Normal generation and Clifford index on algebraic curves

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 \({\mathcal{L}}\).