Exponential rarefaction of maximal real algebraic hypersurfaces

TL;DR

This paper proves that the probability of a real algebraic hypersurface being maximal decreases exponentially with degree, extending prior results from curves to higher dimensions and linking topology of zero loci across degrees.

Contribution

It establishes an exponential decay result for maximal hypersurfaces in high-degree line bundles, generalizing previous surface results to higher dimensions.

Findings

Probability of maximal hypersurfaces tends to zero exponentially as degree increases

A low degree approximation property relates topologies of zero loci across degrees

Extension of Gayet and Welschinger's results to higher dimensions

Abstract

Given an ample real Hermitian holomorphic line bundle $L$ over a real algebraic variety $X$, the space of real holomorphic sections of $L^{\otimes d}$ inherits a natural Gaussian probability measure. We prove that the probability that the zero locus of a real holomorphic section $s$ of $L^{\otimes d}$ defines a maximal hypersurface tends to $0$ exponentially fast as $d$ goes to infinity. This extends to any dimension a result of Gayet and Welschinger valid for maximal real algebraic curves inside a real algebraic surface. The starting point is a low degree approximation property which relates the topology of the real vanishing locus of a real holomorphic section of $L^{\otimes d}$ with the topology of the real vanishing locus a real holomorphic section of $L^{\otimes d'}$ for a sufficiently smaller $d'<d$. Such a statement is inspired by a recent work of Diatta and Lerario.