Cubic surfaces on the singular locus of the Eckardt hypersurface

TL;DR

This paper characterizes the singular locus of a specific model of the Eckardt hypersurface in projective space, revealing its reducible structure and special cubic surfaces with unique symmetries and Eckardt points.

Contribution

It describes the structure of the singular locus of the Eckardt hypersurface model in projective space and identifies its components and their geometric properties.

Findings

The singular locus is a reducible surface with two rational components.

The components intersect along two rational curves.

Special cubic surfaces are distinguished by Eckardt points and automorphisms.

Abstract

The Eckardt hypersurface in $\mathbb{P}^{19}$ parameterizes smooth cubic surfaces with an Eckardt point, which is a point common to three of the $27$ lines on a smooth cubic surface. We describe the cubic surfaces lying on the singular locus of the model of this hypersurface in $\mathbb{P}^4$, obtained via restriction to the space of cubic surfaces possessing a so-called Sylvester form. We prove that inside the moduli of cubics, the singular locus corresponds to a reducible surface with two rational irreducible components intersecting along two rational curves. The two curves intersect in two points corresponding to the Clebsch and the Fermat cubic surfaces. We observe that the cubic surfaces parameterized by the two components or the two rational curves are distinguished by the number of Eckardt points and automorphism groups.