Fano type surfaces with large cyclic automorphisms

Fano hypersurfaces with no finite order birational automorphisms

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.

理性Fano锥面三维流形的自同构群与线性化

Automorphism groups and linearizability of rational Fano conic bundle threefolds

本文研究了Fano锥面三维流形的自同构群与线性化问题，证明了No. 2.18类Fano三维流形对于其自同构群的线性化性质。

Automorphism and Cohomology I: Fano variety of lines and Cubic

In this paper, we prove a general principle of lifting an automorphism from positive characteristic to zero characteristic. We based on the principle to prove the automorphism group of Fano variety of cubic threefold (fourfold) acts on its cohomology group faithfully. We also prove the same result for a singular cubic threefold (fourfold) with one ordinary double singularity. As an application, we classify the automorphism groups of cubic fourfolds with small middle Picard number over complex numbers.

加权超曲面Fano三维簇的自同构群与柱性

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。

Hilbert schemes of lines and conics and automorphism groups of Fano threefolds

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…

Delta-invariants for Fano varieties with large automorphism groups

For a variety $X$, a big $\mathbb{Q}$-divisor $L$ and a closed connected subgroup $G \subset \mathrm{Aut}(X, L)$ we define a $G$-invariant version of the $δ$-threshold. We prove that for a Fano variety $(X, -K_X)$ and a connected subgroup $G \subset \mathrm{Aut}(X)$ this invariant characterizes $G$-equivariant uniform $K$-stability. We also use this invariant to investigate $G$-equivariant $K$-stability of some Fano varieties with large groups of symmetries, including spherical Fano varieties. We also consider the case of $G$ being a finite group.

The Fano variety of lines of a cuspidal cyclic cubic fourfold

We prove that the Fano variety of lines of a cuspidal cyclic cubic fourfold is a symplectic variety with transversal A2-singularities and we study the properties of the nonsymplectic order three automorphism induced by the covering automorphism on the irreducible holomorphic symplectic manifold obtained by blowing up the singular locus.

Fano-Mukai fourfolds of genus $10$ and their automorphism groups

The automorphism groups of the Fano-Mukai fourfold of genus 10 were studied in our previous paper [arXiv:1706.04926]. In particular, we found in [arXiv:1706.04926] the neutral components of these groups. In the present paper we finish the description of the discrete parts. Up to isomorphism, there are two special Fano--Mukai fourfold of genus 10 with the automorphism groups $GL_2(k)\rtimes\mathbb{Z}/2\mathbb{Z}$ and $(\mathbb{G}_a\times\mathbb{G}m)\rtimes\mathbb{Z}/2\mathbb{Z}$, respectively. For any other Fano-Mukai fourfold $V$ of genus 10 one has $\mathrm{Aut}(V)=\mathbb{G}_m^2\rtimes \math…

Examples on Loewy filtrations and K-stability of Fano varieties with non-reductive automorphism groups

It is known that the automorphism group of a K-polystable Fano manifold is reductive. Codogni and Dervan construct a canonical filtration of the section ring, called Loewy filtration, and conjecture that the Loewy filtration destabilizes any Fano variety with non-reductive automorphism group. In this note, we give a counterexample to their conjecture.

On Fano manifolds of Picard number one with big automorphism groups

Let $X$ be an $n$-dimensional smooth Fano complex variety of Picard number one. Assume that the VMRT at a general point of $X$ is smooth irreducible and non-degenerate (which holds if $X$ is covered by lines with index $ >(n+2)/2$). It is proven that $\dim \mathfrak{aut}(X) > n(n+1)/2$ if and only if $X$ is isomorphic to $\mathbb{P}^n, \mathbb{Q}^n$ or ${\rm Gr}(2,5)$. Furthermore, the equality $\dim \mathfrak{aut}(X) = n(n+1)/2$ holds only when $X$ is isomorphic to the 6-dimensional Lagrangian Grassmannian ${\rm Lag}(6)$ or a general hyperplane section of ${\rm Gr}(2,5)$.

奇异循环三次四重簇上直线的Fano簇

The Fano variety of lines on singular cyclic cubic fourfolds

本文研究奇异循环三次四重簇上直线Fano簇的辛解消，特别关注沿具有$A_i$型孤立奇点的三次三重簇分支的情形，并发现覆盖自同构诱导的三阶非辛自同构。

Non-rationality of a three-dimensional Fano variety of index 2 and degree 1

We describe the set of Mori structures for a Fano 3-fold of index 2 and degree 1 (the double cone over the Veronese surface). In partiular, it is proved that such a Fano variety is not rational, the group of birational automorphisms coincides with the group of biregular automorphisms, and there are no structures of conic bundle.

Reductivity of the automorphism group of K-polystable Fano varieties

We prove that K-polystable log Fano pairs have reductive automorphism groups. In fact, we deduce this statement by establishing more general results concerning the S-completeness and $Θ$-reductivity of the moduli of K-semistable log Fano pairs. Assuming the conjecture that K-semistability is an open condition, we prove that the Artin stack parametrizing K-semistable Fano varieties admits a separated good moduli space.

Fano manifolds of index n-2 and the cone conjecture

The Morrison--Kawamata Cone Conjecture predicts that the action of the automorphism group on the effective nef cone and the action of the pseudo-automorphism group on the effective movable cone of a klt Calabi-Yau pair have rational, polyhedral fundamental domains. In a previous paper, we proved the conjecture for certain blowups of Fano manifolds of index n-1. In this paper, we consider the Morrison--Kawamata conjecture for blowups of Fano manifolds of index n-2.

Genus 2 curve configurations on Fano surfaces

We study the configurations of genus 2 curves on the Fano surfaces of cubic threefolds. We establish a link between some involutive automorphisms acting on such a surface S and genus 2 curves on S. We give a partial classification of the Fano surfaces according to the automorphism group generated by these involutions and determine the configurations of their genus 2 curves. We study the Fano surface of the Klein cubic threefold for which the 55 genus 2 curves generate a rank 25=h^{1,1} index 2 subgroup of the Néron-Severi group.

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.

Bounds for the order of automorphism groups of cyclic covering fibrations of a ruled surface

We study the order of automorphism groups of cyclic covering fibrations of a ruled surface. Arakawa and later Chen studied it for hyperelliptic fibrations and gave the upper bound. The purpose of present paper is to pursue the analog for cyclic covering fibrations.

Automorphism groups of cyclic curves defined over finite fields of any characteristics

In this paper we determine automorphism groups of cyclic algebraic curves defined over finite fields of any characteristic.

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.

Integral representations of cyclic groups acting on relative holomorphic differentials of deformations of curves with automorphisms

We study integral representations of holomorphic differentials on the Oort-Sekiguci-Suwa component of deformations of curves with cyclic group actions.

Cyclic groups of automorphisms of complex K3 surfaces

We determine all possible orders of automorphisms of complex K3 surfaces. A positive integer N is the order of an automorphism of a complex K3 surface if and only if $ϕ(N) \leq 20$ where $ϕ$ is the Euler function.

Remarks on Automorphism and Cohomology of Cyclic Coverings

For a smooth finite cyclic covering over a projective space of dimension greater than one, we show that the group of automorphisms acts faithfully on the cohomology except for a few cases. In characteristic zero, we study the equivariant deformation theory and automorphism groups for complex cyclic coverings. The proof uses the decomposition of the sheaf of differential forms due to Esnault and Viehweg. In positive characteristics, a lifting criterion of automorphisms reduces the faithfulness problem to characteristic zero. To apply this criterion, we prove the degeneration of the Hodge-de Rha…

Lifting Problem on Automorphism Groups of Cyclic Curves

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.

椭圆曲面循环覆盖纤维化的自同构群阶的上界

Bounds for the order of automorphism groups of cyclic covering fibrations of an elliptic surface

本文针对椭圆曲面的有限循环覆盖纤维化，利用纤维亏格、基曲线亏格、覆盖次数和相对典范除子平方，给出了自同构群有限子群阶的上界估计。

On the automorphisms of Mukai varieties

Mukai varieties are Fano varieties of Picard number one and coindex three. In genus seven to ten they are linear sections of some special homogeneous varieties. We describe the generic automorphism groups of these varieties. When they are expected to be trivial for dimensional reasons, we show they are indeed trivial, up to three interesting and unexpected exceptions in genera 7, 8, 9, and codimension 4, 3, 2 respectively. We conclude in particular that a generic prime Fano threefold of genus g has no automorphisms for 7 $\le$ g $\le$ 10. In the Appendix by Y. Prokhorov, the latter statement i…

Fano manifolds of index n-1 and the cone conjecture

The Morrison-Kawamata cone conjecture predicts that the actions of the automorphism group on the effective nef cone and the pseudo-automorphism group on the effective movable cone of a klt Calabi-Yau pair $(X, Δ)$ have finite, rational polyhedral fundamental domains. Let $Z$ be an $n$-dimensional Fano manifold of index $n-1$ such that $-K_Z = (n-1) H$ for an ample divisor $H$. Let $Γ$ be the base locus of a general $(n-1)$-dimensional linear system $V \subset |H|$. In this paper, we verify the Morrison-Kawamata cone conjecture for the blow-up of $Z$ along $Γ$.

Quotients of Fano Surfaces

Fano surfaces parametrize the lines of smooth cubic threefolds. In this paper, we study their quotients by some of their automorphism sub-groups. We obtain in that way some interesting surfaces of general type.

Slopes of Fibered Surfaces with a Finite Cyclic Automorphism

We study slopes of finite cyclic covering fibrations of a fibered surface. We give the best possible lower bound of the slope of these fibrations. We also give the slope equality of finite cyclic covering fibrations of a ruled surface and observe the local concentration of the global signature of these surfaces on a finite number of fiber germs. We also give an upper bound of the slope of finite cyclic covering fibrations of a ruled surface.

Explicit Birational Geometry of Fano threefold complete intersections

We complete the analysis on the birational rigidity of quasismooth Fano 3-fold deformation families appearing in the Graded Ring Database as a complete intersection. When such a deformation family $X$ has Fano index at least 2 and is minimally embedded in a weighted projective space in codimension 2, we determine which cyclic quotient singularity is a maximal centre. If a cyclic quotient singularity is a maximal centre, we construct a Sarkisov link to a non-isomorphic Mori fibre space or a birational involution. This allows, in particular, the construction of new examples of Fano 3-folds of co…

Double quadrics with large automorphism groups

We classify three-dimensional nodal Fano varieties that are double covers of smooth quadrics branched over intersections with quartics acted on by finite simple non-abelian groups, and study their rationality.

关于光滑Fano三维簇的自同构群

On the automorphism groups of smooth Fano threefolds

本文研究了亏格不超过10、非普通光滑Gushel-Mukai三维簇的Picard秩1光滑Fano三维簇，利用齐性簇完全交分类及前人拓扑结果，给出其有限自同构群阶的数值限制，证明其整除依赖亏格的明确数。

Smoothing Toric Fano Surfaces Using the Gross-Siebert Algorithm

A toric del Pezzo surface $X_P$ with cyclic quotient singularities determines and is determined by a Fano polygon $P$. We construct an affine manifold with singularities that partially smooths the boundary of $P$; this a tropical version of a Q-Gorenstein partial smoothing of $X_P$. We implement a mild generalization of the Gross-Siebert reconstruction algorithm - allowing singularities that are not locally rigid - and thereby construct (a formal version of) this partial smoothing directly from the affine manifold. This has implications for mirror symmetry: roughly speaking, it implements half…

Automorphisms of blowups of threefolds being Fano or having Picard number $1$

Let $X_0$ be a smooth projective threefold which is Fano or which has Picard number $1$. Let $π:X\rightarrow X_0$ be a finite composition of blowups along smooth centers. We show that for "almost all" of such $X$, if $f\in Aut(X)$ then its first and second dynamical degrees are the same. We also construct many examples of finite blowups $X\rightarrow X_0$, on which any automorphism is of zero entropy. The main idea is that because of the log-concavity of dynamical systems and the invariance of Chern classes under holomorphic automorphisms, there are some constraints on the nef cohomology class…

Determining equations of families of cyclic curves

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.

带$G_m$作用的亏格12素Fano三维簇

Prime Fano threefolds of genus 12 with a $G_m$-action

本文显式构造了带$G_m$作用的亏格12素Fano三维簇，并描述了其同构类与自同构群。

On the action of symplectic automorphisms on the $CH_0$-groups of some hyper-Kähler fourfolds

We prove that for any polarized symplectic automorphism of the Fano variety of lines of a cubic fourfold (equipped with the Plücker polarization), the induced action on the Chow group of 0-cycles is identity, as predicted by Bloch-Beilinson conjecture.

On cyclic covers of the projective line

We construct configuration spaces for cyclic covers of the projective line that admit extra automorphisms and we describe the locus of curves with given automorphism group. As an application we provide examples of arbitrary high genus that are defined over their field of moduli and are not hyperelliptic.

Complete toric varieties with reductive automorphism group

We give equivalent and sufficient criteria for the automorphism group of a complete toric variety, respectively a Gorenstein toric Fano variety, to be reductive. In particular we show that the automorphism group of a Gorenstein toric Fano variety is reductive, if the barycenter of the associated reflexive polytope is zero. Furthermore a sharp bound on the dimension of the reductive automorphism group of a complete toric variety is proven by studying the set of Demazure roots.

曲面上点的 Hilbert 概形的自同构

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。

K3 surfaces with a non-symplectic automorphism and product-quotient surfaces with cyclic groups

We classify all the K3 surfaces which are minimal models of the quotient of the product of two curves $C_1\times C_2$ by the diagonal action of either the group $\Z/p\Z$ or the group $\Z/2p\Z$. These K3 surfaces admit a non-symplectic automorphism of order $p$ induced by an automorphism of one of the curves $C_1$ or $C_2$. We prove that most of the K3 surfaces admitting a non-symplectic automorphism of order $p$ (and in fact a maximal irreducible component of the moduli space of K3 surfaces with a non-symplectic automorphism of order $p$) are obtained in this way.\\ In addition, we show that o…

对称与 Kähler--Einstein Fano 多边形

Symmetric and Kähler--Einstein Fano polygons

本文研究奇异对称与 Kähler--Einstein Fano 多胞体，证明所有对称 Fano 多胞体均为 Kähler--Einstein，并详细刻画了对称与 Kähler--Einstein Fano 多边形的自同构群。

特征正和混合特征下 genus 12 的 Fano 三维簇具有大自同构群

Fano threefolds of genus 12 with large automorphism group in positive and mixed characteristic

本文研究了特征正和混合特征下 genus 12 的 Fano 三维簇具有大自同构群的情况，证明了其存在性条件及 Shafarevich 猜想的成立情况。

On symplectic automorphisms of hyperkähler fourfolds of K3^[2] type

The present paper proves that finite symplectic groups of automorphisms of hyperkähler fourfolds deformation equivalent to the Hilbert scheme of two points on a $K3$ surface are contained in the simple group $Co_1$. Then we give an example of a symplectic automorphism of order 11 on the Fano scheme of lines of a cubic fourfold.

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。

Automorphism Groups of Cyclic p-gonal Pseudo-real Riemann Surfaces

In this article we prove that the full automorphism group of a cyclic $p$-gonal pseudo-real Riemann surface of genus $g$ is either a semidirect product $C_{n}\ltimes C_{p}$ or a cyclic group, where $p$ is a prime $>2$ and $g>(p-1)^{2}$. We obtain necessary and sufficient conditions for the existence of a cyclic $p$-gonal pseudo-real Riemann surface with full\ automorphism group isomorphic to a given finite group. Finally we describe some families of cyclic $p$-gonal pseudo-real Riemann surfaces where the order of the full automorphism group is maximal and show that such families determine some…

The Full Automorphism Group of a Cyclic $p$-gonal Surface

If $p$ is prime, a compact Riemann surface $X$ of genus $g\geq 2$ is called cyclic $p$-gonal if it admits a cyclic group of automorphisms $C_{p}$ of order $p$ such that the quotient space $X/C_{p}$ has genus 0. If in addition $C_{p}$ is not normal in the full automorphism $G$, then we call $G$ a non-normal cyclic $p$-gonal group. In the following we classify all non-normal $p$-gonal groups.

Elliptic curve configurations on Fano surfaces

The elliptic curves on a surface of general type constitute an obstruction for the cotangent sheaf to be ample. In this paper, we give the classification of the configurations of the elliptic curves on the Fano surface of a smooth cubic threefold. That means that we give the number of such curves, their intersections and a plane model. This classification is linked to the classification of the automorphism groups of theses surfaces.

On the Group of Automorphisms of Cyclic Covers of the Riemann Sphere

This note uses some recent calculations of Conder and Bujalance (on classifying finite index group extensions of Fuchsian groups with abelian quotient and torsion free kernel) in order to determine the full automorphism groups of some cylic coverings of the line (including curves of Fermat and Lefschetz type). The answer is complete for cyclic covers that branch over three points.

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。

具有指数为2的循环子群的大自同构群的曲线

Curves with a large automorphism group admitting a cyclic subgroup of index $2$

研究任意特征域上具有指数为2的循环子群的曲线，给出特征\(p\neq2\)的代数闭域上曲线自同构二面体群大小的上界，还得到大于\(4g(\mathcal{X}) + 4\)且具指数为2循环子群的（非二面体）群的分类结果。

K3 surfaces with Picard number two

It is known that an automorphism group of a K3 surface with Picard number two is either infinite cyclic group or infinite dihedral group if it is infinite. In this paper, we study the generators of an automorphism group. We use the eigenvector corresponding to the spectral radius of an automorphism of infinite order to determine the generators. We also determine whether an automorphism group is the infinite cyclic group or the infinite dihedral group with several examples.

The Fano surface of the Klein cubic threefold

We prove that the Klein cubic threefold $F$ is the only smooth cubic threefold which has an automorphism of order 11. We compute the period lattice of the intermediate Jacobian of $F$ and study its Fano surface $S$. We compute also the set of fibrations of $S$ onto a curve of positive genus and the intersection between the fibres of these fibrations. These fibres generate an index 2 sub-group of the Néron-Severi group and we obtain a set of generators of this group. The Néron-Severi group of $S$ has rank $25=h^{1,1}$ and discriminant $11^{10}$.

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…

On the moduli of Kahler-Einstein Fano manifolds

We prove that Kahler-Einstein Fano manifolds with finite automorphism groups form Hausdorff moduli algebraic space with only quotient singularities. We also discuss the limits as Q-Fano varieties which should be put on the boundary of its canonical compactification.