A SageMath program for calculations in the tautological ring of \(\overline{\mathcal{M}}_{g,n}\) and computation of cycles of admissible covers

Vincent Delecroix, Johannes Schmitt and Jason van Zelm

Many functions of the program rely on an earlier implementation of tautological computations by Aaron Pixton. We want to thank Samuel Lelièvre for help and advice.

The tautological ring of the moduli space of stable curves has been studied extensively in the last decades. We present a SageMath implementation of many core features of this ring. This includes lists of generators and their products, intersection numbers and verification of tautological relations. Maps between tautological rings induced by functoriality, that is pushforwards and pullbacks under gluing and forgetful maps, are implemented. Furthermore, many interesting cycle classes, such as the double ramification cycles, strata of k-differentials and hyperelliptic or bielliptic cycles are available.

The admcycles package is available on the Python Package Index (PyPI), where detailed installation instructions are available for a range of systems. It is being developed on GitLab.

Further links and downloads:

- examples.sage - a file with some (commented) example computations
- An online worksheet on CoCalc.com where the above examples can be executed and modified (no registration required)
- CycleDatabase.sage - a compilation of some tautological expressions of admissible cover cycles
- Slides of a presentation I gave at the SageDays 100 in Bonn

If you use the program above, we would very much like to hear any comments or questions you have, bugs you encounter or features that are still missing and would be valuable to you! It's our goal to make the program as accessible and useful as possible.

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