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Hilbert Scheme of a Pair of Skew Lines on Cubic Threefolds

Yilong Zhang

Abstract

A pair of disjoint lines on a smooth cubic threefold determines an irreducible component of the Hilbert scheme. We prove that this component is smooth and isomorphic to the blow-up of the symmetric product of Fano varieties of lines on the diagonal. We also study its relation to the geometry of lines and singularities on the hyperplane sections and its relation to Bridgeland moduli spaces.

Hilbert Scheme of a Pair of Skew Lines on Cubic Threefolds

Abstract

A pair of disjoint lines on a smooth cubic threefold determines an irreducible component of the Hilbert scheme. We prove that this component is smooth and isomorphic to the blow-up of the symmetric product of Fano varieties of lines on the diagonal. We also study its relation to the geometry of lines and singularities on the hyperplane sections and its relation to Bridgeland moduli spaces.

Paper Structure

This paper contains 23 sections, 35 theorems, 50 equations, 12 figures.

Table of Contents

  1. introduction
  2. Hilbert Scheme of a Pair of Skew Lines
  3. Relative Hilbert Scheme
  4. Relation to Bridgeland Moduli Spaces
  5. Preliminaries
  6. Lines on Cubic Threefold
  7. Abel-Jacobi Map
  8. Gauss Map
  9. Hilbert Scheme of a Pair of Skew Lines on Projective Spaces
  10. Hibert Scheme of a Pair of Skew Lines on Cubic Threefolds
  11. Main Theorem
  12. Strategy of the Proof
  13. Type (I) and type (III) schemes
  14. Type (II) and Type (IV) schemes
  15. Proof for Step 1
  16. ...and 8 more sections

Key Result

Theorem 1.2

(cf. Theorem main_theorem) $H(X)$ is smooth and isomorphic to the Hilbert scheme of two points $F^{[2]}$.

Figures (12)

  • Figure 1: Schemes of the Four Types
  • Figure 2: Examples of Singular Cubic Surfaces
  • Figure :
  • Figure :
  • Figure :
  • ...and 7 more figures

Theorems & Definitions (68)

  • Theorem 1.2
  • Proposition 1.3
  • Proposition 1.4
  • Proposition 1.5
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Definition 2.3
  • Proposition 2.4
  • proof
  • ...and 58 more