Let $mathcal H_r=mathbb P (mathcal O_{mathbb P^1}oplus mathcal O_{mathbb P^1}(r))$ be a Hirzebruch surface for some $rinmathbb Z$. 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.)