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  On abelian automorphism groups of hypersurfaces

Zheng, Z. (2022). On abelian automorphism groups of hypersurfaces. Israel Journal of Mathematics, 247(1), 479-498. doi:10.1007/s11856-021-2275-1.

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 Creators:
Zheng, Zhiwei1, Author           
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1Max Planck Institute for Mathematics, Max Planck Society, ou_3029201              

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Free keywords: Mathematics, Algebraic Geometry
 Abstract: Given integers $d\ge 3$ and $N\ge 3$. Let $G$ be a finite abelian group
acting faithfully and linearly on a smooth hypersurface of degree $d$ in the
complex projective space $\mathbb{P}^{N-1}$. Suppose $G\subset PGL(N,
\mathbb{C})$ can be lifted to a subgroup of $GL(N,\mathbb{C})$. Suppose
moreover that there exists an element $g$ in $G$ such that $G/\langle g\rangle$
has order coprime to $d-1$. Then all possible $G$ are determined (Theorem 4.3).
As an application, we derive (Theorem 4.8) all possible orders of linear
automorphisms of smooth hypersurfaces for any given $(d,N)$. In particular, we
show (Proposition 5.1) that the order of an automorphism of a smooth cubic
fourfold is a factor of 21, 30, 32, 33, 36 or 48, and each of those 6 numbers
is achieved by a unique (up to isomorphism) cubic fourfold.

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Language(s): eng - English
 Dates: 2022
 Publication Status: Issued
 Pages: 20
 Publishing info: -
 Table of Contents: -
 Rev. Type: Peer
 Identifiers: arXiv: 2004.09008
DOI: 10.1007/s11856-021-2275-1
 Degree: -

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Title: Israel Journal of Mathematics
Source Genre: Journal
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Publ. Info: Springer
Pages: - Volume / Issue: 247 (1) Sequence Number: - Start / End Page: 479 - 498 Identifier: -