In the paper ["Normal Subgroups in the Cremona Group"](https://link.springer.com/content/pdf/10.1007/s11511-013-0090-1.pdf), it is stated that the induced isometry $f\_{\ast}$ of $f\in J\_d$, where $J\_d$ denote the set of Jonquières transformations of degree $d$, satisfies the equation $$f\_\ast [H]=d[H] - (d-1)[E\_{p\_0}] -\sum\_{i=1}^{2d-2}[E\_{p\_i}]$$ I understand the part where they say that $f$ has a base point $p\_0$ of multiplicity $d-1$ and $2d-2$ distinct points of multiplicity 1 each. However I do not understand how they derived the above action of $f\_\ast$ on the line $[H]$ lying in the (completed) Picard-Manin Space. Would be grateful if anyone points me in the correct direction! Thank you. 



Action of birational map $f$ on the divisor class of line $[H]$
Let us take any birational map $f\colon \mathbb{P}^2\dashrightarrow \mathbb{P}^2$ which is not an isomorphism. Let us denote by $\eta,\rho\colon X\to \mathbb{P}^2$ the blow-up of the base-points of $f$ and $f^{-1}$, so that $\rho=f\circ \eta$ (this is simply a minimal resolution of the birational map $f\colon \mathbb{P}^2\dashrightarrow \mathbb{P}^2$).


Denote by $p\_0,p\_1,\ldots,p\_{n}\in\mathbb{P}^2$ the base-points of $f^{-1}$ and by $H$ a general line of $\mathbb{P}^2$. Then, $\eta^\*(H)$ is linearly equivalent to $$d \rho^\*(H)-\sum\_{i=0}^n m\_i E\_{p\_i}$$ where $E\_i$ is the exceptional divisor associated to $p\_i$, $m\_i\ge 1$ is the multiplicity and $d$ is the degree of the curve $\rho\_\*(\eta^\*(H))$.


The action on the Picard-Manin space then sends $H$ onto $\eta^\*(H)$ and then sees this element via $\rho$ as an element of the Picard-Manin space of $\mathbb{P}^2$. Hence, it sends the class of $H$ onto 
$$d H-\sum\_{i=0}^n m\_i E\_{p\_i}.$$
You obtain the equality desired by replacing the multiplicities by the number you know in the case of de Jonquières elements.


