Syzygies of algebraic varieties through symmetric products of algebraic curves

TL;DR

This survey reviews recent advances in understanding syzygies of algebraic varieties, focusing on conjectures and results related to algebraic curves and their symmetric products, highlighting geometric methods used.

Contribution

It consolidates recent results on syzygies of algebraic varieties, emphasizing the role of symmetric products of algebraic curves in proving key conjectures.

Findings

Proofs of gonality conjecture for weight-one syzygies

Results on syzygies of secant varieties and tangent developable surfaces

Verification of Green's conjecture for canonical curves

Abstract

This is a survey paper on recent work on syzygies of algebraic varieties. We discuss the gonality conjecture on weight-one syzygies of algebraic curves, syzygies of secant varieties of algebraic curves, syzygies of tangent developable surfaces and Green's conjecture on syzygies of canonical curves, and asymptotic syzygies of algebraic varieties. All results considered in this paper were proven using the geometry of symmetric products of algebraic curves.