How curved is a random complex curve?

2024/02/19 by Michele Ancona, Ancona, Michele, Damien Gayet +1
Computer Science · Engineering · #Advanced Numerical Analysis Techniques #Algebraic Geometry (math.AG) #Data Management and Algorithms #FOS: Mathematics #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2402.11972

openalex publication_date 2024/02/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

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.

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