# When does the Hirzebruch surface have a nef anticanonical divisor?

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.)

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