2019/09/12 by Hanieh Keneshlou, Keneshlou, Hanieh
Mathematics · #14J70 #14K10 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1909.05554
openalex publication_date 2019/09/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Eckardt hypersurface in ℙ19 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 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 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.