# Explicit Enriques involutions on the Fermat quartic surface

Let $$X$$ be the complex Fermat quartic surface defined by the polynomial $$x^4+y^4+z^4+w^4$$.

By results of Sertöz, we know that the surface $$X$$ admits at least one Enriques involution, i.e. an involution without fixed points. Is an explicit formula for one such involution already known?

• A remark: any biregular involution of $\mathbb{P}^3$ has two lines of fixed points, hence (generically) eight fixed points on $X$. So one cannot hope to get an Enriques involution by the restriction of a biregular involution of the ambient space. I do not know if there is a construction using a birational involution of $\mathbb{P}^3$. – Francesco Polizzi Feb 6 at 10:46