The cone of curves and the cone of divisors of an algebraic variety play an important role in the study of its birational geometry. Intermediate cones are equally important, but we do not know too many concrete examples of them.

Based on a collaboration with Luca Ugaglia, we will report on some results on intermediate cones of blow-ups of $({\mathbb P}^1)^n$ at very general points. In particular, we will give a system of generators obtained from projections onto $({\mathbb P}^1)^k$ for $k<n$.