An Exponential Rarefaction Result for Sub-Gaussian Real Algebraic Maximal Curves

TL;DR

This paper demonstrates that maximal real algebraic curves linked to sub-Gaussian random sections are exponentially rare, extending previous Gaussian-based results to a broader class of distributions.

Contribution

It generalizes the exponential rarity result of maximal real algebraic curves from Gaussian to sub-Gaussian random sections in complex geometry.

Findings

Maximal real algebraic curves are exponentially rare under sub-Gaussian distributions.

Extends previous Gaussian case results to sub-Gaussian settings.

Provides a broader understanding of the distribution of real algebraic curves.

Abstract

We prove that maximal real algebraic curves associated with sub-Gaussian random real holomorphic sections of a smoothly curved ample line bundle are exponentially rare. This generalizes the result of Gayet and Welschinger \cite{GW} proved in the Gaussian case for positively curved real holomorphic line bundles.