Non Uniform Projections of Surfaces in $\mathbb{P}^3$

Abstract

Consider the projection of a smooth irreducible surface in $\mathbb{P}^3$ from a point. The uniform position principle implies that the monodromy group of such a projection from a general point in $\mathbb{P}^3$ is the whole symmetric group. We will call such points uniform. Inspired by a result of Pirola and Schlesinger for the case of curves, we prove that the locus of non-uniform points of $\mathbb{P}^3$ is at most finite.