Homology of Hirzebruch surfaces

Question

Let $\mathbb{F}_n:=\mathbb{P}(\mathcal{O}(-n)\oplus\mathcal{O}(0))$ be the $n$th Hirzebruch surface, where $\mathcal{O}(k)$ is the canonical line bundle on $\mathbb{P}^1_\mathbb C$, for any $k\in\mathbb{Z}$. I'm trying to compute the intersection form associated to $\mathbb F_n$. In order to do this I need to compute its homology, and in particular, I have to check that the generators of $H_2(\mathbb F_n;\mathbb Z)$ are given by $s_\infty(\mathbb P^1), P_x$, where the notations are those used here. I would follow the paper I linked, but I don't understand what $D$ is and what the notation $D\cup_{id} (-D)$ means. Can someone help me to understand?