Parle m'en !

Let X be a toric* smooth projective toric variety (eg: a product of projective spaces  $\mathbb{P}^{d_1} \times ... \times \mathbb{P}^{d_r}$). To any toric vector bundle E on X, we may associate a combinatorial object : a set of (indexed) polytopes. This is the parliament of polytopes of E. 

This object contains a lot of information on E. We may recover the positivity properties of E, its global sections, if E is stable or not, ... in a visual way thanks to its parliament. We can also describe the parliament of its subbundles. Finally, another proof of the relevance of the parliaments of polytopes is that we can translate operations on toric bundles (such as direct sum, tensorization by line bundles or restriction to an invariant curve of X) in terms of their parliaments.

Parliaments of polytopes were invented in 2014 by Sandra Di Rocco, Kelly Jabbusch et Greg Smith. They have allowed us to give explicit counter examples for many problems. We will try to rediscover some of these!

*i.e. there exists a "torus" $T=(\mathbb{C}^*)^d$ dense in X et such that the action of T on T via multiplication component by component extends to an action of T on X.