Automorphisms of quartic surfaces and Cremona transformations

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.