Action of birational map $f$ on the divisor class of line $[H]$

Question

In the paper "Normal Subgroups in the Cremona Group", 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.

Answer 1

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.