Happy thanksgiving! I’m giving a talk in the student commutative algebra seminar next week on canonical curves and Green’s conjecture. When I first saw it, it seemed like another one of those algebraic conjectures where two niche invariants coincide. But it turns out that there’s actually a fair amount of geometry at work behind the scenes.
The base case of Green’s conjecture – for Veronese curves or, equivalently, canonical models of hyperelliptic curves – is precisely the case that the Eagon-Northcott complex is designed to cater for. Here are my notes supporting the brief allusion to it that I’ll make during my talk.
And my actual talk notes are available here. Canonical rings are naturally motivated for a geometer, who automatically cares about the MMP, and it’s nice to engage more with their underlying commutative algebra to see what geometry is reflected in algebra that’s slightly away from the beaten path.