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

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 $q\to \mathrm{\infty }$ 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.