Random fields and the enumerative geometry of lines on real and complex hypersurfaces

TL;DR

This paper develops formulas linking the average number of real lines on random hypersurfaces to determinants of random matrices, providing new proofs and asymptotic relations for counts of real and complex lines in algebraic geometry.

Contribution

It introduces a novel approach connecting enumerative geometry of lines on hypersurfaces with random matrix theory, including new formulas, proofs, and asymptotic analysis.

Findings

Derived a formula for the average number of real lines in terms of random matrix determinants.

Proved the average number of real lines on a cubic surface is 6√2 - 3.

Established the asymptotic relation lim (log E_n / log C_n) = 1/2.

Abstract

We derive a formula expressing the average number $E_n$ of real lines on a random hypersurface of degree $2n-3$ in $\mathbb{R}\textrm{P}^n$ in terms of the expected modulus of the determinant of a special random matrix. In the case $n=3$ we prove that the average number of real lines on a random cubic surface in $\mathbb{R}\textrm{P}^3$ equals: $$E_3=6\sqrt{2}-3.$$ Our technique can also be used to express the number $C_n$ of complex lines on a generic hypersurface of degree $2n-3$ in $\mathbb{C}\textrm{P}^n$ in terms of the determinant of a random Hermitian matrix. As a special case we obtain a new proof of the classical statement $C_3=27.$ We determine, at the logarithmic scale, the asymptotic of the quantity $E_n$, by relating it to $C_n$ (whose asymptotic has been recently computed D. Zagier). Specifically we prove that: $$\lim_{n\to \infty}\frac{\log E_n}{\log C_n}=\frac{1}{2}.$$ Finally we show that this approach can be used to compute the number $R_n=(2n-3)!!$ of real lines, counted with their intrinsic signs, on a generic real hypersurface of degree $2n-3$ in $\mathbb{R}\textrm{P}^n$.