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 $[H]$. Would be grateful if anyone points me in the correct direction! Thank you.

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.

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 Jonquiéres transformation 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 $[H]$. Would be grateful if anyone point me in the correct direction! Thank you.

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 $[H]$. Would be grateful if anyone points me in the correct direction! Thank you.