On the order of an automorphism of a smooth hypersurface |
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Authors: | Víctor González-Aguilera Alvaro Liendo |
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Institution: | 1. Departamento de Matemáticas, Universidad Técnica Federico Santa María, Casilla 110-V, Valparaíso, Chile 2. Instituto de Matemática y Física, Universidad de Talca, Casilla 721, Talca, Chile
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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 = Φ n+2(1 ? d) where Φ n+2 is the (n+2)-th cyclotomic polynomial. Finally, we provide some applications to intermediate jacobians of Klein hypersurfaces. |
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