Real Lines on Random Cubic Surfaces

Abstract We give an explicit formula for the expectation of number real lines on a random invariant cubic surface, i.e., surface $$Z\subset {\mathbb {R}}{\mathrm {P}}^3$$ Z ⊂ R P 3 defined by gaussian polynomial whose probability distribution is under action orthogonal group O (4) change variables. Such distributions are completely described one parameter $$\lambda \in [0,1]$$ λ ∈ [ 0 , 1 ] and as function this expected equals: $$\begin{aligned} E_\lambda =\frac{9(8\lambda ^2+(1-\lambda )^2)}{2\lambda )^2}\left( \frac{2\lambda ^2}{8\lambda )^2}-\frac{1}{3}+\frac{2}{3}\sqrt{\frac{8\lambda )^2}{20\lambda )^2}}\right) . \end{aligned}$$ E = 9 ( 8 2 + - ) 20 . This result generalizes previous results Basu et al. (Math Ann 374(3–4):1773–1810, 2019) case Kostlan polynomial, which corresponds to =\frac{1}{3}$$ $$E_{\frac{1}{3}}=6\sqrt{2}-3.$$ 6 Moreover, we show that maximized purely harmonic polynomials, =1$$ $$E_1=24\sqrt{\frac{2}{5}}-3$$ 24 5