 Solution

For every maximal cone, one endpoint of the interval of lattice points from part a realizes the corresponding Cartier datum. The toric basepoint-free criterion therefore makes $\mathcal{O}(k{D}_1)$ basepoint-free for $k\ge 0$. Equivalently, part d identifies it with the pullback of the globally generated bundle ${\mathcal{O}}_{{\mathbb{P}}^1}(k)$.