cvolume

cvolume is a SageMath module to compute completed and Masur-Veech volumes of strata of quadratic differentials with odd zeros. You can install it on top of a SageMath installation on your computer (see the instructions below).

cvolume is based on SageMath module admcycles to compute intersection numbers on moduli space of complex curves. This README file is partially borrowed from admcycles page.

Installation

Prerequisites

Make sure that SageMath 9 is installed on your computer. Detailed installation instructions for different operating systems are available here and on the SageMath website.

WARNING: Module will not work with SageMath 8 as that version doesn't support some functionality of Python 3. Make sure to use the last version of SageMath!

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.

Manual installation with pip

Follow the procedure below:

• Download the admcycles package as a .zip file from github.

• Inside a shell console run:

  $ sage -pip install /where/is/the/package.zip --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.

Cloning with git

You can also clone this repository to your computer, if you have versioning software git installed. Go to the folder where you want cvolume to be stored and run the following commands:

$ git init
$ git clone https://github.com/eduryev/cvolume.git
$ cd cvolume
$ git fetch --tags
$ git checkout tags/v1.0 -b v1.0

Use without installation

To use the package without installing, download the package as a .zip file either from github. Unpack the .zip 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 /Amazing/Organized/Folder/System/cvolume, you could do:

$ cd /Amazing/Organized/Folder/System/cvolume
$ sage

Or directly inside a Sage session:

sage: cd /Amazing/Organized/Folder/System/admcycles

Quick start

Documentation

Check this page for documentation. Also see Examples below.

Demonstration

You can find a demo of functionality of cvolume in this jupyter notebook.

Examples

To start using cvolume, start a Sage session: either in the command line:

$ sage

or a Jupyter notebook, or inside one of the online services). Then type:

sage: from cvolume import *

To try a first computation, you can compute the completed volume of Q(1,1,1,1), which coincides with its Masur-Veech volume, since the stratum is principle:

sage: completed_volume([1,1,1,1])
1/15*pi^6

You can also compute the completed volume of Q(3,1), which has Masur-Veech volume 0, since it is empty:

sage: completed_volume([3,1])
23/90*pi^4

Apart from computing volumes you can generate labeled stable graph. Here is a an example, which generated all of labeled stable graphs for stratum [3,1,1,-1]:

sage: stable_lab_graphs([3,1,1,-1])
{Labeled Stable Graph with edges = ((0, 1, 1),), loops = (0, 0), kappa = ((1, 1), (3, -1)),
 Labeled Stable Graph with edges = ((0, 1, 1),), loops = (0, 1), kappa = ((1, 1), (3, -1)),
 Labeled Stable Graph with edges = ((0, 1, 1),), loops = (1, 0), kappa = ((1, 1), (3, -1)),
 Labeled Stable Graph with edges = ((0, 1, 1),), loops = (1, 1), kappa = ((1, 1), (3, -1)),
 Labeled Stable Graph with edges = ((0, 1, 2),), loops = (0, 0), kappa = ((1, -1), (3, 1)),
 Labeled Stable Graph with edges = ((0, 1, 2),), loops = (0, 1), kappa = ((1, -1), (3, 1)),
 Labeled Stable Graph with edges = ((0, 1, 3),), loops = (0, 0), kappa = ((1, 1), (3, -1)),
 Labeled Stable Graph with edges = (), loops = (1,), kappa = ((3, 1, 1, -1),),
 Labeled Stable Graph with edges = (), loops = (2,), kappa = ((3, 1, 1, -1),)}

You can also compute a generalized local polynomial for any stratum:

sage: Nlocal(g=1,n=5,stratum=[5,3,1,1])
2045/64*b1^4 + 1575/16*b1^2*b2^2 + 2045/64*b2^4 + 1575/16*b1^2*b3^2 + 1575/16*b2^2*b3^2 + 2045/64*b3^4 + 1575/16*b1^2*b4^2 + 1575/16*b2^2*b4^2 + 1575/16*b3^2*b4^2 + 2045/64*b4^4 + 1575/16*b1^2*b5^2 + 1575/16*b2^2*b5^2 + 1575/16*b3^2*b5^2 + 1575/16*b4^2*b5^2 + 2045/64*b5^4