delPezzo

This is a Sagemath module for studying flexibility of affine cones over del Pezzo surfaces, see arXiv:2305.06462.

Quickstart

The module delPezzo_cylinders.py is intended to be used with a Sagemath-provided Python interpreter sage -python.

1. install Sagemath locally or in a container, e.g., using a VSCode Dev Container extension with the included .devcontainer/devcontainer.json file.
2. Install the module with sage -pip install . in the source directory.
3. Import the module with from delPezzo import * in your script or a Jupyter notebook.
4. Run your code in one of the following ways:
   • Python script script.py with sage -python script.py
   • Sagemath script script.sage with sage script.sage
   • Jupyter notebook with a kernel provided by Sagemath, see instructions, or in case of the Dev Container, select Python interpreter sage -python (or try to select an available one).

Usage

The following example code imports the module, creates a Surface instance, chooses the subdivision cone B3 and a cylinder collection.

from delPezzo import *
S = Surface(3)
B3 = S.cone_representative('B(3)')
collection = Cylinder.make_type_cuspcubic(S, S.E, S.E[-4:])

The list of available subdivision cones for S is returned by NE_SubdivisionCone.cone_types(S). The rays of polarity and forbidden cones of the collection are returned by collection.Pol.rays() and collection.Forb.rays() respectively. The properties of the collection in the relative interior of a given cone (B3 here) as well as subdivision cones, where the collection is polar and complete, can be checked with the following methods.

collection.is_polar_on(B3) # False
collection.is_complete_on(B3) # True
collection.is_transversal() # True
collection.is_generically_flexible_on(B3) # False
list(collection.compatible_representatives()) # ['B(2)', 'C(2)']
list(collection.compatible_representatives(complete=True)) # ['B(2)', 'C(2)']

The method CylinderGenerator.all_cylinders(S, constructions) returns a collection comprised of all cylinders of certain constructions (e.g., constructions = ['lines','tangent']) corresponding to all choices of the contraction to the projective plane. Such a collection is useful in conjuction with the following methods.

The method collection.make_polar_on(cone) filters out the cylinders that are not polar inside the cone, and collection.reduce() removes abundant cylinders from the collection while keeping the forbidden cone unchanged.

Tests

The tests are implemented in files tests/test_*.py and can be mass-checked with pytest. There is also a notebook tests/tests.ipynb with all tests included for convenience.

Data

The file Lubbes_list.txt' contains a list of geometric representations of weak del Pezzo surfaces from [Lub-algo, Appendix A]. The file Lubbes_parse.py' is a helper parser script.

[Lub-algo] Niels Lubbes, `Algorithms for singularities and real structures of weak Del Pezzo surfaces', Journal of Algebra and Its Applications, 13(05):1350158, 2014.

Authorship

Created by Alexander Perepechko.