Dajano Tossici, Francesca Vetro
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.
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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 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 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.
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.
This paper has been withdrawn by the author. The main result is wrong, as M.Matignon and C. Lehr provided a counterexample to it.
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.
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。
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。
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。
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。
In this article, we focus on a new perspective of automorphisms of complex 2-tori, reviewing previous works from a lattice-theoretic point of view. In particular, we give a classification of families of symplectic and non-symplectic automorphisms of prime order.
We provide explicit families of tame automorphisms of the complex affine three-space which degenerate to wild automorphisms. This shows that the tame subgroup of the group of polynomial automorphisms of $\C^3$ is not closed, when the latter is seen as an infinite dimensional algebraic group.tomorphism group of affine three-space
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.
The non-degeneracy invariant of Brandhorst and Shimada's families of Enriques surfaces
本文研究了Brandhorst和Shimada提出的$(τ,\overlineτ)$-generic Enriques曲面，给出了其非退化不变量的下界，证明多数情况下该不变量值为$10$，并首次发现具有无限自同构群且该不变量不为$10$的复Enriques曲面实例。
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.
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.
We construct one dimensional families of Abelian surfaces with quaternionic multiplication which also have an automorphism of order three or four. Using Barth's description of the moduli space of (2,4)-polarized Abelian surfaces, we find the Shimura curve parametrizing these Abelian surfaces in a specific case. We explicitly relate these surfaces to the Jacobians of genus two curves studied by Hashimoto and Murabayashi. We also describe a (Humbert) surface in Barth's moduli space which parametrizes Abelian surfaces with real multiplication by Z[\sqrt{2}].
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。
We construct algebraic families of exotic affine 3-spheres, that is, smooth affine threefolds diffeomorphic to a non-degenerate smooth complex affine quadric of dimension 3 but non algebraically isomorphic to it. We show in particular that for every smooth topologically contractible affine surface S with trivial automorphism group, there exists a canonical smooth family of pairwise non isomorphic exotic affine 3-spheres parametrized by the closed points of S.
We study the iterative behavior of the family of 3-step linear fractional recurrences and the family of birational maps they define. We determine all the possible periodicities within this family or, equivalently, the birational maps of finite order. This family also contains pseudo-automorphisms of infinite order. One such family consists of completely integrable maps, and another family consists of maps of positive entropy. Both of these families have invariant families of $K3$ surfaces.
We extend the notion of lattice polarization for K3 surfaces to families over a (not necessarily simply connected) base, in a way that gives control over the action of monodromy on the algebraic cycles, and discuss the uses of this new theory in the study of families of K3 surfaces admitting fibrewise symplectic automorphisms. We then give an application of these ideas to the study of Calabi-Yau threefolds admitting fibrations by lattice polarized K3 surfaces.
This paper is the first version of a project of classifying all superelliptic curves of genus $g \leq 48$ according to their automorphism group. We determine the parametric equations in each family, the corresponding signature of the group, the dimension of the family, and the inclussion among such families. At a later stage it will be determined the decomposition of the Jacobians and each locus in the moduli spaces of curves.
Artin-Schreier curves of prescribed p-rank without extra automorphisms
该研究针对奇素数p，构造了无额外自同构的指定p-秩的Artin-Schreier曲线，确定了自同构群仅为Cp的精确数值范围，通过Valentini-Madan定理及相关界完成例外族排除。
Weak del Pezzo surfaces with global vector fields
本文分类了任意特征代数闭域上具有全局向量场的光滑弱 del Pezzo 曲面，描述了曲线构型并计算了自同构概型单位分量，发现在不同特征下分别有53、61和75个曲面族，且证明了非既约自同构概型仅存在于特征2和3中。
Automorphisms of the affine 3-space of degree 3
本文给出仿射三维空间三次自同构的两个显式族，证明其驯服性，列出所有动力度（3个整数和9个二次整数），并分类满足超平面原像为仿射平面的低次态射。
Automorphism groups of curves with simple Jacobians
研究特征为0的代数闭域上具有简单雅可比矩阵的光滑射影连通曲线的自同构群，通过构造曲线族并证明其密度为1的成员集有简单雅可比矩阵，在主极化阿贝尔簇模空间中给出无限多个新的不太可能的交的例子。
Classification of Automorphism Groups of Smooth Cubic Threefolds and Fourfolds
本文分类了光滑三次三维簇与四维簇的自同构群，确定了156个四维簇和40个三维簇的连通族及其群作用和定义方程，方法结合表示论与格论。
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。
Finite subgroups of automorphism groups of Severi--Brauer varieties of prime degree
本文研究了素数维度的非平凡塞弗里-布拉泽变元的自同构群的有限子群分类，并构造了示例，证明了这些变元在特征不等于素数的域上不G-理性刚性。
Marked points of families of hyperbolic automorphisms of smooth complex projective varieties
本文研究正熵自同构族中标记点轨道的稳定性，分别构造几何标准高度与连续势的正闭 $(1,1)$-流，并在曲线情形证明二者一致，还推出 Kawaguchi–Silverman 猜想的特例。
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 …
In this paper we consider the Suzuki curve $y^q + y = x^{q_0}(x^q + x)$ over the field with $q = 2^{2m+1}$ elements. The automorphism group of this curve is known to be the Suzuki group $Sz(q)$ with $q^2(q-1)(q^2+1)$ elements. We construct AG codes over $\mathbb{F}_{q^4}$ from a $Sz(q)$-invariant divisor $D$, giving an explicit basis for the Riemann-Roch space $L(\ell D)$ for $0 < \ell \leq q^2-1$. These codes then have the full Suzuki group $Sz(q)$ as their automorphism group. These families of codes have very good parameters and are explicitly constructed with information rate close to one. …
Dynamics of automorphism groups of projective surfaces: classification, examples and outlook
本文概述紧复曲面自同构群子群动力学的前期研究、开放问题与新分类结果，并深入研究两类示例：平面五边形折叠衍生的示例与Jérémy Blanc提出的具有新动力学特征的族群。
Some families of non-isomorphic maximal function fields
本文研究作为 Hermitian 函数域的 Galois 子域产生的极大函数域族，通过计算其自同构群、Weierstrass 半群和同构类，构造出具有相同亏格和自同构群但不同构的极大函数域实例。
Let $H$ be a connected semisimple linear algebraic group defined over $\mathbb C$ and $X$ a compact connected Riemann surface of genus at least three. Let ${\mathcal M}'_X(H)$ be the moduli space parametrising all topologically trivial stable principal $H$-bundles over $X$ whose automorphism group coincides with the centre of $H$. It is a Zariski open dense subset of the moduli space of stable principal $H$-bundles. We prove that there is a universal principal $H$-bundle over $X\times {\mathcal M}'_X(H)$ if and only if $H$ is an adjoint group (that is, the centre of $H$ is trivial).
Determining the number of singular fibers in a family of varieties over a curve is a generalization of Shafarevich's Conjecture and has implications for the types of subvarieties that can appear in the corresponding moduli stack. We consider families of log canonically polarized varieties over $¶^1$, i.e. families $g:(Y,D)\to ¶^1$ where $D$ is an effective snc divisor and the sheaf $ω_{Y/¶^1}(D)$ is $g$-ample. After first defining what it means for fibers of such a family to be singular, we show that with the addition of certain mild hypotheses (the fibers have finite automorphism group, $\sO_…
A Locally Recoverable Code is a code such that the value of any single coordinate of a codeword can be recovered from the values of a small subset of other coordinates. When we have $δ$ non overlapping subsets of cardinality $r_i$ that can be used to recover the missing coordinate we say that a linear code $\mathcal{C}$ with length $n$, dimension $k$, minimum distance $d$ has $(r_1,\ldots, r_δ)$-locality and denote it by $[n, k, d; r_1, r_2,\dots, r_δ].$ In this paper we provide a new upper bound for the minimum distance of these codes. Working with a finite number of subgroups of cardinality …
Using Weil descent, we give bounds for the number of rational points on two families of curves over finite fields with a large abelian group of automorphisms: Artin-Schreier curves of the form $y^q-y=f(x)$ with $f\in\Fqr[x]$, on which the additive group $\Fq$ acts, and Kummer curves of the form $y^{\frac{q-1}{e}}=f(x)$, which have an action of the multiplicative group $\Fq^\star$. In both cases we can remove a $\sqrt{q}$ factor from the Weil bound when $q$ is sufficiently large.
We construct families of birational involutions on $\mathbb{P}^3$ or a smooth cubic threefold which do not fit into a non-trivial elementary relation of Sarkisov links. As a consequence, we construct new homomorphisms from their group of birational transformations, effectively re-proving their non-simplicity. We also prove that these groups admit a free product structure. Finally, we produce automorphisms of these groups that are not generated by inner and field automorphisms.
Local and 2-Local automorphisms of n-dimensional totally graded filiform Lie algebras
研究最大长度的有限维完全分次复丝状李代数族的局部和 2 - 局部自同构空间，利用代数过滤等方法，通过分析多个序列和族，明确了不同结构中局部及 2 - 局部自同构的形式特点，划分出非线性与严格线性结构的边界。
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。
Intersections of the automorphism and the Ekedahl-Oort strata in $M_2$
研究二型曲线模空间中自同构strata与Torelli映射下埃克尔-奥尔特strata的交集，分析自同构群的可能结构及参数化方法，计算Jacobian变种的固定埃克尔-奥尔特类型strata的维度和不可约分量数量。
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…
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…
Hybrid dynamics of hyperbolic automorphisms of K3 surfaces
本研究利用混合空间理论研究复K3曲面双曲动力学退化族，证明双曲自同构解析族的Cantat不变测度族在t趋于0时，在混合空间上弱收敛于非阿基米德K3曲面上的Filip测度。
Two families of surfaces arise from considering cyclic branched covers of $\mathbb{P}^{2}$ over smooth quartic curves. These consist of degree 2 del Pezzo surfaces with a $\mathbb{Z}/2\mathbb{Z}$ action and $K3$ surfaces with a $\mathbb{Z}/4\mathbb{Z}$ action. We compute the monodromy groups of both families. In the first case, we obtain the Weyl group $W\left(E_{7}\right)$, corresponding to the automorphisms of the $56$ lines contained in a degree $2$ del Pezzo surface. In the second case we obtain an arithmetic lattice: the unitary group $U\left(h_{L_{-}}\right)$ of a type $\left(1, 6\right)…
Any irreducible Dynkin diagram $Δ$ is obtained from an irreducible Dynkin diagram $Δ_h$ of type $\mathrm{ADE}$ by folding via graph automorphisms. For any simple complex Lie group $G$ with Dynkin diagram $Δ$ and compact Riemann surface $Σ$, we give a Lie-theoretic construction of families of quasi-projective Calabi-Yau threefolds together with an action of graph automorphisms over the Hitchin base associated to the pair $(Σ, G)$ . These give rise to Calabi-Yau orbifolds over the same base. Their intermediate Jacobian fibration, constructed in terms of equivariant cohomology, is isomorphic to t…
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