SageMath code

Introduction

This is a repository with the code that produces the table in Chapter 12 of the paper

NCCRs of cones over del Pezzo surfaces

by Anya Nordskova and Michel Van den Bergh.

Running the code

The code assumes that the SageMath package is installed.

If this is the case then two additional required Python packages should be installed using the command:

sage_cmd --python -m pip install -r requirements.txt --upgrade

where here and below sage_cmd stands for the command that invokes SageMath.

Internal consistency of the table

The table in the paper is automatically generated from the file minimal_collections.txt.

Verifying the internal consistency of this file can be done with the command:

sage_cmd validate_minimal_collections.py

The following checks are performed:

• vtjson is used to check that there is no missing data.
• The case identifiers are consecutive lists that have the correct length.
• The exceptional collections are really exceptional.
• The exceptional collections have the required Gram matrix.
• The exceptional collections are minimal.
• The exceptional collections are very strong.
• Paths between cases in the relation sections are valid.
• The relations graph for each del Pezzo is a tree.
• The quivers are correct.
• The ranks are correct.
• There is a certificate for each surface.
• Certificates are valid. Either:
  • The symmetry group is trivial or
  • all reflections give the same NCCR or
  • a valid certificate is supplied.
• All Gram matrices are distinct up to rotation.
• All wild card 3 block cases are valid.

Generation scripts

Minimal block-complete 4-block collections, case 1

The script

sage_cmd 4blocks_case_1.py

implements the algorithm 13.5.1 in the paper. It finishes instantaneously and prints out two Gram matrices.

Minimal block-complete 4-block collections, case 2

The script

sage_cmd 4blocks_case_2.py

implements the algorithm 13.5.2 in the paper. It finishes instantaneously and prints out seven Gram matrices.

Non-existence of minimal block-complete 5-block collections

The script

sage_cmd 5blocks_case_2.py

implements the algorithm 13.6.2 in the paper. It takes about 2000 seconds on a 4 year old laptop and ends with an empty file 5blocks_case_2_data/5blocks_case_2_stage_3.txt, meaning that no solutions were found (as expected).

Testing

The directory test_data contains data files with lists of very strong exceptional collections. We use these for our own research.

The script

sage_cmd testing.py

applies the algorithm in the proof of Theorem 11.2 to samples of exceptional collections extracted from these files and checks that the resulting minimal block-complete collections are in the table.