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

Abstract

We introduce a probabilistic framework for the study of real and complex enumerative geometry of lines on hypersurfaces. This can be considered as a further step in the original Shub–Smale program of studying the real zeros of random polynomial systems. Our technique is general, and it also applies, for example, to the case of the enumerative geometry of flats on complete intersections. 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:

$$\begin{aligned} E_3=6\sqrt{2}-3. \end{aligned}$$

This technique can also be applied 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 expectation of the square of the modulus of the determinant of a random Hermitian matrix. As a special case, we recover 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 in [19]). Specifically we prove that:

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{\log E_n}{\log C_n}=\frac{1}{2}. \end{aligned}$$

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 (as defined in [28]), on a generic real hypersurface of degree $2n-3$ in ${\mathbb {R}}{\text {P}}^n$.