Let $\mathcal H\_r=\mathbb P (\mathcal O\_{\mathbb P^1}\oplus \mathcal O\_{\mathbb P^1}(r))$ be a Hirzebruch surface for some $r\in\mathbb Z$. As a toric variety, the fan structure is spanned by $(-1,0)$, $(0,-1)$, $(1,r)$, and $(0,1)$ in $N\_{\mathbb R}\cong \mathbb R^2$. When does the Hirzebruch surface $\mathcal H\_r$ have a nef anticanonical divisor? (I am not an expert on toric geometry. I hope my question was not too dumb.)



When does the Hirzebruch surface have a nef anticanonical divisor?
If $r \ge 3$ the exceptional section has negative intersection with anticanonical divisors), so the answer is $r \le 2$.


