ completely ramifies at a unique place, and does…

Realisation of abelian varieties as automorphism groups

Let $A/F$ be an abelian variety over a field. Does there exist a smooth projective $F$-variety $X$, such that $A$ is isomorphic to the automorphism group scheme of $X/F$? We show that the answer is positive, if and only if $A$ has only finitely many automorphisms, over an algebraic closure of $F$. When $F=\mathbb C$, this result is due to Lombardo and Maffei. When $F$ is algebraically closed, it was obtained independently by Blanc and Brion.

Mori Del Pezzo纤维 over 一个无理曲线的自同构群

Automorphism groups of Mori Del Pezzo fibrations over an irrational curve

本文研究了Mori Del Pezzo纤维 over 一个正亏曲线的自同构群，并给出了Bir(C×P²)中最大连通代数子群的分类。

Automorphism groups of almost homogeneous varieties

Consider a smooth connected algebraic group $G$ acting on a normal projective variety $X$ with an open dense orbit. We show that Aut($X$) is a linear algebraic group if so is $G$; for an arbitrary $G$, the group of components of Aut($X$) is arithmetic. Along the way, we obtain a restrictive condition for $G$ to be the full automorphism group of some normal projective variety.

Tautological rings on Jacobian varieties of curves with automorphisms

Let $J$ be the Jacobian of a smooth projective complex curve $C$ which admits non-trivial automorphisms, and let $A(J)$ be the ring of algebraic cycles on $J$ with rational coefficients modulo algebraic equivalence. We present new tautological rings in $A(J)$ which extend in a natural way the tautological ring studied by Beauville (Compos Math 140(3):683-688, 2004). We then show there exist tautological rings induced on special complementary abelian subvarieties of $J$.

Some algebraic and geometric properties of hyper-Kähler fourfold symmetries

We complete the classification of order $5$ nonsymplectic automorphisms on hyper-Kähler fourfolds deformation equivalent to the Hilbert square of a K3 surface. We then compute the topological Lefschetz number of natural automorphisms of generalized Kummer fourfolds and we describe the geometry of their fix loci.

A surface in odd characteristic with discrete and non-finitely generated automorphism group

It was proved by Tien-Cuong Dinh and me that there is a smooth complex projective surface whose automorphism group is discrete and not finitely generated. In this paper, we will show that there is a smooth projective surface, birational to some K3 surface, such that the automorphism group is discrete and not finitely generated, over any algebraically closed field of odd characteristic except precisely an algebraic closure of the prime field.

Algebraic surfaces with quotient singularities - including some discussion on automorphisms and fundamental groups

We survey some recent progress in the study of algebraic varieties X with log terminal singularities, especially, the uni-ruledness of the smooth locus X^0 of X, the fundamental group of X^0 and the automorphisms group on (smooth or singular) X when dim X = 2. The full automorphism groups of a few interesting types of K3 surfaces are described, mainly by Keum-Kondo. We conjecture that when X is Q-Fano then X^0 has a finite fundamental group, which had been proved if either dim X < 3 or the Fano index is bigger than dim X - 2. We also conjecture that when X is a log Enriques (e.g. a normal K3 o…

Algebraic curves with many automorphisms

Let $X$ be a (projective, geometrically irreducible, nonsingular) algebraic curve of genus $g \ge 2$ defined over an algebraically closed field $K$ of odd characteristic $p$. Let $Aut(X)$ be the group of all automorphisms of $X$ which fix $K$ element-wise. It is known that if $|Aut(X)|\geq 8g^3$ then the $p$-rank (equivalently, the Hasse-Witt invariant) of $X$ is zero. This raises the problem of determining the (minimum-value) function $f(g)$ such that whenever $|Aut(X)|\geq f(g)$ then $X$ has zero $p$-rank. For {\em{even}} $g$ we prove that $f(g)\leq 900 g^2$. The {\em{odd}} genus case appear…

Automorphism Groups of Danielewski Surfaces

In this note we study the automorphism group of a smooth Danielewski surface $D_p= \{(x,y,z) \in \mathbb{A}^3 \mid xy = p(z) \} \subset \mathbb{A}^3$, where $p \in \mathbb{C}[z]$ is a polynomial without multiple roots and $deg (p) \ge 3$. It is known that two such generic surfaces $D_p$ and $D_q$ have isomorphic automorphism groups. Moreover, $\mathrm{Aut}(D_p)$ is generated by algebraic subgroups and there is a natural isomorphism $ϕ\colon \mathrm{Aut}(D_p) \xrightarrow{\sim} \mathrm{Aut}(D_q)$ which restricts to an isomorphism of algebraic groups $G \xrightarrow{\sim} ϕ(G)$ for any algebraic…

Large odd prime power order automorphism groups of algebraic curves in any characteristic

Let $\mathcal{X}$ be a (projective, geometrically irreducible, nonsingular) algebraic curve of genus $g \ge 2$ defined over an algebraically closed field $\mathbb{K}$ of odd characteristic $p\ge 0$, and let $\rm{Aut}(\mathcal{X})$ be the group of all automorphisms of $\mathcal{X}$ which fix $\mathbb{K}$ element-wise. For any a subgroup $G$ of $\rm{Aut}(\mathcal{X})$ whose order is a power of an odd prime $d$ other than $p$, the bound proven by Zomorrodian for Riemann surfaces is $|G|\leq 9(g-1)$ where the extremal case can only be obtained for $d=3$. We prove Zomorrodian's result for any $\mat…

On the group of automorphisms of a quasi-affine variety

Let K be an algebraically closed field of characteristic zero. We show that if the automorphisms group of a quasi-affine variety X is infinite, then X is uniruled.