Subgroups of odd order in the real plane Cremona group (Q298023)

The Cremona group \(\mathrm{Cr}_n(k)\) is the group of birational automorphisms of \(\mathbb{P}^n_k\). The group \(\mathrm{Cr}_2(\mathbb{C})\) is well understood thanks to the work of many authors.NEWLINENEWLINEThe paper under review deals with finite subgroups of odd order of \(\mathrm{Cr}_2(\mathbb{R})\). Here a regular morphism is a rational morphism defined at all complex points.NEWLINENEWLINEA cornerstone result is that for any subgroup \(G\subseteq\mathrm{Cr}_2(k)\) there exists a rational surface \(X\) such that \(G\) is conjugate to a subgroup of \(\mathrm{Aut}(X)\).NEWLINENEWLINEIn the paper under review, it is proved that , if \(k=\mathbb{R}\) and \(G\) is finite and of odd order, then either \(X\) is \(\mathbb{R}\)-rational and the rank of \(\mathrm{Pic}(X)^G\) is 1, where \(\mathrm{Pic}(X)^G\) are the classes of divisors invariant by the action of \(G\), or \(X=\mathbb{P}^1_{\mathbb{R}}\times\mathbb{P}^1_{\mathbb{R}}\) and \(G\) is the direct product of at most 2 cyclic groups. Recall that a del Pezzo surface is a surface \(X\) that is the blow up of \(\mathbb{P}^2_{\mathbb{R}}\) in at most 8 points general position. Furthermore, in the del Pezzo case, it is proved that \(X\) is the blow up of \(\mathbb{P}^2_{\mathbb{R}}\) in \(d\) points where \(d\in\{0,1,3,4\}\) and in each case all the possibilities for \(G\) are determined. One of the main tools in the proofs is the study of the action of \(G\) on the exceptional curves of \(X_{\mathbb{C}}\rightarrow \mathbb{P}^2_{\mathbb{C}}\).