Real Lines on Random Cubic Surfaces

We give an explicit formula for the expectation of the number of real lines on a random invariant cubic surface, i.e. a surface $Z\subset \mathbb{R}P^3$ defined by a random gaussian polynomial whose probability distribution is invariant under the action of the orthogonal group $O(4)$ by change of variables. Such invariant distributions are completely described by one parameter $\lambda\in [0,1]$ and as a function of this parameter the expected number of real lines equals: \begin{equation} E_\lambda=\frac{9(8\lambda^2+(1-\lambda)^2)}{2\lambda^2+(1-\lambda)^2}\left(\frac{2\lambda^2}{8\lambda^2+(1-\lambda)^2}-\frac{1}{3}+\frac{2}{3}\sqrt{\frac{8\lambda^2+(1-\lambda)^2}{20\lambda^2+(1-\lambda)^2}}\right). \end{equation} This result generalizes previous results by Basu, Lerario, Lundberg and Peterson for the case of a Kostlan polynomial, which corresponds to $\lambda=\frac{1}{3}$ and for which $E_{\frac{1}{3}}=6\sqrt{2}-3.$ Moreover, we show that the expectation of the number of real lines is maximized by random purely harmonic cubic polynomials, which corresponds to the case $\lambda=1$ and for which $E_1=24\sqrt{\frac{2}{5}}-3$.