Narrow Results

Allcock and Freitag recently showed that the moduli space of marked cubic surfaces is a subvariety of a nine dimensional projective space which is defined by cubic equations. They used the theory of automorphic forms on ball quotients to obtain these results. Here we describe the same embedding using Naruki's toric model of the moduli space. We also give an explicit parametrization of the tritangent divisors, we discuss another way to find equations for the image and we show that the moduli space maps, with degree at least ten, onto the unique quintic hypersurface in a five dimensional projective space which is invariant under the action of the Weyl group of the root system E₆.

We classify finite groups acting by birational transformations of a nontrivial Severi–Brauer surface over a field of characteristic zero that are not conjugate to subgroups of the automorphism group. Also, we show that the automorphism group of a smooth cubic surface over a field $𝕂$ of characteristic zero that has no $𝕂$-points is abelian, and find a sharp bound for the Jordan constants of birational automorphism groups of such cubic surfaces.

We consider the action of the group ${\mathrm{\text{PGL}}}_4(K)$ on the smooth cubic surfaces of ${\mathbb{P}}_K^3$ ($K$ an algebraically closed field of characteristic zero). We classify, in an explicit way, all the smooth cubic surfaces with nontrivial stabilizer, the corresponding stabilizers and obtain a geometric description of each group in terms of permutations of the Eckardt points, of the $27$ lines or of the $45$ tritangent planes.

We present a method to construct non-singular cubic surfaces over ℚ with a Galois invariant pair of Steiner trihedra. We start with cubic surfaces in a form generalizing that of Cayley and Salmon. For these, we develop an explicit version of Galois descent.