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

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.

• Some context can be helpful. The underlying group is the Cremona group in dimension 2 (over an algebraically closed field, maybe of characteristic zero). The induced isometry $f_*$ is on the Picard-Manin space. – YCor Jan 17 at 14:29
• One way is to find the degree is to simply write down the formula for $f$: you can find this in Dolgachev's book "Classical Algebraic Geometry" (it's on page 301 in my edition). Another method would be to argue that it lifts to a regular (and not merely birational) involution on the blow-up at the base points, and then the fact that the pushforward preserves the canonical class lets you compute the degree. – Mark Jan 17 at 14:36

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.