# Variations of GIT quotients package v0.6.13

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", both available in the Arxiv. The software package implements a series of algorithms for the study of variations of Geometric Invariant Theory (GIT) quotients of pairs formed by a hypersurface in projective space of dimension n and degree d and a hyperplane embedded in the same projective space. Given a dimension n and a degree d, the code finds all relevant one-parameter subgroups which determine all the GIT quotients for these pairs. In addition, it finds a finite list of candidate 'walls' in the wall-chamber decomposition studied by Dolgachev-Hu and Thaddeus. Furthermore, for each prospective chamber and wall it finds all maximal orbits of non stable and strictly semistable pairs, as well as minimal closed orbits of strictly semistable pairs in terms of families of pairs defined by monomials with non-zero coefficients. Finally, it runs through all the list of prospective chambers and walls eliminating 'false walls' for which the GIT quotient is detected not to vary. Further details can be found at the article "Variations of geometric invariant quotients for pairs, a computational approach". The full worked out output for cubic surfaces and hyperplanes, as reflected in "Moduli of cubic surfaces and their anticanonical divisors", is attached as a text file.

The sotware package is implemented as a Python 2.7 package and it requires several additional libraries to run which are all detailed in Installation file. The package has been thoroughly tested for Windows and it may work on Unix-based platforms with extra work.

Cite this dataset as:

Martinez Garcia, J.,
Gallardo, P.,
2017.
*Variations of GIT quotients package v0.6.13.*
Bath: University of Bath Research Data Archive.
Available from: https://doi.org/10.15125/BATH-00458.

## Export

## Code

vgit-0.6.13-bath … version.zip

application/zip (145kB)

Creative Commons: Attribution-Share Alike 4.0

The source code and data, but not the text of the articles, are released under a Creative Commons CC BY-SA 4.0 license. If you make use of the source code and/or data in an academic or commercial context, you should acknowledge this by including a reference or citation to the article "Variations of geometric invariant quotients for pairs, a computational approach" — in the case of the code — or to "Moduli of cubic surfaces and their anticanonical divisors" — in the case of the data for cubic surfaces. See the file license.txt for the full license text.

## Creators

Jesus Martinez Garcia

University of Bath

Patricio Gallardo

Washington University in St. Louis

## Contributors

University of Bath

Rights Holder, Contributor

## Documentation

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:

Gallardo, P., and Martinez-Garcia, J.,
2016.
Variations of geometric invariant quotients for pairs, a computational approach.
*arXiv*.
Available from: https://doi.org/10.48550/ARXIV.1602.05282.

Gallardo, P., and Martinez-Garcia, J.,
2016.
Moduli of cubic surfaces and their anticanonical divisors.
*arXiv*.
Available from: https://doi.org/10.48550/ARXIV.1607.03697.

## Documentation Files

readme.txt

text/plain (5kB)

Creative Commons: Attribution-Share Alike 4.0

InstallationWindows.txt

text/plain (1kB)

Creative Commons: Attribution-Share Alike 4.0

## Funders

National Science Foundation (NSF)

https://doi.org/10.13039/100000001

RTG: Algebra, Algebraic Geometry, and Number Theory

DMS-1344994

## Publication details

Publication date: 28 November 2017

by: University of Bath

Version: 0.6

DOI: https://doi.org/10.15125/BATH-00458

URL for this record: https://researchdata.bath.ac.uk/id/eprint/458

## Related papers and books

Gallardo, P., and Martinez-Garcia, J.,
2018.
Variations of geometric invariant quotients for pairs, a computational approach.
*Proceedings of the American Mathematical Society*, 146(6), 2395-2408.
Available from: https://doi.org/10.1090/proc/13950.

Gallardo, P., and Martinez-Garcia, J.,
2019.
Moduli of cubic surfaces and their anticanonical divisors.
*Revista Matemática Complutense*, 32(3), 853-873.
Available from: https://doi.org/10.1007/s13163-019-00298-y.

Gallardo, P., Martinez‐Garcia, J., and Spotti, C.,
2020.
Applications of the moduli continuity method to log K‐stable pairs.
*Journal of the London Mathematical Society*, 103(2), 729-759.
Available from: https://doi.org/10.1112/jlms.12390.

## Contact information

Please contact the Research Data Service in the first instance for all matters concerning this item.

Contact person: Jesus Martinez Garcia

Faculty of Science

Mathematical Sciences