REMARKS ON NONSPECIAL LINE BUNDLES ON GENERAL κ-GONAL CURVES

Title & Authors
REMARKS ON NONSPECIAL LINE BUNDLES ON GENERAL κ-GONAL CURVES
CHOI, YOUNGOOK; KIM, SEONJA;

Abstract
In this work we obtain conditions for nonspecial line bundles on general $\small{{\kappa}}$-gonal curves failing to be normally generated. Let L be a nonspecial very ample line bundle on a general $\small{{\kappa}}$-gonal curve X with $\small{{\kappa}{\geq}4}$ and $\small{deg\mathcal{L}{\geq}{\frac{3}{2}}g+{\frac{g-2}{{\kappa}}}+1}$. If L fails to be normally generated, then L is isomorphic to $\small{\mathcal{K}_X-(ng^1_{\kappa}+B)+R}$ for some $\small{n{\geq}1}$, B and R satisfying (1) $\small{h^0(R)=h^0(B)=1}$, (2) $\small{n+3{\leq}degR{\leq}2n+2}$, (3) $\small{deg(R{\cap}F){\leq}1}$ for any $\small{F{\in}g^1_k }$. Its converse also holds under some additional restrictions. As a corollary, a very ample line bundle $\small{\mathcal{L}{\simeq}\mathcal{K}_X-g^0_d+{\xi}^0_e}$ is normally generated if $\small{g^0_d{\in}X^{(d)}}$ and $\small{{\xi}^0_e{\in}X^{(e)}}$ satisfy $\small{d{\leq}{\frac{g}{2}}-{\frac{g-2}{\kappa}}-3}$, supp$\small{(g^0_d{\cap}{\xi}^0_e)={\phi}}$ and deg$\small{(g^0_d{\cap}F){\leq}{\kappa}-2}$ for any $\small{F{\in}g^1_k}$.
Keywords
general $\small{{\kappa}}$-gonal curve;normal generation;nonspecial line bundle;Clifford index;
Language
English
Cited by
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