Let $\varphi\!: S \to S^\prime$ be a birational morphism of projective non-singular irreducible surfaces over a (algebraically closed) field $k$ and let $D \in \mathrm{Div}(S)$. Also, let $\big(\varphi^{-1}\big)^*\!: k(S) \to k(S^\prime)$ be the function field isomorphism associated with the inverse map $\varphi^{-1}$ and $V = \big(\varphi^{-1}\big)^*\big(H^0(S, D)\big)$. How can I describe in explicit geometric terms the (in general non-complete) linear system $\mathbb{P}(V)$ on $S^\prime$?
More precisely, I am interested in the following case: $S^\prime = \mathbb{P}^2$, $S = \mathrm{Bl}_Q(\mathbb{F}_1)$, $\varphi$ is the successive blowing up a point $P \in \mathbb{P}^2$ and a point $Q \in E_P$, where $E_P$ is the exceptional curve on $\mathbb{F}_1 = \mathrm{Bl}_P\big(\mathbb{P}^2\big)$.
