Real Lines on Random Cubic Surfaces

2019/10/16 by Manssour, Rida Ait El, Belotti, Mara, Meroni, Chiara
#Algebraic Geometry (math.AG) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1910.07326

Abstract

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⊂ ℝP³ 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 λ∈ [0,1] and as a function of this parameter the expected number of real lines equals: E_λ=(9(8λ²+(1-λ)²))/(2λ²+(1-λ)²)((2λ²)/(8λ²+(1-λ)²)-(1)/(3)+(2)/(3)√((8λ²+(1-λ)²)/(20λ²+(1-λ)²))). This result generalizes previous results by Basu, Lerario, Lundberg and Peterson for the case of a Kostlan polynomial, which corresponds to λ=(1)/(3) and for which 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 λ=1 and for which E₁=24√((2)/(5))-3.

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