The octanomial normal forms of cubic surfaces with applications to automorphisms

We will show that in any characteristic every nonsingular cubic surface is projectively isomorphic to the surface given by the octanomial normal form. This normal form is discovered by M. Panizzut, E. C. Sert\"oz, and B. Sturmfels in 2020 only in characteristic 0 in a heavily computational way. Our proof is more conceptual and characteristic-free. As an application, we give octanomial normal forms of the strata of the coarse moduli space of cubic surfaces defined by I. Dolgachev and A. Duncan, which preserve most specialization with respect to automorphisms.