  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.
