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

## TL;DR

This paper investigates the points in projective 3-space from which projecting a smooth surface does not produce the full symmetric monodromy group, proving that such non-uniform points are at most finitely many.

## Contribution

It extends the understanding of monodromy groups in surface projections, showing the finiteness of non-uniform points in $ ext{P}^3$, inspired by analogous results for curves.

## Key findings

- The monodromy group is the full symmetric group for a general point.
- The set of non-uniform points in $ ext{P}^3$ is at most finite.
- The result generalizes known curve cases to surfaces.

## 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.