There are several ways of describing a Hirzebruch surface, for example as the blow-up of $\mathbb{P}^2$ at one point or as $\mathbb{P}(\mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1}(n))$.

I am wondering if these surfaces admit a three-dimensional analogue and in particular if the blow-up of $P^3$ at one point is the same thing as $\mathbb{P}(\mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1}(n))$ (if not, is this projectivization even toric ?). I have a lacunar background in algebraic geometry and I don't have any idea on how to attack the problem. Do you have any suggestions ?