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Degeneration of Dual Varieties of Hypersurfaces

Yilong Zhang

Abstract

Consider a one-parameter family of smooth projective varieties X_t which degenerate into a simple normal crossing divisor at t=0. What is the dual variety in the limit? We answer this question for a hypersurface of degree d degenerate to the union of two hypersurfaces of degree d_1 and d-d_1 meeting transversely. We find all the irreducible components of the dual variety in the limit and their multiplicities.

Degeneration of Dual Varieties of Hypersurfaces

Abstract

Consider a one-parameter family of smooth projective varieties X_t which degenerate into a simple normal crossing divisor at t=0. What is the dual variety in the limit? We answer this question for a hypersurface of degree d degenerate to the union of two hypersurfaces of degree d_1 and d-d_1 meeting transversely. We find all the irreducible components of the dual variety in the limit and their multiplicities.
Paper Structure (3 sections, 6 theorems, 21 equations, 3 figures)

This paper contains 3 sections, 6 theorems, 21 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Multiplicity Counting
  3. Proof of Theorem 1

Key Result

Theorem 1.2

The dual variety in the limit $(X^0)^*$ associated to the family $\{F^s=0\}_{s\in\Delta}$ is a reducible hypersurface in $(\mathbb P^{n+1})^*$. Then $(X^0)^*$ consists of following components

Figures (3)

  • Figure :
  • Figure :
  • Figure :

Theorems & Definitions (13)

  • Definition 1.1
  • Theorem 1.2
  • Example 1.3
  • Example 1.4
  • Proposition 2.1
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • ...and 3 more