Normal generation of line bundles of degree \(2g-2h^1(L)-\mathrm{Cliff}(X)-k(k=2,3,4)\) on curves (Q2479946)

Let \(X\) be a smooth projective algebraic curve of genus \(g \geq 4\) defined over an algebraically closed field of characteristic zero, and let \(L\) be a very ample line bundle on \(X\); one says that \(L\) is normally generated if \(X\) is projectively normal under the associated embedding induced by the linear system \(| L| \). \textit{M. Green} and \textit{R. Lazarsfeld} [Invent. Math. 83, 73--90 (1985; Zbl 0594.14010)] proved that if \(\deg(L) \geq 2 g - 2 h^1(L) - \text{Cliff}(X) + 1\) then \(L\) is normally generated; here \[ \text{Cliff}(X) := \min \{ \deg (A) - 2 h^0(A) + 2 \; | \; A \text{ a line bundle such that } h^0(A) \geq 2 \text{ and }h^1(A) \geq 2\} \] is the so-called Clifford index of \(X\). In that same paper, the authors gave necessary and sufficient condition for a very ample line bundle of degree \(2 g - 2 h^1(L) - \text{Cliff}(X)\) to be normally generated. In [Abh. Math. Sem. Univ. Hamburg 75, 77--82 (2005; Zbl 1083.14033)] the author of the present paper determined necessary and sufficient conditions for a very ample line bundle \(L\) of degree \(2 g - 2 h^1(L) - \text{Cliff}(X) - 1\) to be normally generated, and in the present paper he determines necessary conditions for \(L\) to not be normally generated assuming that \(\deg (L) = 2 g - 2 h^1(L) - \text{Cliff}(X) - k\), with \(k \in \{2, 3, 4\}\).