Timeline for Isometry fixes ample class implies it is an automorphism?
Current License: CC BY-SA 4.0
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| Jan 29, 2019 at 13:44 | comment | added | Soby | @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? | |
| Jan 23, 2019 at 18:25 | comment | added | abx | @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. | |
| Jan 23, 2019 at 10:35 | comment | added | Francesco Polizzi | @abx: Does $h_*$ fix the linear system $|D|$, or just the numerical class of $D$? | |
| Jan 23, 2019 at 10:29 | comment | added | abx | 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. | |
| Jan 23, 2019 at 7:36 | history | asked | Soby | CC BY-SA 4.0 |