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

Abstract

We derive a formula expressing the average number $E_n$ of real lines on a random hypersurface of degree $2n-3$ in $\mathbb{R}{\text{P}}^n$ 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 $\mathbb{R}{\text{P}}^3$ 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 $\mathbb{C}{\text{P}}^n$ 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: $$\underset{n\to \infty }{\lim }\frac{\log E_n}{\log C_n}=\frac{1}{2}.$$ 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 $\mathbb{R}{\text{P}}^n$.

24 pages. This version replaces an earlier version by the same authors entitled "The average number of real lines on a random cubic". The title and abstract have changed to reflect substantial additions to the paper

References in corpus (1)

• On the number of connected components of random algebraic hypersurfaces