Syzygies of tangent developable surfaces and K3 carpets via secant varieties

TL;DR

This paper provides geometric proofs for syzygy theorems related to tangent developable surfaces and K3 carpets, leading to new insights on Green's conjecture and the normality of certain algebraic surfaces.

Contribution

It offers simplified geometric proofs of existing syzygy theorems and applies these results to prove Green's conjecture for general curves and establish normality of tangent developable surfaces.

Findings

Simplified proofs of syzygy theorems for tangent developable surfaces and K3 carpets.

A new proof of Green's conjecture for general curves in characteristic zero.

Proof of arithmetic normality of tangent developable surfaces of large degree.

Abstract

We give simple geometric proofs of Aprodu-Farkas-Papadima-Raicu-Weyman's theorem on syzygies of tangent developable surfaces of rational normal curves and Raicu-Sam's result on syzygies of K3 carpets. As a consequence, we obtain a quick proof of Green's conjecture for general curves of genus $g$ over an algebraically closed field $\mathbf{k}$ with $\operatorname{char}(\mathbf{k}) = 0$ or $\operatorname{char}(\mathbf{k}) \geq \lfloor (g-1)/2 \rfloor$. We also show the arithmetic normality of tangent developable surfaces of arbitrary smooth projective curves of large degree.