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