Monodromy of Hyperplane Sections of Curves and Decomposition Statistics over Finite Fields

TL;DR

This paper investigates the distribution of hyperplane sections of algebraic curves over finite fields, establishing a general equidistribution result and analyzing the monodromy to understand decomposition statistics as the field size grows.

Contribution

It introduces a broad equidistribution theorem for hyperplane sections of curves over finite fields and computes the monodromy to derive decomposition statistics in the large field limit.

Findings

Established a general equidistribution result for hyperplane sections.

Calculated monodromy groups of transversal hyperplane sections.

Derived new and existing results on decomposition statistics over finite fields.

Abstract

For a projective curve $C\subset\mathbf{P}^n$ defined over $\mathbf{F}_q$ we study the statistics of the $\mathbf{F}_q$-structure of a section of $C$ by a random hyperplane defined over $\mathbf{F}_q$ in the $q\to\infty$ limit. We obtain a very general equidistribution result for this problem. We deduce many old and new results about decomposition statistics over finite fields in this limit. Our main tool will be the calculation of the monodromy of transversal hyperplane sections of a projective curve.