Igor Dolgachev, Ichiro Shimada
We describe a set of generators and defining relations for the group of birational automorphisms of a general 15-nodal quartic surface in the complex projective 3-dimensional space.
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The $\mathbb{Q}$-factoriality of a nodal quartic 3-fold implies its non-rationality. We prove that a nodal quartic 3-fold with at most 8 nodes is $\mathbb{Q}$-factorial, and we show that a nodal quartic 3-fold with 9 nodes is not $\mathbb{Q}$-factorial if and only if it contains a plane. However, there are non-rational non-$\mathbb{Q}$-factorial nodal quartic 3-folds in $\mathbb{P}^4$. In particular, we prove the non-rationality of a general non-$\mathbb{Q}$-factorial nodal quartic 3-fold that contains either a plane or a smooth del Pezzo surface of degree 4.
We explain a classical construction of a del Pezzo surface of degree d = 4 or 5 as a smooth order two congruence of lines in 3-space whose focal surface is a quartic surface $X_{20-d}$ with 20-d ordinary double points. We also show that $X_{15}$ can be realized as a hyperplane section of the Castelnuovo-Richmond-Igusa quartic hypersurface. This leads to the proof of rationality of the moduli space of 15-nodal quartic surfaces. We discuss some other birational models of $X_{15}$: quartic symmetroids, 5-nodal quartic surfaces, 10-nodal sextic surfaces in $P^4$ and nonsingular surfaces of degree …
We study the rationality problem for nodal quartic double solids. In particular, we prove that nodal quartic double solids with at most six singular points are irrational, and nodal quartic double solids with at least eleven singular points are rational.
We prove that the quartic threefolds defined by $$ \sum_{i=0}^{5}x_i=\sum_{i=0}^{5}x_i^4-t\left(\sum_{i=0}^{5}x_i^2\right)^2=0 $$ in $\mathbb{P}^5$ are rational for $t=\frac{1}{6}$ and $t=\frac{7}{10}$.
For every p >= 5, we determine all Z_p-invariant nonsingular quartic surfaces in the three dimensional projective space over an algebraically closed field of characteristic zero. In some cases, we also determine their full projective automorphism groups.
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.
We consider the class of singular double coverings $X \to \PP^3$ ramified in the degeneration locus $D$ of a family of 2-dimensional quadrics. These are precisely the quartic double solids constructed by Artin and Mumford as examples of unirational but nonrational conic bundles. With such quartic surface $D$ one can associate an Enriques surface $S$ which is the factor of the blowup of $D$ by a natural involution acting without fixed points (such Enriques surfaces are known as nodal Enriques surfaces or Reye congruences). We show that the nontrivial part of the derived category of coherent she…
Quartic surfaces with an outer Galois point and K3 surfaces with an automorphism of order 4
本文建立了具有外伽罗瓦点的光滑四次曲面与具有特定4阶自同构的K3曲面之间的一一对应，并进一步用K3曲面刻画了具有两个及以上外伽罗瓦点的四次曲面。
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.
It is well-known that a nonsingular minimal cubic surface is birationally rigid; the group of its birational selfmaps is generated by biregular selfmaps and birational involutions such that all relations between the latter are implied by standard relations between reflections on an elliptic curve. It is also known that a factorial nodal quartic threefold is birationally rigid and its group of birational selfmaps is generated by biregular ones and certain birational involutions. We prove that all relations between these involutions are implied by standard relations on elliptic curves, complete …
We study quartic surfaces that admit a group of projective automorphisms isomorphic to icosahedron group.
Automorphisms of quartic surfaces and Cremona transformations
本文研究光滑四次曲面自同构是否由Cremona变换诱导，在ρ(S)=2时给出初步步骤，构造反例否定Oguiso问题。
This is a continuation of [Og12], concerning automorphisms of smooth quartic K3 surfaces and birational automorphisms of ambient projective three spaces.
Automorphisms of a $(24_4, 32_3)$-configuration on the Schur quartic
研究舒尔四次曲面上(24_4, 32_3)构型的自同构问题，通过特定模型得出24条线一半的稳定子群同构情况及全射影自同构群阶数，还给出内在着色并缩短关联检查。
For any field k of characteristic at most 5 we exhibit an explicit smooth quartic surface in projective threespace over k with trivial automorphism group over the algebraic closure of k. We also show how this can be extended to higher characteristics. Over the rationals we construct an example on which the set of rational points is Zariski dense.
Costantini and Kappes gave an algebraic equation of the universal family over the Kenyon-Smillie (2,3,4)-Teichmüller curve. This equation gives rise to a family of projective plane quartic curves with three singular members. These singular curves are: (1) an integral nodal quartic, (2) the union of a line and a nodal cubic, and (3) a quadruple line. We determine the limit Weierstrass points on these singular curves.
Sixteen generators of the automorphism group of the Fermat quartic surface
本文解决了复费马四次K3曲面自同构群生成元的长期公开问题，明确给出该群的16个有限阶几何生成元。
We find generators of the group of birational automorphisms of the Hessian surface of a general cubic surface. Its nonsingular minimal model is a K3 surface with the Picard lattice of rank 16. The latter embeds naturally in the even unimodular lattice $II^{1,25}$ of rank 26 and signature $(1,25)$ as the orthogonal complement of a root sublattice of rank 10. Our generators are related to reflections with respect to some Leech roots. A similar observation was made first in the case of quartic Kummer surfaces in the work of S. Kond$\bar {\roman o}$. We shall explain how our generators are related…
Let $S$ be the (minimal) Enriques surface obtained from the symmetric quartic surface $(\sum_{i<j}x_ix_j)^2=kx_1x_2x_3x_4$ in $\mathbb{P}^3$ with $k\neq 0,4,36$, by taking quotient of the Cremona action $(x_i) \mapsto (1/x_i)$. The automorphism group of $S$ is a semi-direct product of a free product $\mathcal{F}$ of four involutions and the symmetric group $\mathfrak{S}_4$. Up to action of $\mathcal{F}$, there are exactly $29$ elliptic pencils on $S$.
Automorphisms of quartic del Pezzo surfaces in characteristic zero
本文对特征零域上四次 del Pezzo 曲面的自同构群进行完整分类，并区分有理、稳定有理与单有理情形，还为每个可实现群给出显式曲面方程。
In this paper we prove that generic small partial smoothings of Kahler-Einstein (KE) Del Pezzo orbifolds with only nodal singularities, and with no non-zero holomorphic vector fields, admit orbifold KE metrics which are close in the Gromov-Hausdorff sense to the original KE metric.
On symmetric plane quartic curves
本文研究具有大自同构群的平面四次曲线的几何，重点分析了在$[2,2,2]$型初等阿贝尔群作用下不变的光滑平面四次曲线以及自同构群阶为$48$的光滑平面四次曲线的几何性质。
Automorphism groups of real rational quartic del Pezzo surfaces
本文通过研究实理性四次德尔皮佐表面作为某些光滑实二次曲面的吹胀，描述了所有可能的自同构群，并利用其对复结构的Gal(C/R)作用，给出了实自同构群的几何描述。
We prove that a nodal quartic threefold $X$ containing no planes is $Q$-factorial provided that it has not more than 12 singular points, with the exception of a quartic with exactly 12 singularities containing a quadric surface. We give some geometrical constructions related to the latter quartic.
Klein's simple group $H$ of order $168$ is the automorphism group of the plane quartic curve $C$, called Klein quartic. By Torelli Theorem, the full automorphism group $G$ of the Jacobian $J=J(C)$ is the group of order $336$, obtained by adding minus identity to $H$. The quotient variety $J/G$ can be alternatively represented as the quotient $\mathbb C^3/\tilde G$ of the complex $3$-space by the complex crystallographic group $\tilde G$, the extension of $G$ by the period lattice of the Klein quartic. Moreover, it turns out that $\tilde G$ is generated by affine complex reflections. According …
Gromov elliptic resolutions of quartic double solid
本文构造节点四次二重立体实例，证明其存在一致有理且在 Gromov 意义下为椭圆的小型代数消解。
Let Y be a complex Enriques surface whose universal cover X is birational to a general quartic Hessian surface. Using the result on the automorphism group of X due to Dolgachev and Keum, we obtain a finite presentation of the automorphism group of Y. A fundamental domain of the action of the automorphism group on the nef cone of Y is described explicitly. The list of elliptic fibrations on Y and the list of combinations of rational double points that can appear on a surface birational to Y are presented. As an application, a set of generators of the automorphism group of the generic Enriques s…
We describe, for various degenerations $S\to Δ$ of quartic $K3$ surfaces over the complex unit disk (e.g., to the union of four general planes, and to a general Kummer surface), the limits as $t\in Δ^*$ tends to 0 of the Severi varieties $V_δ(S_t)$, parametrizing irreducible $δ$-nodal plane sections of $S_t$. We give applications of this to (i) the counting of plane nodal curves through base points in special position, (ii) the irreducibility of Severi varieties of a general quartic surface, and (iii) the monodromy of the universal family of rational curves on quartic $K3$ surfaces.
Conical resolutions and cohomology of the moduli spaces of nodal hypersurfaces
本文利用锥形消解及其推广形式，并结合紧 GIT 商的显式描述，计算节点三次、四次平面曲线及三次曲面方程空间与模空间的有理上同调。
Three-dimensional del Pezzo varieties of degree 2 are double covers of projective space $\mathbb{P}^{3}$ branced in a quadric. In this paper we prove that if a del Pezzo variety of degree 2 has exactly 15 nodes then the corresponding quadric is a hyperplane section of the Igusa quartic or, equivalently, all such del Pezzo varieties are members of one particular linear system on the Coble fourfold. Their automorphism groups are induced from the automorphism group of Coble fourfold. Also we classify all $G$-birationally rigid varieties of such type.
Bitangents to symmetric quartics
本文研究亏格三非奇异对称平面四次曲线的自同构群在28条双切线上诱导的对称性，证明了双切线的G-轨道与曲线选择无关，并利用等变同伦理论技术计算了所有十二类曲线的轨道，揭示了经典模视角不明显的模式。
Recently Oguiso showed the existence of K3 surfaces that admit a fixed point free automorphism of positive entropy. The K3 surfaces used by Oguiso have a particular rank two Picard lattice. We show, using results of Beauville, that these surfaces are therefore determinantal quartic surfaces. Long ago, Cayley constructed an automorphism of such determinantal surfaces. We show that Cayley's automorphism coincides with Oguiso's free automorphism. We also exhibit an explicit example of a determinantal quartic whose Picard lattice has exactly rank two and for which we thus have an explicit descript…
Even nodal surfaces of K3 type
本文研究具有二次曲线丛结构的K3型Fano四重态，构造其与hyperKähler簇的几何联系，描述了作为Kummer四次曲面推广的节点曲面族。
We give conceptual proofs of some results on the automorphism group of an Enriques surface X, for which only computational proofs have been available. Namely, there is an obvious upper bound on the image of Aut(X) in the isometry group of X's numerical lattice, and we establish a lower bound for the image that is quite close to this upper bound. These results apply over any algebraically closed field, provided that X lacks nodal curves, or that all its nodal curves are (numerically) congruent to each other mod 2. In this generality these results were originally proven by Looijenga and Cossec-D…
We prove that there is a smooth quartic K3 surface automorphism that is not derived from the Cremona transformation of the ambient three-dimensional projective space. This gives a negative answer to a question of Professor Marat Giz.atullin
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.
We calculate the automorphism group of certain Enriques surfaces. The Enriques surfaces that we investigate include very general $n$-nodal Enriques surfaces and very general cuspidal Enriques surfaces. We also describe the action of the automorphism group on the set of smooth rational curves and on the set of elliptic fibrations.
We continue our investigation of the connected components of the moduli space of surfaces of general type containing the Burniat surfaces, correcting a mistake in part II. We define the family of extended Burniat surfaces with K_S^2 = 4, resp. 3, and prove that they are a deformation of the family of nodal Burniat surfaces with K_S^2 = 4, resp. 3. We show that the extended Burniat surfaces together with the nodal Burniat surfaces with K_S^2=4 form a connected component of the moduli space. We prove that the extended Burniat surfaces together with the nodal Burniat surfaces with K_S^2=3 form an…
We provide explicit equations of some smooth complex quartic surfaces with many lines, including all 10 quartics with more than 52 lines. We study the relation between linear automorphisms and some configurations of lines such as twin lines and special lines. We answer a question by Oguiso on a determinantal presentation of the Fermat quartic surface.
On Gizatullin's Problem for quartic surfaces of Picard rank $2$
本文研究了 Picard 秩为 $2$ 的一般光滑四次曲面的 Gizatullin 问题，确定了哪些自同构可以限制为 $\mathbb{P}^3$ 的 Cremona 变换。
We give an algebro-geometric construction of some of the non-arithmetic ball quotients constructed by the author, Parker and Paupert. The new construction reveals a relationship between the corresponding orbifold fundamental groups and the automorphism group of the Klein quartic, and also with groups constructed by Barthel-Hirzebruch-Höfer and Couwenberg-Heckman-Looijenga.
Action of the automorphism group on the Jacobian of Klein's quartic curve II: Invariant theta functions
本文证明了由反射生成的不可约复晶体反射群在三维复仿射空间上的商是权重为1,2,4,7的加权射影空间，其中相伴共线群是168阶克莱因单群，主要基于不变theta函数代数的计算。
Somos-4 and a quartic Surface in $\mathbb{RP}^{3}$
本文研究Somos-4序列在附加性质下导出的四次多项式，该多项式定义了$\mathbb{RP}^{3}$中的有理射影曲面，并确定了其双有理自同构所在的$Cr_3(\mathbb{R})$子群的生成元。
A fake quadric is a smooth minimal surface of general type with the same invariants as the quadric in P^3, i.e. K^2=2c_2=8 and q=p_g=0. We study here quaternionic fake quadrics i.e. fake quadrics constructed arithmetically by using some quaternion algebras over real number fields. We provide examples of quaternionic fake quadrics X with a non-trivial automorphism group and compute the invariants of the minimal desingularisation of the quotient of X by this group. In that way we obtain minimal surfaces of general type Z with q=p_g=0 and K^2=4,2 which contain the maximal number of disjoint nodal…
Automorphisms of the moduli space of smooth cubic surfaces and its fundamental group
研究光滑三重曲面模空间的基本群，证明其除子子群是特征子群，并推导出该空间无非平凡复解析轨道自同构。
We classify all possible automorphism groups of smooth cubic surfaces over an algebraically closed field of arbitrary characteristic. As an intermediate step we also classify automorphism groups of quartic del Pezzo surfaces. We show that the moduli space of smooth cubic surfaces is rational in every characteristic, determine the dimensions of the strata admitting each possible isomorphism class of automorphism group, and find explicit normal forms in each case. Finally, we completely characterize when a smooth cubic surface in positive characteristic, together with a group action, can be lift…
Birational rigidity of quartic three-folds with a double point of rank 3
本文证明复射影空间中仅含秩3二重点的一般四次三重流形具有双有理刚性，其双有理自映射群为25个2阶循环群的自由积，并由此推出刚性阶乘四次流形补集的余维数至少为3。
A generalisation of the pencil of Kuribayashi-Komiya quartics
本文将 Kuribayashi-Komiya 四次曲线束推广到素数 p≥5 的情形，确定广义曲线束光滑成员的完全自同构群及其作用，证明无超椭圆成员，并分类具有相同自同构群的亏格 (p-1)(2p-1) 黎曼曲面。
Explicit birational geometry of determinantal quartic 3-folds
研究$\mathbb{P}^4$中一般线性行列式四次三维流形的双有理几何，发现其族包含有理$\mathbb{Q}$-factorial成员并可特化为有理非$\mathbb{Q}$-factorial双二次曲面，且几何由$\mathbb{P}^3$中10次11亏格曲线控制。
Pseudo-automorphisms of rational threefolds and Kummer surfaces
本文将45个经典Kummer曲面自同构提升为$\mathbb{P}^3$沿6点15线爆破所得三维簇的伪自同构，并描述了由这些已知伪自同构生成的群的基本域。
We give an alternative argument for the classification of real bundle pairs over smooth symmetric surfaces and extend this classification to nodal symmetric surfaces. We also classify the homotopy classes of automorphisms of real bundle pairs over symmetric surfaces. The two statements together describe the isomorphisms between real bundle pairs over symmetric surfaces up to deformation.
On Coble surfaces and their automorphisms
研究Coble表面的自同构性质，发现两类满足特定条件的表面，第一类为节点型且自同构恒等，第二类在模空间中为小维度。
We find explicit projective models of a compact Shimura curve and of a (non-compact) surface which are the moduli spaces of principally polarised abelian fourfolds with an automorphism of order five. The surface has a 24-nodal canonical model in IP^4 which is the complete intersection of two S_5-invariant cubics. It is dominated by a Hilbert modular surface and we give a modular interpretation for this. We also determine the L-series of these varieties as well as those of several modular covers of the Shimura curve.
Henn and Komiya-Kuribayashi listed, independently, the groups $G$ for which $\widetilde{M_3^{Pl}(G)}$ is non-empty. In this paper, we determine the loci $\widetilde{M_6^{Pl}(G)}$, corresponding to non-singular degree $5$ projective plane curves, which are non-empty. Also, we present the analogy of Henn's results for quartic curves concerning non-singular plane model equations associated to these loci.
For each finite primitive subgroup $G$ of $\operatorname{PGL}_4(\mathbb{C})$, we find all the smooth $G$-invariant quartic surfaces. We also find all the faithful representations in $\operatorname{PGL}_4(\mathbb{C})$ of the smooth quartic $G$-invariant surfaces by the groups: $\mathfrak{A}_5,\mathfrak{S}_5, \operatorname{PSL_2(\mathbb{F}_7)},\mathfrak{A}_6,\mathbb{Z}_2^4\rtimes\mathbb{Z}_5$ and $\mathbb{Z}_2^4\rtimes D_{10}$. The primitive representation of these groups are precisely the subgroups of $\operatorname{PGL}_4(\mathbb{C})$ for which $\mathbb{P}^3$ is not $G$-super rigid. As a bypro…
Rational surfaces with a non-arithmetic automorphism group
本文构造了具有非算术自同构群的有理曲面例子，这些例子为 Looijenga 对，从而将 Totaro 关于 K3 曲面的相关结果推广到了有理曲面。
On two-elementary K3 surfaces with finite automorphism group
本研究针对皮卡秩11-13、带有限自同构群和二阶截面雅可比椭圆纤维化的复代数K3曲面，证明其存在射影四次超曲面双有理模型，构造椭圆纤维化标架，确定有理曲线对偶图等几何结构。
Transitive automorphism groups of maximal curves
本文分类了亏格至少二的极大曲线上完全几何自同构群在有理点集上传递的所有情形，证明除 Hermite 曲线外仅 $q=5$ 时 Klein 四次曲线例外，并给出完整分类。
Desmic quartic surfaces in arbitrary characteristic
研究Desmic四次曲面作为椭圆曲线自积的Kummer曲面的双有理模型，在任意特征下推广其经典几何，并证明关联的三次线丛是有34个节点的有理Fano三维流形。
We investigate the action of the automorphism group of a closed Riemann surface on its set of theta characteristics (or spin structures). We give criteria for when an automorphism fixes all spin structures, or when it fixes just one. The case of hyperelliptic curves and of the Klein quartic are discussed in detail.
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