Tautological ring on Mbar_g,n

## Project description

## admcycles

admcycles is a SageMath module to compute with the tautological ring of the moduli spaces of complex curves. You can install it on top of a SageMath installation on your computer (see the instructions below). You can alternatively use one of the online services below that have admcycles already installed:

SageMathCell: An online service for SageMath computations that does not require authentification.

CoCalc: A complete computation environment that offers a free plan subscription (with limited resources).

admcycles includes the SageMath module diffstrata to compute with the tautological ring of the moduli space of multi-scale differentials, a compactification of strata of flat surfaces. It is particularly tailored towards computing Euler characteristics of these spaces.

Detailed information on the package is contained in the documents

**NEW: Online database with tautological relations (Feb 2023)**

The latest version of admcycles includes an automated online lookup of pre-calculated tautological relations (see the list of available cases). By default, if you attempt a calculation that needs tautological relations, the program will first try to look them up in your local storage (in the folder .sage/admcycles). If not found there, the program will check the above database and download the relations to your computer in case they are available. If this fails as well, the program will then calculate the relations itself.

If you would like to deactivate this online lookup, execute the following lines in a Sage session:

sage: from admcycles.admcycles import set_online_lookup_default sage: set_online_lookup_default(False)

If you have calculated and stored relations that are not yet in our database, we would be happy if you reach out so we can integrate them!

### Installation

#### Prerequisites

Make sure that SageMath version 9.0 or later is installed on your computer. Detailed installation instructions for different operating systems are available here and on the SageMath website. If you need to use admcycles with an older version of SageMath, use admcycles version 1.3.2.

All the command below must be run inside a console (the last character of the prompt is usally a $). On Windows this is called SageMath Shell while on linux and MacOS this is often referred to as a terminal.

Inside the console, we assume that the command sage launches a Sage session (whose prompt is usually sage:). To quit the Sage session type quit and press Enter.

#### Installation with pip (the Python package manager)

The most convenient way to use admcycles is to add the package to your Sage installation. The exact procedure for this depends on your operating system and how you installed Sage. If the pip installation fails, see the sections Manual installation with pip or Use without installation below.

If you manually installed Sage by downloading it from the website, then run in a shell console:

$ sage -pip install admcycles --user

Here, the --user is an optional argument to pip which, when provided, will install admcycles not inside the Sage folder but in your home directory.

If you have a linux distribution and installed the sagemath package via your package manager then run in a shell console:

$ pip install admcycles --user

The pip command above might have to be changed to pip2 or pip3 depending on your system. Also, on Debian/Ubuntu systems, the following step might be necessary before running the above command:

$ source /usr/share/sagemath/bin/sage-env

#### Manual installation with pip

The automatic installation with pip from Section Installation with pip might fail. One common reason is the lack of SSL support of the SageMath binaries. In such situation you can follow the procedure below that bypass the connection of pip to the web.

Download the admcycles package as a tar.gz-file or .zip file either from PyPI or from gitlab.

Inside a shell console run:

$ sage -pip install /where/is/the/package.tar.gz --user

Here, the --user is an optional argument to pip which, when provided, will install admcycles not inside the Sage folder but in your home directory.

#### Installation of the development version

If you want to install the development version, you need to have the versioning software git installed. The only change in the procedure is to replace admcycles in the any of the command above by git+https://gitlab.com/modulispaces/admcycles.

#### Upgrade

If you have already installed admcycles and a new version appears on PyPI, you can update your installation by appending the option --upgrade above.

#### Use without installation

To use the package without installing, download the package as a .zip or .tar.gz-file either from PyPI or from gitlab. Unpack the .zip or .tar.gz file. This creates a folder which should contain a file setup.py. In order to use the module, you need to run Sage from this folder. For example, if the full path of this folder is /u/You/Downloads/admcycles, you could do:

$ cd /u/You/Downloads/admcycles $ sage

Or directly inside a Sage session:

sage: cd /u/You/Downloads/admcycles

If you run Sage in Windows using cygwin, the path above should be a cygwin path and will looks something like /cygdrive/c/Users/You/Downloads/admcycles-master.

### Example

To start using admcycles, start a Sage session (either in the command line, or a Jupyter notebook, or inside one of the online services). Then type:

sage: from admcycles import *

To try a first computation, you can compute the degree of the class kappa_1 on Mbar_{1,1} by:

sage: kappaclass(1,1,1).evaluate() 1/24

You can have a look at the above computation directly in SageMathCell.

Here is a more advanced computation:

sage: t1 = 3*sepbdiv(1,(1,2),3,4) - psiclass(4,3,4)^2 sage: t1 Graph : [1, 2] [[1, 2, 5], [3, 4, 6]] [(5, 6)] Polynomial : 3* <BLANKLINE> Graph : [3] [[1, 2, 3, 4]] [] Polynomial : (-1)*psi_4^2

To use diffstrata, the package must be imported separately. Type:

sage: from admcycles.diffstrata import *

To try a first computation, you can compute the Euler characteristic of the minimal stratum H(2) in genus 2:

sage: X = Stratum((2,)) sage: X.euler_characteristic() -1/40

Here is a more advanced computation:

sage: X = Stratum((1,1)) sage: (X.xi^2 * X.psi(1) * X.psi(2)).evaluate() -1/720

### Building documentation

The documentation is available online at https://modulispaces.gitlab.io/admcycles/

You can alternatively build the documentation as follows. Go in the repository docs/ and then run in a console:

$ sage -sh (sage-sh)$ make html (sage-sh)$ exit

The documentation is then available in the folder docs/build/html/. Note that you need the package nbsphinx to compile the full documentation including the example Jupyter notebooks. On most systems, you should be able to install nbsphinx by typing:

$ sage -pip install nbsphinx

### Running doctests

To run doctests, use the following command:

$ sage -t --force-lib admcycles/ docs/source

If it succeeds, you should see a message:

All tests passed!

### License

admcycles is distributed under the terms of the GNU General Public License (GPL) published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. See http://www.gnu.org/licenses/.

### Funding

Johannes Schmitt was supported by the grant SNF-200020162928 and has received funding from the European Research Council (ERC) under the European Union Horizon 2020 research and innovation programme (grant agreement No 786580). He also profited from the SNF Early Postdoc.Mobility grant 184245 and also wants to thank the Max Planck Institute for Mathematics in Bonn for its hospitality. Vincent Delecroix was a guest of the Max-Planck Institut and then of the Hausdorff Institut for Mathematics during the development of the project. Jason van Zelm was supported by the Einstein Foundation Berlin during the course of this work.

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