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arXiv · 2610.39627

Automorphism groups of smooth quartic threefolds

Zigang Zhu
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Abstract

We classify finite groups of automorphisms of smooth quartic threefolds over $\mathbb{C}$. Every such group embeds into one of $17$ maximal groups, which we realize by explicit equations and matrix generators. Among finite subgroups of $\mathrm{PGL}(5,\mathbb{C})$ 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 $\mathrm{PGL}(4,\mathbb{C})$.

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Zigang Zhu. 2026-10-29. Automorphism groups of smooth quartic threefolds. https://arxiv.org/abs/2610.39627

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