Real Lines on Random Cubic Surfaces

Abstract

AbstractWe give an explicit formula for the expectation of the number of real lines on a random invariant cubic surface, i.e., a surface $$Z\subset {\mathbb {R}}{\mathrm {P}}^3$$ Z ⊂ 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 $$\lambda \in [0,1]$$ λ ∈ [ 0 , 1 ] and as a function of this parameter the expected number of real lines equals: $$\begin{aligned} E_\lambda =\frac{9(8\lambda ^2+(1-\lambda )^2)}{2\lambda ^2+(1-\lambda )^2}\left( \frac{2\lambda ^2}{8\lambda ^2+(1-\lambda )^2}-\frac{1}{3}+\frac{2}{3}\sqrt{\frac{8\lambda ^2+(1-\lambda )^2}{20\lambda ^2+(1-\lambda )^2}}\right) . \end{aligned}$$ E λ = 9 ( 8 λ 2 + ( 1 - λ ) 2 ) 2 λ 2 + ( 1 - λ ) 2 2 λ 2 8 λ 2 + ( 1 - λ ) 2 - 1 3 + 2 3 8 λ 2 + ( 1 - λ ) 2 20 λ 2 + ( 1 - λ ) 2 . This result generalizes previous results by Basu et al. (Math Ann 374(3–4):1773–1810, 2019) for the case of a Kostlan polynomial, which corresponds to $$\lambda =\frac{1}{3}$$ λ = 1 3 and for which $$E_{\frac{1}{3}}=6\sqrt{2}-3.$$ E 1 3 = 6 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 $$\lambda =1$$ λ = 1 and for which $$E_1=24\sqrt{\frac{2}{5}}-3$$ E 1 = 24 2 5 - 3 .