\(G\)-solid rational surfaces (Q6564863)

In the paper under review the author studies \(G\)-solid rational surfaces. Let \(S\) be a rational surface and \(G\) be a finite group acting faithfully and bi-regularly on \(S\). We denote by \(\rho^{G}(S)\) the rank of the \(G\)-invariant part of the Picard group \(S\). The \(G\)-equivariant Minimal Model Programme, applied to a resolution of singularities of \(S\), implies that \(S\) is a \(G\)-birational to \(G\)-Mori fibre space, i.e., a \(G\)-surface is one of the following cases:\N\begin{itemize}\N\item[a)] A \(G\)-del Pezzo surface \(S\), i.e., \(S\) is a dell Pezzo with \(-K_{S}\) being ample and \(\rho^{G}(S)=1\).\N\item[b)] A \(G\)-conic bundle, i.e., there exists a \(G\)-equivariant morphism \(S \rightarrow \mathbb{P}^{1}\) with general fibre isomorphic to \(\mathbb{P}^{1}\) and such that \(\rho^{G}(S)=2\).\N\end{itemize}\NWe say that \(S\) is a \(G\)-solid surface if it is not \(G\)-birational to any \(G\)-conic bundle.\N\NMain Theorem. Let \(S\) be a \(G\)-del Pezzo surface of degree \(d = K_{S}^{2}\). Then \(S\) is \(G\)-solid if and only if\N\begin{itemize}\N\item[a)] \(d\leq 3\),\N\item[b)] \(d=4\) and \(G\) does not fix a point on \(S\) in general position.\N\item[c)] \(d=5\) and \(G\) is not isomorphic to \(\mathbb{Z}_{5}\) or \(D_{5}\).\N\item[d)] \(d=6\) and \(G\) is not isomorphic to \(\mathbb{Z}_{6}\), \(\mathfrak{S}_{3}\), or \(D_{6}\).\N\item[e)] \(d=8\), \(S \cong \mathbb{P}^{1} \times \mathbb{P}^1\), and, up to conjugation in \(\mathrm{Aut}(S)\), either\N\begin{itemize}\N\item[--] \(G\) has a subgroup isomorphic to \(\mathfrak{A}_{4}\), or\N\item[--] \(G_{4} \subset G\) and \(G \not\subset G_{16}\) , for some specific groups \(G_{4}\) and \(G_{16}\) presented there.\N\end{itemize}\N\item[f)] \(S \cong \mathbb{P}^{2}\), the group \(G\) does not fix any point of \(S\), and it is not isomorphic to \(\mathfrak{S}_{4}\) or \(\mathfrak{A}_{4}\).\N\end{itemize}