Remarks on normal generation of line bundles on algebraic curves (Q1766179)

Let \(L\) be a very ample line bundle of degree \(d\) on a smooth projective curve \(\mathcal X\) of genus \(g\geq 2\) over an algebraically closed field of characteristic zero. \textit{M. Green} and \textit{R. Lazarsfeld} [Invent. Math. 83, 73--90 (1986; Zbl 0594.14010)] showed that \(L\) is normally generated whenever \[ d\geq 2g+1-2h^1(L)-\text{Cliff}(\mathcal X). \] For \(d=2g,2g-1\) they also characterized the normally generated property of \(L\). A similar result for nonspecial linear bundles with \(d=2g-2,2g-3\) was proved by \textit{T. Kato, C. Keem} and \textit{A. Ohbuchi} [Abh. Math. Sem. Univ. Hamburg 69, 319--333 (1999; Zbl 0969.14017)]. The first goal of this paper is to extend the aforementioned properties to the cases \(d=2g-4,2g-5\). The second goal is to extend results in a paper of the author [Tsukuba J. Math. 22, 213--225 (1998; Zbl 0965.14015)] and Kato et al. [loc. cit.] concerning the characterization of the normally generated property of special line bundles of degree \(d=2g-7,2g-8,2g-9\).