For a smooth curve C it is known that a very ample line bundle L">L on C is normally generated if Cliff(L">L ) < Cliff(C) and there exist extremal line bundles L">L (:non-normally generated very ample line bundle with Cliff(L">L ) = Cliff(C)) with h1(L)≤1">h1(L)≤1 . However it has been unknown whether there exists an extremal line bundle L">L with h1(L)≥2">h1(L)≥2 . In this paper, we prove that for any positive integers (g, c) with g =？ 2c +？ 5 and c≡0">c≡0 (mod 2) there exists a smooth curve of genus g and Clifford index c carrying an extremal line bundle L">L with h1(L)=2">h1(L)=2 . In fact, a smooth quadric hypersurface section C of a general projective K3 surface always has an extremal line bundle L">L with h1(L)=2">h1(L)=2 . More generally, if C has a line bundle M">M computing the Clifford index c of C with (3c/2)+3<degM≤g−1">(3c/2)+3<degM≤g？1 , then C has such an extremal line bundle L">L .
