I am reading through an article by Matsumura and Monsky on Automorphisms of Hypersurfaces in which they state that there exist quartic surfaces is $\mathbb{P}^{3}$ which have infinite automorphism groups, this is Theorem 4 in the linked article.
They construct an example of such a surface but it is not explicit, they say;
"Let $F$ be a non-singular (smooth) quartic surface in $\mathbb{P}^{3}$ containing a non-singular curve $C$ of genus 2 and of degree 6."
I was wondering if there was an explicit formula for such a surface?
I have read around and found that a possible construction of a non-singular qaurtic surface in $\mathbb{P}^{3}$ is to take the intersection of 2 quadrics in $\mathbb{P}^{4}$, and also a construction of a non-singular curve of genus 2 and degree 6 is to take a polynomial $f(x)$ in one variable with 6 distinct roots and then consider the hyper-elliptic curve $y^{2}=f(x)$. However I do not know how to combine these two constructions together to give the surface I am looking for.
Finally they say that the result is classical and cite 3 articles claiming they contain other examples/constructions, one due to G. Fano (Sopra alcune superficie del 4 ordine rappresentabili sal piano doppio (1906)), and two due to F. Severi (Complementi alla teoria della base per la totalita delle curve di una superficie algebric (1910), and Geometria dei sistemi algebrici (1958)) all of which are very old and also in Italian. I cannot get a hold of any of the three articles (nor can I read Italian), so these do not provide much help.
Any help about a possible construction or a pointer to an article that isn't in Italian would be greatly appreciated. Thanks.