Given any Fano variety X, the cone of divisors of X is polyhedral. Better than that, it admits a decomposition into polyhedral chambers, for the equivalence relation "D ~ D' if large multiples of D and D' define the same rational map".

This cone lives in a real vector space of dimension r, where r is the Picard number of the Fano variety. This dimension r can be quite large, for instance r = 7 in the case of a cubic surface, so the polyhedral decomposition is not easy to visualize in general. On the contrary if r = 1 or 2 the decomposition is more or less trivial.

In the intermediate case r = 4, we have a cone in R⁴. Such a polyhedral cone is completely determined by an affine section, which is a polyhedral covex body in dimension 3. This is a nice object to visualize!

We consider del Pezzo surfaces (another name for Fano varieties in dimension 2), of Picard rank 4, and defined over the field Q of rational numbers. Here note that by Picard number we mean the equivariant Picard number, with respect to the action of the absolute Galois of Q.

One way to get such a surface is to blow-up 3 orbits of points of respective size a,b,c, such that the sum a+b+c is at most 8. By following the links at the top of this page you can visualize all such constructions: We computed the polyhedral cone and its chamber decomposition for each admissible triplets a b c (for instance the preview on the right if for a b c = 2 3 3). You can have a preview of all cases here, and here you can explore the inner structure of the decomposition by moving the pieces appart.

Reference: "Signature morphisms from the Cremona group over a non-closed field", S. Lamy and S. Zimmermann, to appear in Journal of the European Mathematical Society.