A note on embeddings of $S_4$ and $A_5$ into the Cremona group and versal Galois coverings

Abstract

In this paper we introduce two versal $S_4$ coverings, and show that the two are birationaly distinct, i.e. there exists no $S_4$ equivariant birational map between them. We also introduce two versal $A_5$ coverings and show that the two are birationaly distinct also. As a corollary we obtain that there are at least three non-conjugate embbedings of $S_4$ (resp. $A_5$) into $Cr_2(\mathbb{C})$, the Cremona group.