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 Zsubset {mathbb {R}}{mathrm {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: Eλ=9(8λ2+(1-λ)2)2λ2+(1-λ)22λ28λ2+(1-λ)2-13+238λ2+(1-λ)220λ2+(1-λ)2.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} E_\\lambda =\\frac{9(8\\lambda ^2+(1-\\lambda )^2)}{2\\lambda ^2+(1-\\lambda )^2}\\left( \\frac{2\\lambda ^2}{8\\lambda ^2+(1-\\lambda )^2}-\\frac{1}{3}+\\frac{2}{3}\\sqrt{\\frac{8\\lambda ^2+(1-\\lambda )^2}{20\\lambda ^2+(1-\\lambda )^2}}\\right) . \\end{aligned}$$\\end{document}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} and for which E_{frac{1}{3}}=6sqrt{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=24sqrt{frac{2}{5}}-3.
Highlights
A classical result from complex algebraic geometry tells that on a generic cubic surface in complex projective space there are exactly 27 lines. This is still true for a generic real cubic surface, i.e., on the zero set in complex projective space of a real cubic polynomial, these lines might not be real
The number of real lines on the real zero locus Z (P) ⊂ RP3, for a generic P ∈ R[x0, . . . , x3](3) in the space of real homogeneous polynomials of degree 3 can be either 27, 15, 7 or 3, depending on the coefficients of the chosen polynomial [33]. This is a typical phenomenon in real algebraic geometry, where in general there is no “generic” answer to such counting problems
There is a recent interest into looking at these questions from the probabilistic point of view, replacing the word generic with “random”, which in the case of the current paper means asking for the expectation of the number of real lines on a random real cubic surface
Summary
A classical result from complex algebraic geometry tells that on a generic cubic surface in complex projective space there are exactly 27 lines. There is a recent interest into looking at these questions from the probabilistic point of view, replacing the word generic with “random”, which in the case of the current paper means asking for the expectation of the number of real lines on a random real cubic surface. In this direction, [18, Theorem 2] gives a way to perform a well defined enriched count but, in the spirit of the current paper, it makes sense to ask for the expected number of Qp-lines on a random p-adic cubic This question has been studied by the first named author of this paper together with Lerario in [2] and the answer is p3. For example looking at question 23 and putting a probability distribution on RP19, instead of asking for a semialgebraic description of the set of smooth hyperbolic cubics in RP19, one could seek the probability of a smooth cubic to be hyperbolic
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