Journal of the
Korean Mathematical Society JKMS

In this work we obtain conditions for nonspecial line bundles on general $k$-gonal curves failing to be normally generated. Let $\mathcal L$ be a nonspecial very ample line bundle on a general $k$-gonal curve $X$ with $k\ge 4$ and $\deg\mathcal L\ge\frac{3}{2}g+\frac{g-2}{k}+1$. If $\mathcal L$ fails to be normally generated, then $\mathcal L$ is isomorphic to $\mathcal K_X-(n g^1_k+B)+R$ for some $n\ge 1$, $B$ and $R$ satisfying (1) $h^0(R)=h^0(B)=1$, (2) $n+3\le \deg R\le 2n+2$, (3) $\deg({R}\cap F)\le 1$ for any $F \in g^1_k$. Its converse also holds under some additional restrictions. As a corollary, a very ample line bundle $\mathcal L\simeq \mathcal K_X-g^0_d+\xi^0_e$ is normally generated if $g^0_d\in X^{(d)}$ and $\xi^0_e\in X^{(e)}$ satisfy $d\le \frac{g}{2}-\frac{g-2}{k}-3$, supp$(g^0_d\cap \xi^0_e)=\emptyset$ and $\deg(g^0_d\cap F)\le k-2$ for any $F\in g^1_k$.

Keywords: general $k$-gonal curve, normal generation, nonspecial line bundle, Clifford index

MSC numbers: 14H30, 14H51, 14C20