Cubic surfaces on the singular locus of the Eckardt hypersurface

Abstract

The Eckardt hypersurface in P¹⁹ is the closure of the locus of 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 P⁴, 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 at two points representing 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.