Normal generation and Clifford index

Abstract

Let $C$ be a smooth curve of genus $g\ge 4$ and Clifford index $c$. In this paper, we prove that if $C$ is neither hyperelliptic nor bielliptic with $g\ge 2c+5$ and $\mathcal{M}$ computes the Clifford index of $C$, then either $°\mathcal{M}\le \frac{3c}{2}+3$ or $|\mathcal{M}|=|g_{c+2}^1+h_{c+2}^1|$ and $g=2c+5$. This strengthens the Coppens and Martens' theorem (\cite{CM}, Corollary 3.2.5). Furthermore, for the latter case (1) $\mathcal{M}$ is half-canonical unless $C$ is a $\frac{c+2}{2}$-fold covering of an elliptic curve, (2) $\mathcal{M}(F)$ fails to be normally generated with $\cli(\mathcal M(F))=c$, $h^1(\mathcal{M}(F))=2$ for $F\in g_{c+2}^1$. Such pairs $(C,\mathcal{M})$ can be found on a $K3$-surface whose Picard group is generated by a hyperplane section in ${\mathbb{P}}^r$. For such a $(C,\mathcal{M})$ on a K3-surface, $\mathcal{M}$ is normally generated while $\mathcal{M}(F)$ fails to be normally generated with $\cli(\mathcal M)=\cli(\mathcal M(F))=c$.

15pages, 2figures