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The Eckardt point configuration of cubic surfaces revisited
(Springer, 2022) Betten, Anton; Karaoğlu, Fatma
The classification problem for cubic surfaces with 27 lines is concerned with describing a complete set of the projective equivalence classes of such surfaces. Despite a long history of work, the problem is still open. One approach is to use a coarser equivalence relation based on geometric invariants. The Eckardt point configuration is one such invariant. It can be used as a coarse-grain case distinction in the classification problem. We provide an explicit parametrization of the equations of cubic surfaces with a given Eckardt point configuration over any field. Our hope is that this will be a step towards the bigger goal of classifying all cubic surfaces with 27 lines.
The number of cubic surfaces with 27 lines over a finite field
(Springer, 2021) Karaoğlu, Fatma; Betten, Anton
We determine the number of cubic surfaces with 27 lines over a finite field F-q. This is based on exploiting the relationship between non-conical six-arcs in a projective plane embedded in projective three-space and cubic surfaces with 27 lines. We revisit this classical relationship, which goes back to work of Clebsch in the nineteenth century. Our result can be used as an enumerative check for a computer classification of cubic surfaces with 27 lines over finite fields.