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

• Published in: Israel journal of mathematics 2023-06, Vol.255 (1), p.283-310, Article 283
• Main Authors: González-Aguilera, Víctor, Liendo, Alvaro, Montero, Pedro
• Format: Article
• Language: English
• Subjects: Algebra; Analysis; Applications of Mathematics; Automorphisms; Group Theory and Generalizations; Hyperspaces; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Prime numbers; Subgroups; Theoretical
• ISSN: 0021-2172; 1565-8511
• DOI: 10.1007/s11856-022-2417-0

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 .