On the discriminant locus of a generic projection

TL;DR

This paper studies the geometric properties of the discriminant locus of a generic projection of a smooth projective variety, revealing its duality relation and topological features over complex numbers.

Contribution

It establishes the duality of the discriminant locus with a linear section of the dual variety and analyzes the fundamental group of the branch divisor complement.

Findings

Discriminant locus is projectively dual to a linear section of the dual variety.

A purity statement is deduced for the discriminant.

The fundamental group surjects onto a braid group via braid monodromy.

Abstract

For a smooth projective variety $X\subseteq \mathbb P^N$ over an algebraically closed field of char $0$, we show that the discriminant locus of a generic projection of $X$ is projectively dual to a general linear section of the dual variety, and deduce a purity statement for the discriminant. Over $\mathbb C$, we also show that the fundamental group of the complement of the branch divisor arising from generic projection of a normal hypersurface surjects onto a braid group via braid monodromy.