Given a point $P$ in $\mathbb{P}^2$ and a natural number $m$ we consider the linear system $\mathcal{L}$ of curves of degree $d$ passing $m$ times through $P$. If $H$ is the line class of the plane, this is a subsystem of $|dH|$.
Now denote by $\mathbb{F}_1$ the blowup of $\mathbb{P}^2$ at P, by $E$ the exceptional divisor and by $H'$ the pullback of $H$. It now seems to be the case that $\mathcal{L}$ transforms to the linear system $|dH' - mE|$. Why is that?
I see that the proper transform of a curve from $\mathcal{L}$ will intersect $E$ $m$-times, but why is that the same as the proper transform belonging to $|dH' - mE|$?
update: After reading the Chapter (4.3.1) in Shafarevich I understand better, but one question still remains: Why contains $|dH' - mE|$ elements at all? $Pic(\mathbb{F}_1) = \mathbb{Z} \oplus \mathbb{Z}$ which, as far as I understand it, is generated by $H'$ and $E$, i.e. $H' = (1,0)$ and $E = (0,1)$. Then it seems to me, that the elements of $|dH' - mE|$ are the ones of the form $(d,-m)$. But how can an element of this form be effective?
My guess is, that I'm wrong as to which divisors are contained in $|dH'|$. If it were $(d,m) \in |dH'|$ there wouldn't be a problem. But how to explain that and would it then still be true that $H' = (1,0)$?