Smooth Hyperbolic Cubics

Setup

A real smooth cubic surface in $\mathbb{P}_{\mathbb{R}}^3$ has either one or two connected components. In the latter case, the cubic is hyperbolic. More precisely, the two connected components are an ovaloïd and a pseudo-hyperplane. The ovaloïd bounds a convex body that is of interest in optimization.

Question

Can we find a semialgebraic description for open set of smooth hyperbolic cubics in $\mathbb{P}_{\mathbb{R}}^{19}$ ? How to express this case distinction in terms of the six fundamental invariants ?

References

A. Buckley and T. Košir, Determinantal Representations of Smooth Cubic Surfaces, Geometriae Dedicata (2007), article.

People working on the problem

Cédric Le Texier, PhD student, University of Oslo

(Feel free to add your name if you work on this problem)

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