Terminal Fano 3-fold hypersurfaces in rank 2 toric varieties database

Terminal Fano 3-fold hypersurfaces in rank 2 toric varieties database This is the dataset which provides the list of all pairs $(\mathbb{F},\mathcal{O}(d))$ such that 1. $\mathbb{F}$ is a simplicial, projective, torsion-free, rank $2$ toric $4$-fold.2. $d\in \mathbb{Z}^2$ such that the general element $X_d$ in the linear system $|\mathcal{O}(d)|$ is a nef hypersurface which is not a linear cone.3. $(\mathbb{F},X_{d})$ is a Fano and terminal pair. For details see [V26]. For each entry, the essential characteristic data is the weight matrix $A$ and degree_of_hypersurface $d$; everything else follows (with the exception of the ID). $X$ refers to a general hypersurface of degree $d$ in the toric variety $\mathbb{F}$ given by weight matrix $A$ and character $\chi$. Brief description of an entry id: a unique integer ID for this entry,weight_matrix: 2*6 integer matrix A describing a toric variety F as a GIT quotient of C^6 by (C*)^2, degree_of_hypersurface: degree d of the hypersurface in the toric variety given by A,character: Character chi to be taken for GIT quotient of the toric variety,basket: multiset of quotient singularities,genus: h^0(X,-K)-2,codimension: GRDB codimension of the Hilbert series of X (from [grdb])degree: -K^3grdb_id: GRDB id corresponding to the Hilbert series of X (from [grdb])topological_euler_characteristic: Topological euler characteristic of X Example entry id: 1weight_matrix: '[[1, 2, 3, 1, 0, 0], [0, 1, 2, 1, 2, 3]]'degree_of_hypersurface: '[[6], [6]]'character: '[[1], [3]]'basket: '1/2(1,1,1),1/2(1,1,1),1/4(1,1,3),1/9(1,4,5)'genus: 0codimension: 8degree:'47/36'grdb_id:6149topological_euler_characteristic: -10 main_table.sql The files "main_table.sql" contains sqlite-formatted version of the data described above, and can be imported into an sqlite database via, for example: $ cat main_table.sql | sqlite3 fano3.db This can then be easily queried. For example $ sqlite3 main_table.db >SELECT primary_id FROM main_table WHERE degree= 19/6 19 References [V26] Thamarai Valli Venkatachalam, "Terminal Fano $3$-fold hypersurfaces in rank $2$ toric varieties", PhD Thesis, University College London, In preparation [grdb] Gavin Brown and Alexander Kasprzyk, "The Fano $3$-fold database", Zenodo, https://doi.org/10.5281/zenodo.5820337秩2环面簇（toric variety）中的终端法诺3维超曲面数据集 本数据集收录所有满足如下条件的配对$(mathbb{F},mathcal{O}(d))$： 1. $mathbb{F}$为单纯、射影、无挠的秩2环面4维簇（toric 4-fold）； 2. $din mathbb{Z}^2$，使得线性系$|mathcal{O}(d)|$中的一般元$X_d$为非线性锥的数值有效（nef）超曲面； 3. $(mathbb{F},X_d)$为法诺终端配对。 详细信息参见文献[V26]。 对于每条数据条目，核心特征数据为权重矩阵$A$与超曲面次数$d$；其余信息均可由二者推导得出（条目ID除外）。此处$X$指代由权重矩阵$A$与特征标$chi$定义的环面簇$mathbb{F}$中次数为$d$的一般超曲面。 数据条目简要说明 id：本条目的唯一整数标识符； weight_matrix：2×6整数矩阵$A$，通过$(mathbb{C}^*)^2$对$mathbb{C}^6$的几何不变量商（Geometric Invariant Theory, GIT）定义环面簇$F$； degree_of_hypersurface：由矩阵$A$定义的环面簇中，超曲面的次数$d$； character：环面簇几何不变量商所需的特征标$chi$； basket：商奇点的多重集合； genus：$h^0(X,-K)-2$； codimension：$X$的希尔伯特级数的GRDB余维数（数据源自文献[grdb]）； degree：$-K^3$； grdb_id：对应$X$希尔伯特级数的GRDB标识符（数据源自文献[grdb]）； topological_euler_characteristic：$X$的拓扑欧拉示性数。 示例条目 id: 1 weight_matrix: '[[1, 2, 3, 1, 0, 0], [0, 1, 2, 1, 2, 3]]' degree_of_hypersurface: '[[6], [6]]' character: '[[1], [3]]' basket: '1/2(1,1,1),1/2(1,1,1),1/4(1,1,3),1/9(1,4,5)' genus: 0 codimension: 8 degree:'47/36' grdb_id:6149 topological_euler_characteristic: -10 main_table.sql 文件`main_table.sql`包含上述数据的SQLite格式版本，可通过如下命令导入至SQLite数据库： $ cat main_table.sql | sqlite3 fano3.db 随后即可便捷地进行数据查询，示例如下： $ sqlite3 main_table.db > SELECT primary_id FROM main_table WHERE degree = 19/6 得到结果：19 参考文献 [V26] Thamarai Valli Venkatachalam，《秩2环面簇中的终端法诺3维超曲面》，伦敦大学学院博士学位论文，待出版 [grdb] Gavin Brown与Alexander Kasprzyk，《法诺3维簇数据库》，Zenodo平台，https://doi.org/10.5281/zenodo.5820337