Finite groups of essential dimension 2 (Q373480)

Fix an algebraically closed field \(k\) of characteristic \(0\). The essential dimension of a finite group \(G\), denoted \(\mathrm{ed}(G)\), is defined as the least dimension of a versal \(G\)-variety. Here a \(G\)-variety \(X\) over \(k\) is versal, if it is faithful and for every \(G\)-torsor \(T\) over a field extension \(K/k\) the set of \(K\)-rational points in the twisted variety \({}^T X\) is dense. The only (finite) group of essential dimension \(0\) is the trivial group and groups of \(\mathrm{ed}(G)=1\) are precisely the non-trivial cyclic and odd dihedral groups as shown by \textit{J. Buhler} and \textit{Z. Reichstein} [Compos. Math. 106, No. 2, 159--179 (1997; Zbl 0905.12003)].NEWLINENEWLINEThe paper under review classifies all finite groups of essential dimension 2. A similar classification was made by \textit{H. Kraft} and \textit{G. W. Schwarz} for covariant dimension instead of essential dimension [J. Algebra 313, No. 1, 268--291 (2007; Zbl 1118.14055)]. In general \(\mathrm{ed(G)}\leq \mathrm{covdim}(G)\) and if \(G\) has a non-trivial center, the two values coincide as shown by the above two authors and the reviewer in [\textit{H. Kraft} et al., J. Algebra 322, No. 1, 94--107 (2009; Zbl 1173.14036)]. This permits the author to consider only groups \(G\) with trivial center.NEWLINENEWLINEIn order to complete the classification of groups of essential dimension \(2\) the author studies minimal rational surfaces with a versal action of a finite group. These have been classified by \textit{Yu. I. Manin} [Math. USSR, Sb. 1, 141--168 (1968); translation from Mat. Sb., n. Ser. 7 (114; Zbl 0182.23701)] and \textit{V. A. Iskovskikh} [Math. USSR, Izv. 14, 17--39 (1980; Zbl 0427.14011)] based on work of Enriques. The list of groups of \(\mathrm{ed}(G)=2\) the author finally obtains includes the groups \(S_5\), \(\mathrm{PSL}_2(\mathbb{F}_7)\), subgroups of \(\mathrm{GL}_2(\mathbb{C})\) that are neither cyclic nor odd dihedral and certain finite subgroups of \((\mathbb{C}^\times)^2 \rtimes D\), where \(D\) is a maximal finite subgroup of \(\mathrm{GL}_2(\mathbb{Z})\).