Aristides Kontogeorgis
This paper has been withdrawn by the author. The main result is wrong, as M.Matignon and C. Lehr provided a counterexample to it.
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Birational automorphism groups in families of hyper-Kähler manifolds
本文研究了投影超kähler流形族中理查德自动群的行为。
In this paper we show that an affine space is determined by the abstract group structure of its group of regular automorphisms in the category of connected affine varieties. To prove this we study commutative subgroups of the group of automorphisms of affine varieties.
We construct k-parameter families of rational surface automorphisms for any k. These are automorphisms of surfaces X, which are constructed from iterated blowups over the projective plane. In certain cases: we are able to determine the exact automorphism group of X, as well as when two of the surfaces X are inequivalent.
We prove that the closure (for the Zariski topology) of the set of polynomial automorphisms of the complex affine plane whose polydegree is (cd-1,b,a) contains all triangular automorphisms of degree cd+a, where a,b >1 and c>0 are integers and d=ab-1. When b=2, this result gives a family of counterexamples to a conjecture of Furter.
In this article we determine the maximal possible order of the automorphism group of the form $ag + b$, where $a$ and $b$ are integers, of a complex three and four-dimensional family of compact Riemann surfaces of genus $g$, appearing for all genus. In addition, we construct and describe explicit complex three and four-dimensional families possessing these maximal number of automorphisms.
In this article we extend results of Zomorrodian to determine upper bounds for the order of a nilpotent group of automorphisms of a complex $d$-dimensional family of compact Riemann surfaces, where $d \geqslant 1.$ We provide conditions under which these bounds are sharp. In addition, for the one-dimensional case we construct and describe an explicit family attaining the bound for infinitely many genera. We obtain similar results for the case of $p$-groups of automorphisms.
Automorphism groups and cylindricity of weighted hypersurface Fano threefolds
本文针对拟光滑终端加权超曲面Fano三维簇剩余的20个形变族，研究其每个成员的柱性与完全自同构群，并探讨这些族的形式的柱性，补充了相关分类研究的空白。
We construct for any smooth projective curve of genus $q\ge 2$ with a fixed point free automorphism a nonisotrivial family of curves. Moreover we study the space of modular curves and that of parameters.
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。
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 $\mathbb{C}^2$.
Families of automorphisms of abelian varieties
本文研究复单位圆盘参数化的极化阿贝尔簇族上自同构动力学的代数性质，证明无分圆因子时映射可正则化当且仅当族不退化，并证明平移族总可正则化，同时刻画了轨道闭包。
We provide the existence, for every complex rational smooth affine curve $Γ$, of a linear action of $\mathrm{Aut}(Γ)$ on the affine 3-dimensional space $\mathbb{A}^3$, together with a $\mathrm{Aut}(Γ)$-equivariant closed embedding of $Γ$ into $\mathbb{A}^3$. It is not possible to decrease the dimension of the target, the reason for this obstruction is also precisely described.
In previous work we determined automorphism groups of cyclic algebraic curves defined over fields of any odd characteristic. In this paper we determine parametric equations of families of curves for each automorphism group for such 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.
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.
We determine all complex hyperelliptic curves with many automorphisms and decide which of their jacobians have complex multiplication.
We study the subvariety of fixed points of an automorphism of a Calogero-Moser space induced by a regular element of finite order of the normalizer of the associated complex reflection group $W$. We determine some of (and conjecturally all) the ${\mathbb{C}}^\times$-fixed points of its unique irreducible component of maximal dimension in terms of the character table of $W$. This is inspired by the mysterious relations between the geometry of Calogero-Moser spaces and unipotent representations of finite reductive groups, which will be the theme of a forthcoming paper.
We show that all the automorphisms of the symmetric product Sym^d(X), d>2g-2, of a smooth projective curve X of genus g>2 are induced by automorphisms of X.
The maximal number of automorphisms of a Mumford curve over a non-archimedean valued field of positive characteristic is known. In this note, the unique family of curves that attains this bound in genus not equal to 5,6,7 or 8 is explicitly determined, as is its automorphism group.
The canonical ideal and the deformation theory of curves with automorphisms
该研究以典范理想为工具开展曲线形变理论研究，将带自同构曲线的提升问题转化为线性表示提升问题，为相关形变问题研究提供了新的转化思路。
For each group $G$, $(|G| > 2)$ \, which acts as a full automorphism group on a genus 3 hyperelliptic curve, we determine the family of curves which have 2-Weierstrass points. Such families of curves are explicitly determined in terms of the absolute invariants of binary octavics. The 1-dimensional families that we discover have the property that they contain only genus 0 components.
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.
Let X be a smooth projective hyperelliptic curve over an algeraically closed field k of prime characteristic p. The aim of this note is to find necessary and sufficient conditions on the automorphism group of the curve X to be lifted to characteristic zero. The results will be generalised for a certain family of curves that we call cyclic curves.
In this paper, we determine the Weierstrass semigroup $H(P_{\infty})$ and the full automorphism group of a certain family of curves $\mathcal{X}_{n,r}$, which was recently introduced by Borges and Conceição.
In this paper we compute the automorphism group of the curves $\mathcal{X}_{a,b,n,s}$ and $\mathcal{Y}_{n,s}$ introduced in Tafazolian et al. in 2016 as new examples of maximal curves which cannot be covered by the Hermitian curve. They arise as subcovers of the first generalized GK curve (GGS curve). As a result, a new characterization of the GK curve, as a member of this family, is obtained.
In this paper we determine automorphism groups of cyclic algebraic curves defined over finite fields of any characteristic.
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.
A Mumford curve of genus g=5,6,7 or 8 over a non-archimedean field of characteristic p (such that if p=0, the residue field characteristic exceeds 5) has at most 12(g-1) automorphisms. In this paper, all curves that attain this bound and their automorphism groups (called of Lame type) are explicitly determined.
Let $\mathcal{X}$ be an irreducible algebraic curve defined over a finite field $\mathbb{F}_q$ of characteristic $p>2$. Assume that the $\mathbb{F}_q$-automorphism group of $\mathcal{X}$ admits as an automorphism group the direct product of two cyclic groups $C_m$ and $C_n$ of orders $m$ and $n$ prime to $p$ such that both quotient curves $\mathcal{X}/C_n$ and $\mathcal{X}/C_m$ are rational. In this paper, we provide a complete classification of such curves, as well as a characterization of their full automorphism groups.
In this paper, an explicit hierarchy of differential equations for the $τ$-functions defining the moduli space of curves with automorphisms as a subscheme of the Sato Grassmannian is obtained. The Schottky problem for Riemann surfaces with automorphisms consists of characterizing those p.p.a.v. that are Jacobian varieties of a curve with a non-trivial automorphism. A characterization in terms of hierarchies of p.d.e. for theta functions is also given.
We study integral representations of holomorphic differentials on the Oort-Sekiguci-Suwa component of deformations of curves with cyclic group actions.
The aim of the paper is twofold. First, some results of Shiota and Plaza-Martin on Prym varieties of curves with an involution are generalized to the general case of an arbitrary automorphism of prime order. Second, the equations defining the moduli space of curves with an automorphism of prime order as a subscheme of the Sato Grassmannian are given.
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.
We provide a method to compute the dimension of the tangent space to the global infinitesimal deformation functor of a curve together with a subgroup of the group of automorphisms. The computational techniques we developed are applied to several examples including Fermat curves, $p$-cyclic covers of the affine line and to Lehr-Matignon curves.
In this paper, we explicitly determine the automorphism group of every nonhyperelliptic superspecial curve of genus $4$ over $\mathbb{F}_{11}$. Our algorithm determining automorphism groups works for any nonhyperelliptic curves of genus $4$ over finite fields. With this computation, we show the compatibility between the enumeration of superspecial curves of genus $4$ over $\mathbb{F}_{11}$ obtained computationally by the first and second authors in 2017 and an enumeration by Galois cohomology theory.
Automorphism groups of hyperelliptic curves of $2$-rank zero
本文研究双秩为零的小亏格超椭圆曲线的约简自同构群，通过Magma计算得出其群结构，并提出两个猜想，类比Oort关于一般极化奇异阿贝尔簇自同构群的猜想。
Let K be an algebraically closed field of characteristic 2, and let X be a curve over K of genus g>1 and 2-rank r>0. For 2-subgroups S of the K-automorphism group Aut(X) of X, the Nakajima bound is |S| < 4g-3. For every g=2^h+1>8, we construct a curve X attaining the Nakajima bound and determine its relevant properties: X is a bielliptic curve with r=g, and its K-automorphism group has a dihedral K-automorphism group of order 4(g-1) which fixes no point in X. Moreover, we provide a classification of 2-groups S of K-automorphisms not fixing a point of X and such that |S|> 2g-1.
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.
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.
Let X be an irreducible smooth complex projective curve of genus g at least 4. Let M(r,Λ) be the moduli space of stable vector bundles over X or rank r and fixed determinant Λ, of degree d. We give a new proof of the fact that the automorphism group of M(r,Λ) is generated by automorphisms of the curve X, tensorization with suitable line bundles, and, if r divides 2d, also dualization of vector bundles.
Let $X$ be a smooth complex projective curve, and let $x\in X$ be a point. We compute the automorphism group of the moduli space of framed vector bundles on $X$ of rank $r \geq 2$ with a framing over $x$. It is shown that this automorphism group is generated by the following three: (1) pullbacks using automorphisms of the curve $X$ that fix the marked point $x$, (2) tensorization with certain line bundles over $X$ and (3) the action of $\operatorname{PGL}_r(\mathbb{C})$ through composition with the framing.
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 $q = g, g-1$.
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.
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
Large orders of automorphisms of smooth curves in $\mathbb P^1\times \mathbb P^1$
本文研究了$\mathbb P^1\times \mathbb P^1$中双度$(a,b)$的光滑曲线的自同构阶，并探讨了自同构与几何结构的关系。
Automorphism Groups of Smooth Cubic Fourfolds through Lattice Theory
本文利用格论和OSCAR枚举算法，完成了42族光滑四次三维簇的所有可能自同构群分类，基于周期映射已知结果。
The Heisenberg curve is defined to be the curve corresponding to an extension of the projective line by the Heisenberg group modulo $n$, ramified above three points. This curve is related to the Fermat curve and its group of automorphisms is studied. Also we give an explicit equation for the curve $C_3$.
We reduce the study of the Krull dimension d of the deformation ring of the functor of deformations of curves with automorphisms to the study of the tangent space of the deformation functor of a class of matrix representations of the p-part of the decomposition groups at wild ramified points, and we give a method in order to compute d.
Maximal Subcovers of the Skabelund Curve: Uniqueness via Genus and Automorphism Groups
通过亏格和自同构群的组合，Skabelund曲线的中间覆盖在同构意义下被唯一确定。
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.
Let X be an irreducible smooth projective curve, of genus at least two, defined over an algebraically closed field of characteristic different from two. If X admits a nontrivial automorphism σthat fixes pointwise all the order two points of Pic}^0(X), then we prove that X is hyperelliptic with σbeing the unique hyperelliptic involution. As a corollary, if a nontrivial automorphisms σ' of X fixes pointwise all the theta characteristics on X, then X is hyperelliptic with σ' being its hyperelliptic involution.
We determine the automorphism group of the modular curve $X_0^*(N)$, obtained as the quotient of the modular curve $X_0(N)$ by the group of its Atkin-Lehner involutions, for all square-free values of $N$.
We give an alternative approach to the computation of the dimension of the tangent space of the deformation space of curves with automorphisms. A homological version of the local-global principle similar to the one of J.Bertin, A. Mézard is proved, and a computation in the case of ordinary curves is obtained, by application of the results of S. Nakajima for the Galois module structure of the space of 2-holomorphic differentials on them.
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.
The automorphism group of a curve is studied from the viewpoint of the canonical embedding and Petri's theorem. A criterion for identifying the automorphism group as an algebraic subgroup the general linear group is given. Furthermore the action of the automorphism group is extended to an action of the minimal free resolution of the canonical ring of the curve $X$.
We study the automorphism groups of the reduction $X_0(N) \times \bar{\mathbb{F}}_p$ of a modular curve $X_0(N)$ over primes $ p\nmid N$.
Let $\mathcal{X}$ be a (projective, non-singular, irreducible) curve of even genus $g(\mathcal{X}) \geq 2$ defined over an algebraically closed field $K$ of characteristic $p$. If the $p$-rank $γ(\mathcal{X})$ equals $g(\mathcal{X})$, then $\mathcal{X}$ is ordinary. In this paper, we deal with large automorphism groups $G$ of ordinary curves. Under the hypotheses that $p = 2$, $g(\mathcal{X})$ is even and $G$ is solvable, we prove that $|G| < 35(g(\mathcal{X}) +1)^{3/2}$.
Remarks on Lattice Actions Corresponding to Automorphism Groups of Smooth Cubic Fourfolds
本文完成了复数域上光滑三次四维簇156个分类族的本原整上同调格上自同构群作用的计算，区分相似族、确定作用极大情形并完成列表，还确定了各族的一般代数格。
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。
A complex compact surface which carries an automorphism of positive topological entropy has been proved by Cantat to be either a torus, a K3 surface, an Enriques surface or a rational surface. Automorphisms of rational surfaces are quite mysterious and have been recently the object of intensive studies. In this paper, we construct several new examples of automorphisms of rational surfaces with positive topological entropy. We also explain how to define and to count parameters in families of birational maps of the complex projective plane and in families of rational surfaces.
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