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Normal generation and Clifford index on algebraic curves

Authors:	Youngook Choi Seonja Kim Young Rock Kim

Institution:	(1) Department of Mathematics Education, Yeungnam University, 214-1 Daedong Gyeongsan, 712-749 Gyeongsangbuk-do, South Korea;(2) Department of Digital Broadcasting and Electronics Engineering, Chungwoon University, Chungnam, 350-701, South Korea;(3) Graduate School of Education, Hankuk University of Foreign Studies, 130-791 Seoul, South Korea;(4) Division of Foundational Mathematics, National Institute for Mathematical Sciences, 305-340 Daejeon, South Korea

Abstract:	For a smooth curve C it is known that a very ample line bundle ${\mathcal{L}}$ on C is normally generated if Cliff( ${\mathcal{L}}$ ) < Cliff(C) and there exist extremal line bundles ${\mathcal{L}}$ (:non-normally generated very ample line bundle with Cliff( ${\mathcal{L}}$ ) = Cliff(C)) with ${h^{1}(\mathcal{L}) \le 1}$ . However it has been unknown whether there exists an extremal line bundle ${\mathcal{L}}$ with ${h^{1}(\mathcal{L}) \ge 2}$ . In this paper, we prove that for any positive integers (g, c) with g = 2c + 5 and ${c \equiv 0}$ (mod 2) there exists a smooth curve of genus g and Clifford index c carrying an extremal line bundle ${\mathcal{L}}$ with ${h^{1}(\mathcal{L}) = 2}$ . In fact, a smooth quadric hypersurface section C of a general projective K3 surface always has an extremal line bundle ${\mathcal{L}}$ with ${h^{1}(\mathcal{L}) = 2}$ . More generally, if C has a line bundle ${\mathcal{M}}$ computing the Clifford index c of C with ${(3c/2) + 3 < {\deg} \mathcal{M} \leq g-1}$ , then C has such an extremal line bundle ${\mathcal{L}}$ . For all authors, this work was supported by Korea Research Foundation Grant funded by Korea Government (MOEHRD, Basic Reasearch Promotion Fund)(KRF-2005-070-C00005).

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