Variations of GIT quotients package v0.6.13

Data collection method:

This software package is a complement to the articles "Moduli of cubic surfaces and their anticanonical divisors" and "Variations of geometric invariant quotients for pairs, a computational approach" authored by Patricio Gallardo and Jesus Martinez-Garcia. It uses algorithms developed in the former paper to solve several problems in variations of GIT quotients, in particular the ones in the latter paper.

Data processing and preparation activities:

A precise description of the coded algorithms is available in the article "Variation of geometric invariant quotients for pairs a computational approach." (https://arxiv.org/abs/1602.05282) 1. The program returns a list $t_i$ including all GIT walls and a representative for each GIT chamber. 2. The program returns the fundamental set of one-parameter subgroups. The program also tells you which one-parameter subgroups are not necessary for the GIT analysis. 3. For each $t_i$, the program returns a list of one-parameter subgroups that generate non-stable maximal sets. The program recognizes the changes in the non-stable maximal sets with respect the previous $t_{i-1}$. There are three types of outputs: A. If a non-stable set of monomial has appeared for an earlier $t_i$, then it is denoted as "same". B. If it becomes semistable, then it is denoted as "same but becomes semistable." C. If it is a new set of monomials, then it is denoted as "new". Finally, the program tells the reader if a maximal non-stable set has disappeared when to compare it with the previous $t_{i-1}$. 4. For each one-parameter subgroup that it is not denoted as "same", the program returns the monomials in the maximal non-stable sets. 5. The program recognizes if the configuration is semi-stable or not. In the former case, it returns the potential close orbit of that pairs.

Technical details and requirements:

The code requires: 1. Python 2.7.13 or higher: when installing for Windows, it is recommended to select the options "Install for all users" and "Add Python.exe to PATH"; 2. libraries NumPy (v.1.11.3 or higher), SciPy (v.0.18.1 or higher), SymPy (v.1.0 or higher) and Mpmath (v.1.0 or higher), which are available from the Python Package Index (PyPI): it is recommended to install these using the pip utility. See InstallationWindows.txt for full installation details.

Additional information:

The output of the program with the parameters used in "Moduli of cubic surfaces and their anticanonical divisors" (https://arxiv.org/abs/1607.03697) is available here in text format in the file cubicsurfaces.txt. In the file run.py, the reader can change the dimension, and degree of the VGIT problem. The code in the folder VGIT has the following files: - The file " __init__.py" is required to make Python treat the directories as containing packages. Then, we can import the modules 'VGIT', 'Hypersurfaces', 'Monomials', and 'OPSubgroups'. - The file "Monomials.py" contains the sub-package Monomials with the class Monomials to deal with one monomial and the class MonFrozenSet to deal with a set of monomials. In this file, the reader can find the Hilbert_Mumford function and the function that generates the normalized one-parameter subgroups from a list of monomials. - The file OPSubgroups.py contains the sub-package OPSubgroups with the class OPS and all the methods to work with them. It imports from the class "Monomials", "associated1ps" and "MonFrozenSet". - The file "VGIT.py" contains the metaclasses "Problem", "Solution" and "Solution_t" that defines the VGIT problem and it stores the solutions. - The file "Hypersurface.py" contains the classes "Problem" and VGIT.Hypersurfaces.Solution_t". We store the VGIT problem in the class "Problem." We store the solution to our VGIT problem in the class "VGIT.Hypersurfaces.Solution_t".

Methodology link: