How curved is a random complex curve?

TL;DR

This paper investigates the curvature distribution of random complex plane curves, providing uniform probability bounds for curvature deviations, and extends these results to curves on projective surfaces, contributing to the understanding of random complex hypersurfaces.

Contribution

It offers the first uniform bounds on the probability of curvature deviations in random complex curves, extending previous work on metric statistics to broader classes of curves.

Findings

Lower bounds on probability of small curvature regions are uniform in degree d.

Upper bounds for similar curvature probabilities are also established.

Results extend to curves on projective surfaces with ample line bundles.

Abstract

In this paper, we study the curvature properties of random complex plane curves. We bound from below the probability that a uniform proportion of the area of a random complex degree $d$ plane curve has a curvature smaller than $-d/8$. Our lower bound is uniform, in the sense that it does not depend on $d$. We also provide uniform upper bounds for similar probabilities. These results extend to random complex curves of projective surfaces equipped with an ample line bundle. This paper can be viewed as a sequel of [1], where other metric statistics were given. On a larger time scale, it joins the general program initiated in [11] of understanding random complex hypersurfaces of projective manifolds.