Strongly incompressible curves (Q2810695)

A \(k\)-variety \(X\) is a geometrically reduced not necessarily irreducible scheme of finite type over the field \(k\). Let the finite group \(G\) act on \(X\). The action is \textit{generically free} if there exists a dense \(G\)-stable open subscheme with trivial (scheme-theoretic) stabilizers. A \(G\)-\textit{compression} of \(X\) is a \(G\)-equivariant dominant rational map \(X \longrightarrow Y\), where \(X\) and \(Y\) are generically free \(k\)-varieties with \(G\)-actions. The \(G\)-variety \(X\) over \(k\) is \textit{strongly incompressible} if every \(G\)-compression of \(X\) is birational.NEWLINENEWLINEThe author discusses the existence of strongly incompressible \(G\)-curves over \(K\), depending on the respective nature of \(G\) and \(k\). To avoid technical difficulties, he restricts to the case of characteristic 0.NEWLINENEWLINEIf \(G\) cannot act faithfully on (irreducible) curves of genus 0, there do exist strongly incompressible \(G\)-curves (Theorem 1.1). As this exhausts almost all finite groups, the remaining ones (cyclic, dihedral, \(A_4\), \(A_5\), \(S_4\)) may be handled case-by-case. For each of the groups \(G\) in question, the author finds necessary and sufficient conditions on \(k\) to allow strongly incompressible \(G\)-curves. These conditions are related to the (non-)existence of roots of unity in \(k\) and its cohomological 2-dimension \(\mathrm{cd}_2(k)\).