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