Generic Green's Conjecture and Generic Geometric Syzygy Conjecture in Positive Characteristic

TL;DR

This paper proves a version of Green's Conjecture for canonical curves over fields of positive characteristic and establishes a key case of the Geometric Syzygy Conjecture for even-genus curves, advancing understanding of syzygies in algebraic geometry.

Contribution

It provides a new proof of Green's Conjecture in positive characteristic and verifies a significant case of the Geometric Syzygy Conjecture for even-genus curves.

Findings

Proved Green's Conjecture for p ≥ (g+4)/2.

Established a key case of the Geometric Syzygy Conjecture for even-genus curves when p > g.

Extended syzygy results to positive characteristic fields.

Abstract

We study the syzygies of canonical curves of genus $g\geq 3$ over an algebraically closed field $\mathbb{F}$ of characteristic $p>0$. We provide a new proof of generic Green's Conjecture for $p\geq\frac{g+4}{2}$. Using the techniques from the even-genus case, we establish a significant case of the Geometric Syzygy Conjecture for the last syzygy space of a general even-genus canonical curve (assuming $p>g$). In characteristic 0, it was shown in prior work that this case implies the full conjecture.