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 in Panizzut et al. (LeMatematiche 75(2), 2020) only in characteristic 0 by exhaustive computer search. We offer a conceptual explanation that has the added benefit of being characteristic free. As an application, we give octanomial normal forms of the strata of the coarse moduli space of cubic surfaces defined in Dolgachev and Duncan (Compos Math 25(1):1–59, 1972) which preserve most specialization with respect to automorphisms.