A Sextic with 35 Cusps

The Modular Form of the Barth-Nieto Quintic

Barth and Nieto have found a remarkable quintic threefold which parametrizes Heisenberg invariant Kummer surfaces which belong to abelian surfaces with a (1,3)-polarization and a lecel 2 structure. A double cover of this quintic, which is also a Calabi-Yau variety, is birationally equivalent to the moduli space {\cal A}_3(2) of abelian surfaces with a (1,3)-polarization and a level 2 structure. As a consequence the corresponding paramodular group Γ_3(2) has a unique cusp form of weight 3. In this paper we find this cusp form which is Δ_1^3. The form Δ_1 is a remarkable weight 1 cusp form with …

A discrete extension of Γ_{1,p}^{\circ}(2) in SP(4,R) and the modular form of the Barth-Nieto quintic

We construct a maximal discrete extension of the paramodular group with a full level-2 structure. The corresponding Siegel variety parametrizes (birationally) the space of Kummer surfaces associated to (1,p)-polarized abelian surfaces with a level-2 structure. In the case p=3 this is related to the Barth-Nieto quintic and in this case we also determine the space of cusp forms of weight 3.

带有18个尖点的五次曲面

Quintic surfaces with 18 cusps

该研究在$\boldsymbol{P}^3$中构造出含18个普通尖点的五次曲面，基于Barth–Rams分解，经有限域与特征零提升验证，为代数几何中奇异五次曲面的研究提供了新实例。

维曼六次曲线的显式Belyi映射与非同余子群的尖点形式

An Explicit Belyi Map for the Wiman Sextic and Cusp Forms for a Noncongruence Subgroup

该研究针对维曼六次曲线的光滑射影模型确定显式Belyi函数，结合非同余子群$\boldsymbol{\tilde{W}}$的模函数描述其复单值化，通过$\boldsymbol{F}_5$上的退化与Puiseux级数展开，给出权2下非同余子群无界分母猜想的不同证明。

Cusps and Codes

We study a construction, which produces surfaces $Y \subset P_3$ with cusps. For example we obtain surfaces of degree six with 18, 24 or 27 three-divisible cusps. For sextic surfaces in a particular family of up to 30 cusps the codes of these sets of cusps are determined explicitly.

K3 surfaces with 9 cusps in characteristic p

We study K3 surfaces with 9 cusps, i.e. 9 disjoint $A_2$ configurations of smooth rational curves, over algebraically closed fields of characteristic $p\neq 3$. Much like in the complex situation studied by Barth, we prove that each such surface admits a triple covering by an abelian surface. Conversely, we determine which abelian surfaces with order three automorphisms give rise to K3 surfaces. We also investigate how K3 surfaces with 9 cusps hit the supersingular locus.

$\mathbb{P}^3$ 中的节点曲面与编码理论

Nodal surfaces in $\mathbb{P}^3$ and coding theory

本文证明 $\mathbb{P}^3$ 中具有 65 个节点的六次曲面（如 Barth 曲面）关联的二元线性码唯一，并给出可能达到节点数上界的七次曲面的候选码。

Explicit computation of symmetric differentials and its application to quasi-hyperbolicity

We develop explicit techniques to investigate algebraic quasi-hyperbolicity of singular surfaces through the constraints imposed by symmetric differentials. We apply these methods to prove that rational curves on Barth's sextic surface, apart from some well-known ones, must pass through at least four singularities, and that genus 1 curves must pass through at least two. On the surface classifying perfect cuboids, our methods show that rational curves, again apart from some well-known ones, must pass through at least seven singularities, and that genus 1 curves must pass through at least two. W…

A sextic surface cannot have 66 nodes

Let S be a surface in complex projective 3-space, having only nodes as singularities. Suppose that S has degree 6. We show that the maximum number of nodes which S can have is 65. An abbreviated history of this is as follows. Basset showed that S can have at most 66 nodes. Catanese and Ceresa and Stagnaro constructed sextic surfaces having 64 nodes. Barth has recently exhibited a 65 node sextic surface. We complete the story by showing that S cannot have 66 nodes. Let f: S~ --> S be a minimal resolution of singularities. A set N of nodes on S is even if there exists a divisor Q on S~ such that…

On sextic surfaces having only nodes (preliminary report)

Let S be a surface in CP^3, having only nodes as singularities. Let pi: S~ --> S be a minimal resolution of singularities. A set N of nodes on S is EVEN if there exists a divisor Q on S~ such that 2Q ~ pi^{-1}(N). Suppose that S has degree 6. It is known (Basset) that S cannot have 67 or more nodes. It is also known (Barth) that S can have 65 nodes. It is not known if S can have 66 nodes. Likewise, it is not known exactly what sizes can occur for an even set of nodes on S. We show that an nonempty even set of nodes on S must have size 24, 32, 40, 56, or 64. We do not know if the sizes 56 and 6…

节点曲面的簇、编码理论与三次超曲面的判别式。第一部分：一般理论与节点K3曲面。第二部分：三次超曲面及其关联判别式。第三部分：节点五次曲面。第四部分：节点六次曲面

Varieties of Nodal surfaces, coding theory and Discriminants of cubic hypersurfaces. Part 1: Generalities and nodal K3 surfaces. Part 2: Cubic Hypersurfaces, associated discriminants. Part 3: Nodal quintics. Part 4: Nodal sextics

本文通过为射影节点曲面附加二元码，系统研究了P^3中节点曲面的Severi簇的不可约分量及其关联层级，完整分类了节点K3曲面，并揭示了Togliatti五次曲面及最大节点数六次曲面与三次超曲面判别式之间的深刻联系。

Dessins d'Enfants and Hypersurfaces with Many $A_j$-Singularities

We show the existence of surfaces of degree $d$ in $\dP^3(\dC)$ with approximately ${3j+2\over 6j(j+1)} d^3$ singularities of type $A_j, 2\le j\le d-1$. The result is based on Chmutov's construction of nodal surfaces. For the proof we use plane trees related to the theory of Dessins d'Enfants. Our examples improve the previously known lower bounds for the maximum number $μ_{A_j}(d)$ of $A_j$-singularities on a surface of degree $d$ in most cases. We also give a generalization to higher dimensions which leads to new lower bounds even in the case of nodal hypersurfaces in $\dP^n, n\ge5$. To conc…