Automorphisms of quartic surfaces and Cremona transformations

TL;DR

This paper investigates which automorphisms of smooth quartic surfaces in projective 3-space are induced by Cremona transformations, providing examples and partial results that challenge previous assumptions about automorphisms of finite order.

Contribution

It offers initial insights into the relationship between automorphisms of quartic surfaces and Cremona transformations, especially for surfaces with Picard number 2, and presents counterexamples to a longstanding question.

Findings

Automorphism groups can be generated by involutions without being induced by Cremona transformations.

Counterexamples show not all finite order automorphisms are induced by Cremona transformations.

Provides partial classification for automorphisms of quartic surfaces with Picard number 2.

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.