# Every projective variety is a quiver Grassmannian: the algorithm

Little over a year ago Markus Reineke published a note on the arXiv titled Every projective variety is a quiver Grassmannian about how quiver Grassmannians can be used to describe every possible projective variety. Based on this note Lieven Le Bruyn wrote a blogpost Quiver Grassmannians can be anything, which explains the result using an easy to grasp example. An interesting discussion in the comments section ensued, which was lost in the move to Drupal but apparently restored yesterday (unrelated to my interest in this).

Yesterday I attended a talk by Sarah Scherotzke in which this result was mentioned, and this renewed my interest for it. To entertain myself tonight I implemented some code that determines the information as described in Lieven's example for a general projective variety. The implementation is based on Reineke's original proof, not on Michel Van den Bergh's alternative proof. It is nothing extraordinary, it is really just a programming exercise.

The code is written in SAGE, and can be found in the gist pbelmans/5645902. Its output can be found at Pastebin. If you have any comments, feel free to post them. The output for the original example would be