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Real Lines on Random Cubic Surfaces

Ait El Manssour, Rida; Belotti, Mara; Meroni, Chiara

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: Eλ=9(8λ2+(1−λ)2)2λ2+(1−λ)2(2λ28λ2+(1−λ)2−13+238λ2+(1−λ)220λ2+(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 λ=13 and for which E13=62–√−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=2425−−√−3.
Published in: Arnold Mathematical Journal, 10.1007/s40598-021-00182-y, Springer Nature