Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the discriminant locus of a generic projection

Si-Yang Liu, Yilong Zhang

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.

On the discriminant locus of a generic projection

Abstract

For a smooth projective variety over an algebraically closed field of char , we show that the discriminant locus of a generic projection of is projectively dual to a general linear section of the dual variety, and deduce a purity statement for the discriminant. Over , 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.
Paper Structure (35 sections, 38 theorems, 91 equations, 6 figures, 1 table)

This paper contains 35 sections, 38 theorems, 91 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Duality of discriminant locus
  3. Purity of discriminant locus
  4. Degree of discriminant
  5. Monodromy and braid groups.
  6. Preliminaries
  7. Linear algebra
  8. Projective duality
  9. Plücker formula
  10. Discriminant of an algebraic map
  11. Smooth discriminant
  12. Finite branched cover
  13. Duality of smooth discriminant
  14. The setup
  15. The smooth critical locus
  16. ...and 20 more sections

Key Result

Theorem 1.3

Let $X\subseteq \mathbb P^N$ be a smooth projective variety, and $\mathbb PV$ is a general $k$-dimensional linear subspace in the dual space $(\mathbb P^N)^{\vee}$. Let $\Delta^{\perp}$ denote the dual variety of the discriminant locus of the generic projection Intro_eqn_gen-proj. Under the natural In particular, the discriminant loci of two families Intro_eqn_univ-family/PV and Intro_eqn_gen-pro

Figures (6)

  • Figure 1: Linear section of $X$ and generic projection of $X^{\perp}$
  • Figure 2: Discriminant of generic projection of the Veronese threefold
  • Figure :
  • Figure :
  • Figure :
  • ...and 1 more figures

Theorems & Definitions (92)

  • Definition 1.1: Discriminant locus
  • Theorem 1.3: Theorem \ref{['thm_mainthmequiv']}
  • Theorem 1.4: Theorem \ref{['Thm2proof']}
  • Corollary 1.5: Corollary \ref{['cor_smooth-fibration']}
  • Remark 1.6: Singular total space
  • Proposition 1.7
  • Theorem 1.8
  • Remark 1.9
  • Example 1.10
  • Lemma 2.1
  • ...and 82 more