  Let $X^{1,n}_r$ be the blow-up of $\mathbb{P}^1\times\mathbb{P}^n$ in $r$
general points. We describe the Mori cone of $X^{1,n}_r$ for $r\leq n+2$ and
for $r = n+3$ when $n\leq 4$. Furthermore, we prove that $X^{1,n}_{n+1}$ is log
Fano and give an explicit presentation for its Cox ring.
