# Isometry fixes ample class implies it is an automorphism?

My question is related to lemma 5.12 in ["Normal Subgroups in the Cremona Group"]. Let $$h$$ be a birational transformation of a projective surface $$X$$ and $$[D']\in N^1(X)$$ be an ample class. We define $$[D]=[D']/\sqrt{[D']\cdot[D']}$$ so that $$[D] \in \mathbb{H}_{\overline{\mathcal{Z}}}$$ where $$\mathbb{H}_{\overline{\mathcal{Z}}}$$ is the hyperboloid of one sheet derived from the (completed) Picard Manin space $$\overline{\mathcal{Z}}(X)$$ of $$X$$.

In the last line of the proof, the author stated that the induced isometry $$h_\ast$$ by $$h$$ on $$\overline{\mathcal{Z}}(X)$$ fixes the ample class $$[D]$$, and since $$[D]$$ is ample then $$h$$ is an automorphism of $$X$$.

My question how does $$h_\ast$$ fixing an ample class $$[D]$$ allow us to deduce that $$h$$ must be an automorphism on $$X$$? Any insight or resources proving the result will be greatly appreciated.

• Replacing $D$ by a multiple, you can assume that $D$ is very ample, i.e. defines an embedding $\varphi _{D}: X\hookrightarrow \lvert D\rvert^*$, with $\lvert D\rvert=\mathbb{P}(H^0(X,\mathcal{O}_{X}(D)))$. Since $h_{*}$ fixes $[D]$, it induces an automorphism of the linear system $\lvert D\rvert$ which preserves $\varphi _{D}(X)$ and induces $h$ on $X$. Therefore $h$ is an automorphism. – abx Jan 23 at 10:29
• @abx: Does $h_*$ fix the linear system $|D|$, or just the numerical class of $D$? – Francesco Polizzi Jan 23 at 10:35
• @Francesco Polizzi: Good point. I was assuming that the surface $X$ is regular, since I think this is the case considered in the quoted paper. – abx Jan 23 at 18:25
• @abx Thank you very much for the input! May I also ask the following question: later on the authors in the proof of proposition 5.13 that if one has $h\in Bir(X)$ s.t. $h_\ast$ preserves the axis of $g_\ast$, then $h_\ast [D']$ is an ample class and so $h$ is an automorphism of $X$. May I know how did they arrive at the image of $[D']$ under $h_\ast$ is ample and how this implies that $h$ is an automorphism? – thedilated Jan 29 at 13:44