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arXiv · 2508.18059

Automorphisms of punctual Hilbert schemes and symmetric powers of varieties

Ashima Bansal·Supravat Sarkar·Shivam Vats
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Abstract

We classify complex smooth projective surfaces whose punctual Hilbert scheme has a non-natural automorphism preserving the big diagonal. This completely answers a question raised by Belmans, Oberdieck and Rennemo, and extends previous works by Boissi{\`e}re-Sarti, Hayashi, Sasaki, Girardet and Wang. We reduce this to studying the existence of non-natural automorphisms of symmetric powers. We study this question for higher dimensional varieties too, giving some sufficient conditions guaranteeing every automorphism of a symmetric power to be natural. As a corollary, we characterize smooth projective surfaces of Kodaira dimension $\geq 1$ whose punctual Hilbert scheme has a non-natural automorphism, this time not assuming the automorphism preserves the big diagonal. We also address the question, when a smooth projective variety is determined up to isomorphism by its punctual Hilbert scheme.

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Ashima Bansal, Supravat Sarkar, Shivam Vats. 2025-08-25. Automorphisms of punctual Hilbert schemes and symmetric powers of varieties. https://arxiv.org/abs/2508.18059

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