Automorphism groups of smooth quintic threefolds

Oguiso, Keiji; Yu, Xun

doi:10.4310/ajm.2019.v23.n2.a2

Cited by 18 publications

(9 citation statements)

References 13 publications

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“…There are many classification results on automorphism groups of hypersurfaces of small degree, in particular, cubic hypersurfaces (see [Hos97], [Hos02], [Dol12, §9.5], [Adl78], [GAL11], [OY15]). Also, Theorem 3.2.5 has the following recent generalization.…”

Section: Finiteness Resultsmentioning

confidence: 99%

Hilbert schemes of lines and conics and automorphism groups of Fano threefolds

2018

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We discuss various results on Hilbert schemes of lines and conics and automorphism groups of smooth Fano threefolds of Picard rank 1. Besides a general review of facts well known to experts, the paper contains some new results, for instance, we give a description of the Hilbert scheme of conics on any smooth Fano threefold of index 1 and genus 10. We also show that the action of the automorphism group of a Fano threefold X of index 2 (respectively, 1) on an irreducible component of its Hilbert scheme of lines (respectively, conics) is faithful if the anticanonical class of X is very ample except for some explicit cases.We use these faithfulness results to prove finiteness of the automorphism groups of most Fano threefolds and classify explicitly all Fano threefolds with infinite automorphism group. We also discuss a derived category point of view on the Hilbert schemes of lines and conics, and use it to identify some of them.

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Section: Finiteness Resultsmentioning

confidence: 99%

Hilbert schemes of lines and conics and automorphism groups of Fano threefolds

2018

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“…The possible automorphism groups of smooth cubic surfaces over an algebraically closed field of arbitrary characteristic were classified by Dolgachev and Duncan [10]. Moreover, the linear automorphism groups of smooth cubic threefolds and smooth quintic threefolds were classified by works of Wei and Yu [35] and Oguiso and Yu [28], respectively. The Fermat cubic fourfold is also known to have the largest possible automorphism group by a result of Laza and Zheng [26].…”

Section: Bounds On Linear Automorphism Groupsmentioning

confidence: 99%

Automorphisms of weighted projective hypersurfaces

Esser

2023

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We prove several results concerning automorphism groups of quasismooth complex weighted projective hypersurfaces; these generalize and strengthen existing results for hypersurfaces in ordinary projective space. First, we establish a Grothendieck-Lefschetz type theorem for these hypersurfaces. We next provide a characterization of when their linear automorphism group is finite and find explicit uniform upper bounds on the size of this group. Finally, we describe the automorphisms of a generic quasismooth hypersurface with given weights and degree.

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“…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. In this paper, we examine automorphisms of X that give Galois points.…”

Section: Lemma 26 Letmentioning

confidence: 99%

Linear automorphisms of smooth hypersurfaces giving Galois points

Hayashi

2021

Preprint

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Let X be a smooth hypersurface X of degree d ≥ 4 in a projective space P n+1 . We consider a projection of X from p ∈ P n+1 to a plane H ∼ = P n . This projection induces an extension of function fields C(X)/C(P n ). The point p is called a Galois point if the extension is Galois. In this paper, we will give a necessary and sufficient conditions for X to have Galois points by using linear automorphisms.

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Automorphism groups of smooth quintic threefolds

Abstract: We study automorphism groups of smooth quintic threefolds. Especially, we describe all the maximal ones with explicit examples of target quintic threefolds. There are exactly 22 such groups.

Cited by 18 publications

References 13 publications

Hilbert schemes of lines and conics and automorphism groups of Fano threefolds

Hilbert schemes of lines and conics and automorphism groups of Fano threefolds

Automorphisms of weighted projective hypersurfaces

Linear automorphisms of smooth hypersurfaces giving Galois points

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