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Automorphism groups of smooth quartic threefolds

  • Zigang Zhu
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Abstract

We classify finite groups of automorphisms of smooth quartic threefolds over ℂ. Every such group embeds into one of $17$ maximal groups, which we realize by explicit equations and matrix generators. Among finite subgroups of PGL(5,ℂ) preserving a smooth quartic threefold, there are exactly $24$ maximal ones up to conjugacy, and we use them to determine the $194$ groups that occur as full automorphism groups. As an application, we classify the finite groups of projective automorphisms of smooth quartic surfaces and show that there are exactly $12$ maximal ones up to conjugacy in PGL(4,ℂ).

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Cite

BibTeXRIS
BibTeX
@misc{zhu2026automorphism,
  title = {{Automorphism groups of smooth quartic threefolds}},
  author = {Zigang Zhu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.39627},
  eprint = {2610.39627},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.39627},
}
RIS
TY  - UNPB
TI  - Automorphism groups of smooth quartic threefolds
AU  - Zigang Zhu
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2610.39627
UR  - https://arxiv.org/abs/2610.39627
AB  - We classify finite groups of automorphisms of smooth quartic threefolds over ℂ. Every such group embeds into one of $17$ maximal groups, which we realize by explicit equations and matrix generators. Among finite subgroups of PGL(5,ℂ) preserving a smooth quartic threefold, there are exactly $24$ maximal ones up to conjugacy, and we use them to determine the $194$ groups that occur as full automorphism groups. As an application, we classify the finite groups of projective automorphisms of smooth quartic surfaces and show that there are exactly $12$ maximal ones up to conjugacy in PGL(4,ℂ).
ER  -