On the liftability of the automorphism group of smooth hypersurfaces of the projective space

Abstract

Let X be a smooth hypersurface of dimension n ≥ 1 and degree d ≥ 3 in the projective space given as the zero set of a homogeneous form F. If (n, d) ≠ (1, 3), (2, 4) it is well known that every automorphism of X extends to an automorphism of the projective space, i.e., Aut(X) ⊆ PGL(n + 2, ℂ). We say that the automorphism group Aut(X) is liftable if there exists a subgroup of GL(n + 2, ℂ) projecting isomorphically onto Aut(X) and leaving F invariant. Our main result in this paper shows that the automorphism group of every smooth hypersurface of dimension n and degree d is 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 Aut(X) are trivial or cyclic of order p.