Normal generation of special line bundles on multiple coverings (Q6594876)

Let \(f : X\to Y\) be a covering of smooth projective curves defined over an algebraically closed field of characteristic \(0\). Let \(g\) be the genus of \(X\). A very ample line bundle \(L\) on \(X\) is said to be normally generated if the image of \(X\) by its embedding is projectively normal. For larger pairs of degrees of degrees and genera it is better to only look at the quadrics in the embedding. An effective divisor \(R\) is said to be minimally RIIF it fails for the quadrics, but all \(R'\subsetneq R\) give independent conditions to quadrics. In [\textit{Y. Choi} and \textit{S. Kim}, Commun. Algebra 50, No. 11, 4592--4609 (2022; Zbl 1495.14009)] considered the classifications of minimally RIIF for non-special line bundles on \(X\). \N\NIn the paper under review the authors consider the case of special line bundles. They give the right definition for this task, the Clifford index of \(f\) which should also be used elsewhere, since it is a nice notion. We recall that the Clifford index \(\mathrm{Cliff}(X)\) of \(X\) is the maximal integer \(\mathrm{Cliff}(L):= \deg(L) -2h^0(L)+2\) among all line bundles \(L\) with \(h^0(X,L)\ge 2\) and \(h^1(X,L)\ge 2\). The Clifford index \(\Delta_f\) of \(f\) is the minimum integer \(m\le g-1\) such that for all base point free ample \(L\) with \(\deg(L)\le g-1\) and \(\mathrm{Cliff}(L)\le m\) there is a line bundle \(R\) on \(X\) with \(L =f^\ast(R)\) and \(h^0(X,L) =h^0(Y,R)\). Their main theorem says that, under certain numerical assumptions, every very ample special line bundle \(L\) on \(X\) with \(\deg(L) >(3g-3)/2\) and \(\mathrm{Cliff}(L)\le \Delta_f\) is normally generated.They proved a general result for RIIF even for non-special line bundles.