Real Lines on Random Cubic Surfaces
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arXiv
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| Format: | Preprint |
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2019
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| _version_ | 1866910974058954752 |
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| author | Manssour, Rida Ait El Belotti, Mara Meroni, Chiara |
| author_facet | Manssour, Rida Ait El Belotti, Mara Meroni, Chiara |
| contents | We give an explicit formula for the expectation of the number of real lines on a random invariant cubic surface, i.e. a surface $Z\subset \mathbb{R}P^3$ 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 $λ\in [0,1]$ and as a function of this parameter the expected number of real lines equals: \begin{equation}
E_λ=\frac{9(8λ^2+(1-λ)^2)}{2λ^2+(1-λ)^2}\left(\frac{2λ^2}{8λ^2+(1-λ)^2}-\frac{1}{3}+\frac{2}{3}\sqrt{\frac{8λ^2+(1-λ)^2}{20λ^2+(1-λ)^2}}\right). \end{equation} This result generalizes previous results by Basu, Lerario, Lundberg and Peterson for the case of a Kostlan polynomial, which corresponds to $λ=\frac{1}{3}$ and for which $E_{\frac{1}{3}}=6\sqrt{2}-3.$ 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 $E_1=24\sqrt{\frac{2}{5}}-3$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1910_07326 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Real Lines on Random Cubic Surfaces Manssour, Rida Ait El Belotti, Mara Meroni, Chiara Algebraic Geometry Probability We give an explicit formula for the expectation of the number of real lines on a random invariant cubic surface, i.e. a surface $Z\subset \mathbb{R}P^3$ 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 $λ\in [0,1]$ and as a function of this parameter the expected number of real lines equals: \begin{equation} E_λ=\frac{9(8λ^2+(1-λ)^2)}{2λ^2+(1-λ)^2}\left(\frac{2λ^2}{8λ^2+(1-λ)^2}-\frac{1}{3}+\frac{2}{3}\sqrt{\frac{8λ^2+(1-λ)^2}{20λ^2+(1-λ)^2}}\right). \end{equation} This result generalizes previous results by Basu, Lerario, Lundberg and Peterson for the case of a Kostlan polynomial, which corresponds to $λ=\frac{1}{3}$ and for which $E_{\frac{1}{3}}=6\sqrt{2}-3.$ 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 $E_1=24\sqrt{\frac{2}{5}}-3$. |
| title | Real Lines on Random Cubic Surfaces |
| topic | Algebraic Geometry Probability |
| url | https://arxiv.org/abs/1910.07326 |