Logarithmic Derivation Module of Hyperplane Arrangement

Authors

Developed by Junyan Chu and Shizuo Kaji
JC was supported by the China Scholarship Council.

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Overview

This module provides tools for working with the logarithmic derivation modules of hyperplane arrangements. It is implemented in SageMath and includes both function definitions and illustrative examples in a single Jupyter Notebook.

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Prerequisites

To use this module, you need to have access to SageMath. You can:

1. Install SageMath locally on your computer.
2. Use an online service such as CoCalc, which provides an environment to run SageMath.

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Usage Instructions

1. Open the Jupyter Notebook named LogarithmicVectorFieldsOfArrangements.ipynb in your SageMath environment.
2. Follow the instructions provided within the notebook. It contains both:
   • Function definitions for operations on hyperplane arrangements.
   • Examples demonstrating the module's functionality.

Example

Here’s a brief example of how to use the module:

A = HyperPlaneArr([[1, 0, 0], [0, 1, 0], [0, 0, 1], [1, 1, 1]])
print(gendic(A.minimal_generators))
print(A.free_resolution())
A.plot_vfield(A.minimal_generators[1], xlim=(-2, 2), ylim=(-2, 2))

For detailed examples and additional functionality, refer to the Examples section in the notebook.

References

Junyan Chu, Free resolution of the logarithmic derivation modules of close to free arrangements