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@YCor because the automorphism $g:X\rightarrow X$ is chosen on the Kummer surface $X$. Since $X$ is rational to $CP^2$ then we can say the same for $G_C$. My question is how does the criterion laid put by the authors suggests that $g$ can be chosen to be real?
@YCor Thank you for your response. I think i got it Just a small doubt that I still have. How can we choose $g$ to be real? This is because $g$ is chosen in $\text{Bir}(X)$. How can we ensure the image of $g$ in $G_C$ is real as well?
@YCor the fact that $N$ is proper iff it intersects trivially with $PGL_3(C)$ follows from Noether-Castelnuovo. Can we say that same for real case? Since Noether-Castelnuovo need not hold for the real plane Cremona group. (Correct me if im wrong.)
@Mark Are there any papers giving a detailed study of this particular construction? I can't seem to find any. I can only find papers detailing constructions where they consider the quotient by an involution.
@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?
