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Syzygies and Koszul modules in geometry

Gavril Farkas

Abstract

We describe the progress in the last 10 years related to Koszul modules and syzygies of algebraic varieties. Topics discussed include the general theory of Koszul modules and resonance varieties, applications to Chen ranks of Kähler and hyperplane arrangement groups (Suciu's Conjecture) and connections related to syzygies of algebraic curves. Developments related to Green's Conjecture, the Secant Conjecture and the Gonality Conjecture on the resolution of line bundles on algebraic curves are also presented. Open question are proposed throughout the text.

Syzygies and Koszul modules in geometry

Abstract

We describe the progress in the last 10 years related to Koszul modules and syzygies of algebraic varieties. Topics discussed include the general theory of Koszul modules and resonance varieties, applications to Chen ranks of Kähler and hyperplane arrangement groups (Suciu's Conjecture) and connections related to syzygies of algebraic curves. Developments related to Green's Conjecture, the Secant Conjecture and the Gonality Conjecture on the resolution of line bundles on algebraic curves are also presented. Open question are proposed throughout the text.
Paper Structure (20 sections, 12 theorems, 107 equations, 2 tables)

This paper contains 20 sections, 12 theorems, 107 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Koszul modules
  3. The BGG correspondence and Koszul modules
  4. Resonance varieties
  5. Koszul modules with vanishing resonance
  6. The Chow form of the Grassmannian
  7. Koszul modules of geometric nature
  8. The Chen ranks of Koszul modules
  9. The scheme structure of the resonance
  10. Chen ranks in geometry and topology
  11. Vanishing Chen invariants
  12. Suciu's conjecture for hyperplane arrangement groups
  13. Koszul modules and Green's Conjecture
  14. Green's Conjecture
  15. Green's Conjecture via the tangent developable
  16. ...and 5 more sections

Key Result

Proposition 2.2

One has the set-theoretic equality $\mathrm{supp }\ W(V,K)=\mathcal{R}(V,K)$

Theorems & Definitions (30)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Theorem 2.3
  • Remark 2.4
  • Remark 2.5
  • Theorem 2.6
  • Definition 3.1
  • Remark 3.2
  • Remark 3.3
  • ...and 20 more