A note on syzygies and normal generation for trigonal curves

Abstract

Let $C$ be a trigonal curve of genus $g\ge 5$ and let $T$ be the unique trigonal line bundle inducing a map $\pi :C\stackrel{3:1}{⟶}{\mathbb{P}}^1$. 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 $C\subset \mathbb{P}(H^0(C,K_C\otimes T^{-n}{)}^{\ast })$ for the residual line bundle $K_C\otimes T^{-n}$ for $n\ge 1$ and $g\ge 3n+4$.

9 pages, funding added