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.

  • 2
    $\begingroup$ 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. $\endgroup$ – YCor Jan 17 at 14:29
  • $\begingroup$ 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. $\endgroup$ – 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.


Your Answer

By clicking "Post Your Answer", you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.