A dataset of 8-dimensional Q-factorial Fano toric varieties of Picard rank 2

This is a dataset of randomly generated 8-dimensional Q-factorial Fano toric varieties of Picard rank 2.

The data is divided into four plain text files:

• bound_7_terminal.txt
• bound_7_non_terminal.txt
• bound_10_terminal.txt
• bound_10_non_terminal.txt

The numbers 7 and 10 in the file names indicate the bound on the weights used when generating the data. Those varieties with at worst terminal singularities are in the files "bound_N_terminal.txt", and those with non-terminal singularities are in the files "bound_N_non_terminal.txt". The data within each file is de-duplicated, however the data in different files may contain duplicates (for example, it is possible that "bound_7_terminal.txt" and "bound_10_terminal.txt" contain some identical entries).

Each line of a file specifies the entries of a (2 x 10)-matrix. For example, the first line of "bound_7_terminal.txt" is:

[[5,6,7,7,5,2,5,3,2,2],[0,0,0,1,1,2,6,4,3,3]]

and this corresponds to the 8-dimensional Q-factorial Fano toric variety with weight matrix

5  6  7  7  5  2  5  3  2  2

0  0  0  1  1  2  6  4  3  3

and stability condition given by the sum of the columns, which in this case is

44

20

It can be checked that, in this case, the corresponding variety has at worst terminal singularities. In this example the largest occurring weight in the matrix is 7.

The number of entries in each file is:

• bound_7_terminal.txt: 5000000
• bound_7_non_terminal.txt: 5000000
• bound_10_terminal.txt: 10000000
• bound_10_non_terminal.txt: 10000000

For details, see the paper:

"Machine learning detects terminal singularities", Tom Coates, Alexander M. Kasprzyk, and Sara Veneziale. Neural Information Processing Systems (NeurIPS), 2023.

Magma code capable of generating this dataset is in the file "terminal_dim_8.m". The bound on the weights is set on line 142 by adjusting the value of 'k' (currently set to 10). The target dimension is set on line 143 by adjusting the value of 'dim' (currently set to 8). It is important to note that this code does not attempt to remove duplicates. The code also does not guarantee that the resulting variety has dimension 8. Deduplication and verification of the dimension need to be done separately, after the data has been generated.

If you make use of this data, please cite the above paper and the DOI for this data:

doi:10.5281/zenodo.10046893