Long Wang
Let $X$ be a smooth projective hypersurface of dimension at least three. We show that every automorphism of the Hilbert square $X^{[2]}$ of $X$ is induced by some automorphism of $X$.
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On automorphism groups of smooth hypersurfaces
本文研究了复射影空间中光滑超曲面的自同构群问题，证明了具有最大自同构群的光滑超曲面同构于Fermat超曲面，并显式给出了少数例外情形的定义方程与自同构群。
Automorphisms of weighted projective hypersurfaces
本文研究拟光滑复加权射影超曲面的自同构群，证明自同构在多数情形下可延拓到外围空间，给出线性自同构群有限的刻画及一致上界，并描述了给定权和次数的通用超曲面的自同构。
Hypersurfaces with large automorphism groups
本研究确定了复射影空间中各维度和次数超曲面自同构群阶数的严格上界，证明达上界的超曲面同构唯一且给出其自同构群的显式生成元。
We use the specialization homomorphism for the birational automorphism group to study finite order birational automorphisms. For a family of varieties over a DVR, we prove that a birational automorphism of order coprime to the residue characteristic cannot specialize to the identity. As an application, we show that very general $n$-dimensional hypersurfaces of degree $d \geq 5 \lceil (n+3)/6 \rceil$ have no finite order birational automorphisms.
We study the connected component of the automorphism group of a cubic hypersurface over complex numbers. When the cubic hypersurface has nonzero Hessian, this group is usually small. But there are examples with unusually large automorphism groups: the secants of Severi varieties. Can we characterize them by the property of having unusually large automorphism groups? We study this question from the viewpoint of prolongations of the Lie algebras. Our result characterizes the secants of Severi varieties, among cubic hypersurfaces with nonzero Hessian and smooth singular locus, in terms of prolong…
Automorphism groups of rigid complete intersections
研究射影空间中次数严格递增的超曲面完全交的自同构群，利用组合刚性条件证明其自同构群是定义超曲面自同构群的交集，并应用此原理于两个超曲面完全交的自然族，确定了相关自同构群。
Finite automorphism groups of hypersurfaces via Bott vanishing
本文通过Bott消失定理给出上同调判据，证明驯特征下光滑Deligne-Mumford栈中超曲面的自同构群概形离散，并应用于环面轨道簇，获得拟光滑加权射影超曲面及射影空间乘积中超曲面的有限性结果，同时指出方法失效的例外窗口。
Trinomial hypersurfaces form a natural class of affine algebraic varieties closely connected with varieties admitting a torus action of complexity one. We investigate orbits of the automorphism group on these hypersurfaces. We prove that each nonrigid trinomial variety has finite number of orbits. We investigate singular orbits and this gives us description of all orbits for some classes of flexible trinomial hypersurfaces. Also we obtain a description of orbits for a class of hypersurfaces having a unique variable with power one in the equation.
Automorphisms of prime power order of weighted hypersurfaces
研究加权射影空间中准光滑超曲面的自同构，扩展经典结果至加权情形，推导素数幂阶自同构的有效判据和素数阶上限。
The Russell cubic is a smooth contractible affine complex threefold which is not isomorphic to affine three-space. In previous articles, we discussed the structure of the automorphism group of this variety. Here we review some consequences of this structure and generalize some results to other hypersurfaces which arise as deformations of Koras-Russell threefolds.
On a Torelli Principle for automorphisms of Klein hypersurfaces
本研究改进Oguiso和Yu的微分方法，给出光滑超曲面自同构的广义三角矩阵表示条件，计算了多类Klein超曲面的自同构群，引入极极化Hodge结构并验证部分情形下自同构群与Torelli原理预测一致。
Let $X$ be a smooth hypersurface $X$ of degree $d\geq4$ in a projective space $\mathbb P^{n+1}$. We consider a projection of $X$ from $p\in\mathbb P^{n+1}$ to a plane $H\cong\mathbb P^n$. This projection induces an extension of function fields $\mathbb C(X)/\mathbb C(\mathbb P^n)$. The point $p$ is called a Galois point if the extension is Galois. In this paper, we will give a necessary and sufficient conditions for $X$ to have Galois points by using linear automorphisms.
In this paper we classify three-dimensional singular cubic hypersurfaces with an action of a finite group $G$, which are not $G$-rational, are not $G$-birationally isomorphic to a quadric and have no birational structure of $G$-Mori fiber space with the base of positive dimension. Also we prove the $A_{5}$ -birational superrigidity of the Segre cubic.
In this paper we give an effective criterion as to when a positive integer q is the order of an automorphism of a smooth hypersurface of dimension n and degree d, for every d>2, n>1, (n,d)\neq (2,4), and \gcd(q,d)=\gcd(q,d-1)=1. This allows us to give a complete criterion in the case where q=p is a prime number. In particular, we show the following result: If X is a smooth hypersurface of dimension n and degree d admitting an automorphism of prime order p then p<(d-1)^{n+1}; and if p>(d-1)^n then X is isomorphic to the Klein hypersurface, n=2 or n+2 is prime, and p=Φ_{n+2}(1-d) where Φ_{n+2} i…
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。
Automorphisms of Smooth Hypersurfaces with Fixed Loci of Codimension at Most Two
本文研究了光滑超曲面在固定 locus 余维数至多为二时自同构的阶数限制及其商空间的有理性。
We prove that every biregular automorphism of the affine algebraic variety ${\mathbb P}^M\setminus S$, $M\geqslant 3$, where $S\subset {\mathbb P}^M$ is a hypersurface of degree $m\geqslant M+1$ with a unique singular point of multiplicity $(m-1)$, resolved by one blow up, is a restriction of some automorphism of the projective space ${\mathbb P}^M$, preserving the hypersurface $S$; in particular, for a general hypersurface $S$ the group $\mathop{\rm Aut} ({\mathbb P}^M\setminus S)$ is trivial.
We study a kind of modification of an affine domain which produces another affine domain. First appeared in passing in the basic paper of O. Zariski (1942), it was further considered by E.D. Davis (1967). The first named author applied its geometric counterpart to construct contractible smooth affine varieties non-isomorphic to Euclidean spaces. Here we provide certain conditions which guarantee preservation of the topology under a modification. As an application, we show that the group of biregular automorphisms of the affine hypersurface $X \subset C^{k+2}$ given by the equation $uv=p(x_1,..…
Given integers $d\ge 3$ and $N\ge 3$. Let $G$ be a finite abelian group acting faithfully and linearly on a smooth hypersurface of degree $d$ in the complex projective space $\mathbb{P}^{N-1}$. Suppose $G\subset PGL(N, \mathbb{C})$ can be lifted to a subgroup of $GL(N,\mathbb{C})$. Suppose moreover that there exists an element $g$ in $G$ such that $G/\langle g\rangle$ has order coprime to $d-1$. Then all possible $G$ are determined (Theorem 4.3). As an application, we derive (Theorem 4.8) all possible orders of linear automorphisms of smooth hypersurfaces for any given $(d,N)$. In particular, …
Koras-Russell threefolds are certain smooth contractible complex hypersurfaces in affine complex four-space which are not algebraically isomorphic to affine three-space. One of the important examples is the cubic Russell threefold, defined by the equation x^2y+z^2+t^3+x=0. In a previous article by A. Dubouloz, P.M. Poloni and the author, the automorphism group of the Russell cubic threefold was studied. It was shown, in particular, that all automorphisms of this hypersurface extend to automorphisms of the ambient space. This has several interesting consequences, including the fact that one can…
Small-Subgroup Criteria for Liftability of Automorphism Groups of Smooth Hypersurfaces
本文为光滑超曲面的自同构群建立可提升性与F-可提升性的小子群判据，通过检验低阶p-子群即可判定，并给出最优界。
In this paper we give an effective criterion as to when a prime number p is the order of an automorphism of a smooth cubic hypersurface of P^{n+1}, for a fixed n > 1. We also provide a computational method to classify all such hypersurfaces that admit an automorphism of prime order p. In particular, we show that p<2^{n+1} and that any such hypersurface admitting an automorphism of order p>2^n is isomorphic to the Klein n-fold. We apply our method to compute exhaustive lists of automorphism of prime order of smooth cubic threefolds and fourfolds. Finally, we provide an application to the moduli…
Automorphisms of rigid hypersurfaces with separable variables
本研究针对具有可分变量的刚性超曲面，刻画其自同构群的结构，证明该自同构群是其唯一极大环面的有限扩张，丰富了刚性代数簇自同构群的相关理论。
We determine the automorphism group of the Hilbert scheme of two points on a generic projective K3 surface of any polarization. We obtain in particular new examples of Hilbert schemes of points having non-natural non-symplectic automorphisms. The existence of these automorphisms depends on solutions of Pell's equation.
Twisted forms of classical hypersurfaces
该研究在实数域和有限域上计数费马、德尔萨特、克莱因超曲面的扭曲形式，推广微分方法到正特征域，计算其正特征自同构群，重现了费马超曲面实形式的相关定理。
We show that when $d \geq 3$, the Hilbert scheme $Hilb_{dT+1-\binom{d-1}{2}}(G(k,n))$ has 2 components, even though elements in both components have the same cohomology class. Moreover, we show that the Hilbert scheme associated to the Hilbert polynomial $\binom{T+n}{n}-\binom{T+n-d}{n}$ in Grassmannian has at most 2 connected components.
Uniform lower bound of arithmetic Hilbert--Samuel function of hypersurfaces
本文给出了射影超曲面算术Hilbert-Samuel函数具有最优主导项的显式一致下界，并将其应用于行列式方法中。
Nonnatural automorphisms of the Hilbert scheme of two points of some simple abelian variety
本研究通过考察阿贝尔簇n次乘积的自同构，构造出某类单阿贝尔簇的两点希尔伯特概型中保持大对角的非自然自同构，为相关代数几何研究提供新的自同构实例。
We show that when $d \geq 3$ and $m>2$, the Nef cone of the Hilbert scheme $Hilb_{P_{d,m}(T)}(G(k,n))$ is a cone spanned by 6 classes in general case, where $P_{d,m}(T)=\binom{T+m}{m}-\binom{T+m-d}{m}$.
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.
Automorphisms of Hilbert schemes of Cayley's K3 surfaces
本研究针对皮卡数为2的Cayley K3曲面的希尔伯特平方，证明其自同构群是三个2阶循环群的自由积，生成元为三个Beauville对合，推进了K3曲面希尔伯特概型自同构群的相关研究。
In this paper, we show that the average size of the automorphism group over $\mathbb{F}_q$ of a smooth degree $d$ hypersurface in $\mathbb{P}^{n}_{\mathbb{F}_q}$ is equal to $1$ as $d\rightarrow \infty$. We also discuss some consequence of this result for the moduli space of smooth degree $d$ hypersurfaces in $\mathbb{P}^n$.
From the degree zero part of logarithmic vector fields along an algebraic hypersurface singularity we indentify the maximal multihomogeneity of a defining equation in form of a maximal algebraic torus in the embedded automorphism group. We show that all such maximal tori are conjugate and in one-to-one correspondence to maxmimal tori in the degree zero jet of the embedded automorphism group. The result is motivated by Kyoji Saito's characterization of quasihomogeneity for isolated hypersurface singularities and extends its formal version and a result of Hauser and Mueller.
Sylow Criteria for Liftability of Automorphism Groups of Smooth Hypersurfaces
研究任意域上射影线性群有限子群及特定多项式相关有限子群的可提升性，通过结合群上同调的限制与上限制及归约到克莱因方法，建立了可提升性的西罗准则。
We present a sufficient condition for the punctural Hilbert scheme of length two of a K3 surface with finite automorphism group to have automorphism group of infinite order in geometric terms (Theorem 2.1). We then give concrete examples (Theorem 1.2). We also discuss about Mori dream space (MDS) structures under an extermal crepant resolution (Theorems 1.2, 5.2, 4.1) from the viewpoint of automorphisms. These affirmatively answer a question of Doctor Malte Wandel.
A construction of multiplicity class of hypersurfaces from Hesselink stratification of a Hilbert scheme
本文利用射影超曲面最大重数与虚拟 1-参数子群长度的正相关关系，定义超曲面重数类，并借助 Hilbert 概形的 Hesselink 分层给出其构造。
Let $X$ be a smooth hypersurface of dimension $n\geq 1$ and degree $d\geq 3$ in the projective space given as the zero set of a homogeneous form $F$. If $(n,d)\neq (1,3), (2,4)$ it is well known that every automorphism of $X$ extends to an automorphism of the projective space, i.e., $\operatorname{Aut}(X)\subseteq \operatorname{PGL}(n+2,\mathbb{C})$. We say that the automorphism group $\operatorname{Aut}(X)$ is $F$-liftable if there exists a subgroup of $\operatorname{GL}(n+2,\mathbb{C})$ projecting isomorphically onto $\operatorname{Aut}(X)$ and leaving $F$ invariant. Our main result in this …
Derived-natural automorphisms on Hilbert schemes of points on generic K3 surfaces
本文从导出范畴的角度研究一般K3曲面上点的Hilbert概形的双有理和双正则自同构，刻画了由底层K3曲面的导出范畴的自等价诱导的双有理和双正则对合。
In this paper we try to further explore the linear model of the moduli of rational maps. Our attempt yields following results. Let $X\subset \mathbf P^n$ be a generic hypersurface of degree $h$. Let $R_d(X, h)$ denote the open set of the Hilbert scheme parameterizing irreducible rational curves of degree $d$ on $X$. We obtain that (1) If $4\leq h\leq n-1$, $R_d(X, h)$ is an integral, local complete intersection of dimension \begin{equation} (n+1-h)d+n-4. \end{equation} (2) If furthermore $(h^2-n)d+h\leq 0$ and $h\geq 4$, in addition to part (1), $R_d(X, h)$ is also rationally connected.
Automorphisms of punctual Hilbert schemes and symmetric powers of varieties
本文分类了复光滑射影曲面，回答了Belmans等人的问题并扩展相关工作，简化为对称幂自同构研究，给出高维簇对称幂自同构自然的条件，刻画Kodaira维数≥1的曲面并探讨簇由点Hilbert概形确定的问题。
Automorphisms of Hilbert schemes of points on surfaces
本文研究曲面上点的 Hilbert 概形自同构的自然性，证明弱 Fano 或一般类型情形除曲线乘积且 n=2 外均由曲面自同构诱导，并刻画唯一例外及 P^n 的两点情形。
We discuss various results on Hilbert schemes of lines and conics and automorphism groups of smooth Fano threefolds with Picard rank 1. Besides a general review of facts well known to experts, the paper contains some new results, for instance, we give a description of the Hilbert scheme of conics on any smooth Fano threefold of index 1 and genus 10. We also show that the action of the automorphism group of a Fano threefold $X$ of index 2 (respectively, 1) on an irreducible component of its Hilbert scheme of lines (respectively, conics) is faithful if the anticanonical class of $X$ is very ampl…
The Koras-Russell threefold is the hypersurface X of the complex affine four-space defined by the equation x^2y+z^2+t^3+x=0. It is well-known that X is smooth contractible and rational but that it is not algebraically isomorphic to affine three-space. The main result of this article is to show that there exists another hypersurface Y of the affine four-space, which is isomorphic to X as an abstract variety, but such that there exists no algebraic automorphism of the ambient space which restricts to an isomorphism between X and Y. In other words, the two hypersurfaces are inequivalent. The proo…
Automorphisms of hypersurfaces in polarized varieties
本文针对极化簇中的一般超曲面，利用截面环的分次 Betti 数给出统一判据，保证其自同构可延拓至全簇，推广了经典结果至更广簇类。
Let $W\subset \mathbb {P}^n$, $n\ge 3$, be a degree $k$ hypersurface. Consider a "general" reducible, but connected, curve $Y\subset \mathbb {P}^n$, for instance a sufficiently general connected and nodal union of lines with $p_a(Y)=0$, i.e. a tree of lines. We study the Hilbert function of the set $Y\cap W$ with cardinality $k°(Y)$ and prove when it is the expected one. We give complete classification of the exceptions for $k=2$ and for $n=k=3$. We apply these results and tools to the case in which $Y$ is a smooth curve with $\mathcal {O}_Y(1)$ non-special.
We study automorphisms of the Hilbert scheme of $n$ points on a generic projective K3 surface $S$, for any $n \geq 2$. We show that the automorphism group of $S^{[n]}$ is either trivial or generated by a non-symplectic involution and we determine numerical and divisorial conditions which allow us to distinguish between the two cases. As an application of these results we prove that, for any $n \geq 2$, there exist infinite values for the degree of $S$ such that $S^{[n]}$ admits a non-natural involution. This provides a generalization of results by Boissière--Cattaneo--Nieper-Wisskirchen--Sarti…
Automorphisms of Hilbert Schemes of Points on Abelian Surfaces
本文构造 Abel 曲面任意点数 Hilbert 概形上保持例外除子的非自然自同构，并证明 Picard 秩 1 的主极化或极化自交为平方数时两点情形的所有自同构均自然。
On Cox Rings of Calabi-Yau hypersurfaces
本文研究了平滑反canonical Calabi-Yau超曲面在平滑的toric Fano变种中的Cox环，通过研究环境Fano多面体的原始对的组合学以及嵌入变种的Cox环描述，确定了几种配置使得超曲面是Mori梦想空间，并获得了其Cox环的显式表示。此外，本文还展示了某些组合配置迫使birational自同构群为无限，从而在三维和四维中揭示了Cox环有限生成与无限birational自同构群之间的二元对立。最后，对于一类非Mori梦想的例子，证明了Morrison-Kawamata锥猜想对于可动锥。
In this paper, we develop a new method to classify abelian automorphism groups of hypersurfaces. We use this method to classify (Theorem 4.2) abelian groups that admit a liftable action on a smooth cubic fourfold. A parallel result (Theorem 5.1) is obtained for quartic surfaces.
In this paper, we will give an upper bound and a lower bound of the arithmetic Hilbert-Samuel function of projective hypersurfaces, which are uniform and explicit. These two bounds have the optimal dominant terms. As an application, we use the lower bound to get an estimate of the density of rational points with small heights in a hypersurface.
Ample cones of Hilbert schemes of points on hypersurfaces in $\mathbb{P}^3$
本文研究 $\mathbb{P}^3$ 中一般高次曲面上点的 Hilbert 概形，计算小 $n$ 时的丰富锥及含线情形的 nef 锥切片，并揭示五次情形中高重超平面截面的作用。
We classify the group of birational automorphisms of Hilbert schemes of points on algebraic K3 surfaces of Picard rank one. We study whether these automorphisms are symplectic or non-symplectic and if there exists a hyperkähler birational model on which they become biregular. We also present new geometrical constructions of these automorphisms.
The present paper provides several results on automorphisms of hyperkähler (or irreducible holomorphic symplectic) manifolds. In particular it focuses on the symplectic case and contains a classification of prime order symplectic automorphisms on deformations of the Hilbert scheme of two points on a K3 surface and, with additional hypothesis, also on Hilbert schemes of n points on a K3 surface. We also give numerical conditions to establish whether a symplectic automorfism of order 2,3 and 5 on a deformation of the Hilbert scheme of two points on a K3 surface is standard. Moreover we also give…
We use Maxima to compute the moment matrices of hypersurfaces. After that, we compute the Hilbert-Mumford numerical criterion for plane cubics and plane quartics, and give the stability of these curves.
It is shown that an irreducible cubic hypersurface with nonzero Hessian and smooth singular locus is the secant variety of a Severi variety if and only if its Lie algebra of infinitesimal linear automorphisms admits a nonzero prolongation.
Affine complements in the projective space with the trivial group of automorphisms
该研究构造了一类新的仿射补集，证明当 $M$ 为不小于5的奇数且 $S$ 为有理超曲面时，其自同构群为平凡群。
We study double line structures in projective spaces and quadric hypersurfaces, and investigate the geometry of irreducible components of Hilbert scheme of curves and moduli of stable sheaves of pure dimension 1 on a smooth quadric threefold.
The Danielewski hypersurfaces are the hypersurfaces $X_{Q,n}$ in $\mathbb{C}^3$ defined by an equation of the form $x^ny=Q(x,z)$ where $n\geq1$ and $Q(x,z)$ is a polynomial such that $Q(0,z)$ is of degree at least two. They were studied by many authors during the last twenty years. In the present article, we give their classification as algebraic varieties. We also give their classification up to automorphism of the ambient space. As a corollary, we obtain that every Danielewski hypersurface $X_{Q,n}$ with $n\geq2$ admits at least two non-equivalent embeddings into $\mathbb{C}^3$.
For a Zariski general (regular) hypersurface $V$ of degree $M$ in the $(M+1)$-dimensional projective space, where $M$ is at least 16, with at most quadratic singularities of rank at least 13, we give a complete description of the structures of rationally connected (or Fano-Mori) fibre space: every such structure over a positive-dimensional base is a pencil of hyperplane sections. This implies, in particular, that $V$ is non-rational and its groups of birational and biregular automorphisms coincide. The set of non-regular hypersurfaces has codimension at least $\frac12(M-11)(M-10)-10$ in the na…
The variety of all smooth hypersurfaces of given degree and dimension has the Fermat hypersurface as a natural base point. In order to study the period map for such varieties, we first determine the integral polarized Hodge structure of the primitive cohomology of a Fermat hypersurface (as a module over the automorphism group of the hypersurface). We then focus on the degree 3 case and show that the period map for cubic fourfolds as analyzed by R. Laza and the author gives complete information about the period map for cubic hypersurfaces of lower dimension dimension. In particular, we thus rec…
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