Remarks on normal generation of special line bundles on multiple coverings (Q525187)

Let \(X\) be a smooth curve of genus \(g\) and \(L\) a very ample line bundle on \(X\). The author proves that \(L\) is normally generated if \(g>78\), \(h^1(L) =1\) and \(\deg (L) =2g-10\), unless either \(X\) is a double covering or a triple covering of a plane degree \(4\) curve with \(L\) related to the covering. If \(h^1(L)\geq 2\) and \(\deg (L) \geq 2g-3-6h^1(L)\) (resp. \(\deg (L) = 2g-4-6h^1(L)\)), the author proves that \(L\) is normally generated if \(X\) is not a double covering (resp. neither a double covering nor a triple covering). In the introduction the author also describes the state of the art on this topic.