Syzygies and Koszul modules in geometry

TL;DR

This paper reviews recent advances in the study of Koszul modules and syzygies of algebraic varieties, highlighting theoretical developments, applications to algebraic groups, and connections to conjectures in algebraic geometry.

Contribution

It provides a comprehensive overview of progress in understanding Koszul modules and syzygies, including new results and open questions in the field.

Findings

Progress on Green's Conjecture and the Secant Conjecture

Applications to Chen ranks of Kähler and hyperplane arrangement groups

Connections between syzygies and algebraic curve invariants

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 K\"ahler 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.