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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.)

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If $r \ge 3$ the exceptional section has negative intersection with anticanonical divisors), so the answer is $r \le 2$.

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