A note on syzygies and normal generation for trigonal curves

TL;DR

This paper proves normal generation for residual line bundles on trigonal curves of genus at least 7 and computes their minimal free resolutions, advancing understanding of syzygies and embeddings of such curves.

Contribution

It provides a simple proof of normal generation for residual line bundles on trigonal curves and explicitly computes their minimal free resolutions for certain genera and line bundle powers.

Findings

Normal generation holds for residual line bundles on trigonal curves with genus ≥ 7.

Explicit minimal free resolutions are obtained for residual line bundles on these curves.

Results apply to curves with genus g ≥ 3n+4 for n ≥ 1.

Abstract

Let $C$ be a trigonal curve of genus $g\ge 5$ and let $T$ be the unique trigonal line bundle inducing a map $\pi: C \stackrel{3:1}{\longrightarrow} \mathbb{P}^1$. This note provides a short and easy proof of the normal generation for the residual line bundle $K_C\otimes T^{-1}$ for curves of genus $g\ge 7$. Moreover, we compute the minimal free resolution of the embedded curve $C\subset \mathbb{P}(H^0(C,K_C\otimes T^{-n})^*)$ for the residual line bundle $K_C\otimes T^{-n}$ for $n\ge 1$ and $g \ge 3n+4$.