On stable conjugacy of finite subgroups of the plane Cremona group. I (Q2440525)

The Cremona group \(\mathrm{Cr}_n(\Bbbk )\) is the group of birational automorhpisms of the projective \(n\)-space \(\mathbb{P}^n\) over an algebraically closed field \(\Bbbk\) of characteristic zero. Subgroups \(G_1,\;G_2\) in the Cremona group are said to be {\textit{stably conjugate}} if they become conjugate after adding variables, namely, in the tower \(\cdots \subset \mathrm{Cr}_n(\Bbbk ) \subset \mathrm{Cr}_{n+1}(\Bbbk )\cdots\). A subgroup \(G\) is linearizable if it is induced from a linear action. A basic observation is that the group \(H^1(G, \mathrm{Pic}(X))\) is a stable birational invariant, independent of the nonsingular projective rational variety \(X\) with the biregular \(G\)-action. In particular, if \(G\) is stably linearizable, then \(H^1(G, \mathrm{Pic}(X))=0\). In the paper, the authors compute the group when \(n=2\) and \(G\) is cyclic of prime order with a pointwise fixed curve of positive genus. Together with the classification result by Dolgachev-Iskovskikh, this shows that \(G\) is stably linearizable if and only if it does not fix a curve of positive genus pointwisely. As a corollary, we see that de Jonquières, Bertini and Geiser involutions are not stably linearizable.