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 ?
Solution
Segre gave the solution to this problem in terms of the Sylvester Pentahedron:
A generic cubic surface $F$ can be represented by an equation of the type $\sum\limits_{i=0}^4 \lambda_i x_i^3 = 0$ with the $\lambda_i$'s being non-zero and where the $x_i$'s are superabundant homogeneous coordinates connected by the identity $\sum\limits_{i=0}^4 x_i = 0$.
We set $a_i = \frac{1}{|\sqrt{\lambda_i}|}$.
For $\pi_i$ the plane given by $x_i=o$, we can distinguish 3 cases : either all those planes are real, or three of them are real and two are complex conjugates, or one is real and the others form two pairs of complex conjugated planes.
In the first case, $F$ is hyperbolic if and only if $a_1+a_2+a_3-a_4 < a_0 < a_1 + a_2 +a_3+a_4$. In the second case, it is hyperbolic if and only if either $\pi_0 , \pi_1$ complex conjugated and $a_0 + a_1 < a_2+a_3+a_4$, or $\pi_1 , \pi_2$ complex conjugated and $a_1 + a_2 - a_3 -a_4 < a_0 < a_1 + a_2 - a_3+a_4$, or $\pi_2 , \pi_3$ (resp. $\pi_3 , \pi_4$) complex conjugated and $a_1 + a_2 - a_3 -a_4 < a_0 < - a_1 + a_2 + a_3+a_4$. In the third case finally, we cannot get an hyperbolic cubic surface.
Now if $F$ is non-generic, it is either cyclic (and in that case it cannot be hyperbolic, as its real part will be homeomorphic to $\mathbb{R} \mathbb{P}^2$) or representable by an equation of the form:
- either $(x_1^3 + x_2^3 + x_3^3) - x_0^2 (\lambda_0 x_0 + 3\lambda_1 x_1 + 3\lambda_2 x_2 + 3\lambda_3 x_3) = 0$ with $\lambda_0 + 2 (\lambda_1^{3/2} + \lambda_2^{3/2} + \lambda_3^{3/2})\neq 0$;
- or $2\lambda x_0^3 + x_1^3 + x_2^3 - 3 x_0 (\mu x_0 x_1 + x_0 x_2 + x_3^2) = 0$ with $\mu \neq 0$ and $\lambda + \mu^{3/2} \pm 1 \neq 0$.
We can notice that the surface $F'$ represented by $(\frac{1}{\varepsilon} - \lambda_0)x_0^3 + x_1^3 + x_2^3 + x_3^3 - \frac{1}{\varepsilon} (x_0 + \varepsilon \lambda_1 x_1 + \varepsilon \lambda_2 x_2 + \varepsilon \lambda_3 x_3)^3 = 0$ tends to the surface $F$ represented by the first equation when $\varepsilon$ tends to zero; so that if $\varepsilon$ is real, non-zero and sufficiently close to zero, the topological type of $F$ is the same as that of $F'$. We can then reduce the equation we gave for $F'$ to the canonical form, and then use the conditions above.
Similarly, we can notice that the surface $F'$ represented by $(2\lambda + \frac{1}{\varepsilon^2}) x_0^3 + x_1^3 + x_2^3 - \frac{1}{2\varepsilon^2} (x_0 + \varepsilon^2 \mu x_1 + \varepsilon^2 x_2 + \varepsilon x_3)^3 - \frac{1}{2\varepsilon^2} (x_0 + \varepsilon^2 \mu x_1 + \varepsilon^2 x_2 - \varepsilon x_3)^3 = 0$ tends to the surface $F$ when $\varepsilon$ tends to zero. We can then again, for $\varepsilon$ real, non-zero and sufficiently close to zero, reduce the equation we gave for $F'$ to the canonical form, and use the conditions above.
References
Segre, Beniamino. The non-singular cubic surfaces: A new method of investigation with special reference to questions of reality. The Clarendon Press, 1942.