On the order of an automorphism of a smooth hypersurface

Abstract

Abstract. In this paper we give an effective criterion as to when a positive integer q is the order of an automorphism of a smooth hypersurface of dimension n and degree d, for every d ≥ 3, n ≥ 2, (n, d) = (2, 4), and gcd(q, d) = gcd(q, d − 1) = 1. This allows us to give a complete criterion in the case where q = p is a prime number. In particular, we show the following result: If X is a smooth hypersurface of dimension n and degree d admitting an automorphism of prime order p then p < (d − 1) n+1 ; and if p > (d − 1) n then X is isomorphic to the Klein hypersurface, n = 2 or n + 2 is prime, and p = Φn+2(1 − d) where Φn+2 is the (n + 2)-th cyclotomic polynomial. Finally, we provide some applications to intermediate jacobians of Klein hypersurfaces.

Automorphisms of weighted projective hypersurfaces

“…By analogy with Collins' computations of Jordan constants, we might expect that unusual behavior such as in the n = 1 case occurs only for small n. Many partial results in this direction are known for smooth hypersurfaces in ordinary projective space. For instance, we have a fairly complete picture of the possible orders of automorphisms that can occur [13,37]. The possible automorphism groups of smooth cubic surfaces over an algebraically closed field of arbitrary characteristic were classified by Dolgachev and Duncan [10].…”

Linear automorphisms of smooth hypersurfaces giving Galois points

“…For a smooth hypersurface X ⊂ P n+1 , orders of automorphisms of X and the structure of the group Aut(X) are studied for n ≥ 1 ( [2,9,7,8,20,26]). Also, as in [12,20], the structures of subgroups of Aut(X) are also investigated based on the way they act on X.…”

On lattice polarizable cubic fourfolds

“…For instance, cubic threefolds are unirational but not rational [CG72], and the development of topics related to smooth cubic hypersurfaces can be found in [Huy23]. The study of their automorphism groups Aut(X) has a long and rich history, see [Seg42], [Adl78], [Hos97], [Rou09], [GL11], [Dol12], [Pro12], [GL13], [Mo13], [BCS16], [Fu16], [DD19], [HM19], [WY20], [LZ22], [Zhe22], [GLM23], etc. All possible subgroups of Aut(X) have been classified for cubic surfaces (see [Seg42], [Hos97], [Dol12]) and for cubic threefolds ( [WY20]).…”