Automorphisms of quartic surfaces and Cremona transformations

Daniela Paiva; Ana Quedo

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arXiv · 2302.09014

Automorphisms of quartic surfaces and Cremona transformations

Daniela Paiva·Ana Quedo

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Abstract

In this paper, we consider the problem of determining which automorphisms of a smooth quartic surface $S \subset \mathbb{P}^3$ are induced by a Cremona transformation of $\mathbb{P}^3$. We provide the first steps towards a complete solution of this problem when $\rho(S)=2$. In particular, we give several examples of quartics whose automorphism groups are generated by involutions, but no non-trivial automorphism is induced by a Cremona transformation of $\mathbb{P}^3$, giving a negative answer for Oguiso's question of whether every automorphism of finite order of a smooth quartic surface $S\subset \mathbb{P}^3$ is induced by a Cremona transformation.

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Daniela Paiva, Ana Quedo. 2023-02-17. Automorphisms of quartic surfaces and Cremona transformations. https://arxiv.org/abs/2302.09014

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