Cones of Cycles on blowups of $({\mathbb{P}}^1{)}^n$

We study cones of pseudoeffective cycles on the blow up of $({\mathbb{P}}^1{)}^n$ at points in very general position, proving some results concerning their structure. In particular we show that in some cases they turn out to be generated by exceptional classes and fiber classes relatively to the projections onto a smaller number of copies of projective lines.