Monodromy of general hypersurfaces

TL;DR

This paper proves that for a general complex hypersurface in projective space, the monodromy group of projection from any point outside the hypersurface is symmetric, extending known results from plane curves to higher dimensions.

Contribution

It generalizes Cukierman's result by showing all points outside a general hypersurface induce symmetric monodromy groups, regardless of the hypersurface's degree.

Findings

All points outside the hypersurface are uniform.

The monodromy group is isomorphic to the symmetric group.

Generalization from plane curves to higher-dimensional hypersurfaces.

Abstract

Let $X$ be a general complex projective hypersurface in $\mathbb{P}^{n+1}$ of degree $d>1$. A point $P$ not in $X$ is called uniform if the monodromy group of the projection of $X$ from $P$ is isomorphic to the symmetric group. We prove that all the points in $\mathbb{P}^{n+1}$ are uniform for $X$, generalizing a result of Cukierman on general plane curves.