The automorphism groups of the Galois expansions of $m$-canonical coverings of the projective spaces

The talk is devoted to the automorphism groups of
algebraic manifolds $\widetilde X_m$ which are the Galois coverings $\widetilde f_m:\widetilde X_m\to \mathbb P^{\dim X}$ induced by $m$-canonical generic coverings $f_m:X\to \mathbb P^{\dim X}$ of the projective space $\mathbb P^{\dim X}$. It will be shown that if $m$
is big enough, then in dimensions one and two the Galois group $Gal(\widetilde X_m/\mathbb P^{\dim X})\simeq S_d$, where $d=\deg f_m$, coincides with $Aut(\widetilde X_m)$ and the action of the symmetric group $S_d$ on $\widetilde X_m$ is not deformable to an action of $S_d$ which is not the full automorphism group of the manifold obtained under the deformation.