On the liftability of the automorphism group of smooth hypersurfaces of the projective space
Abstract
Let $X$ be a smooth hypersurface of dimension $n\geq 1$ and degree $d\geq 3$ in the projective space given as the zero set of a homogeneous form $F$. If $(n,d)\neq (1,3), (2,4)$ it is well known that every automorphism of $X$ extends to an automorphism of the projective space, i.e., $\operatorname{Aut}(X)\subseteq \operatorname{PGL}(n+2,\mathbb{C})$. We say that the automorphism group $\operatorname{Aut}(X)$ is $F$liftable if there exists a subgroup of $\operatorname{GL}(n+2,\mathbb{C})$ projecting isomorphically onto $\operatorname{Aut}(X)$ and leaving $F$ invariant. Out main result in this paper shows that the automorphism group of every smooth hypersurface of dimension $n$ and degree $d$ is $F$liftable if and only if $d$ and $n+2$ are relatively prime. We also provide an effective criterion to compute all the integers which are a power of a prime number and that appear as the order of an automorphism of a smooth hypersurface of dimension $n$ and degree $d$. As an application, we give a sufficient condition under which some Sylow $p$subgroups of $\operatorname{Aut}(X)$ are trivial or cyclic of order $p$.
 Publication:

arXiv eprints
 Pub Date:
 April 2020
 arXiv:
 arXiv:2004.12455
 Bibcode:
 2020arXiv200412455G
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Group Theory
 EPrint:
 16 pages. Comments are welcome!