On the projective normality of line bundles of extremal degree (Q1945154)

Let \(C\) be a general smooth curve of genus \(g\) and \(L\) a general line bundle on \(C\) with Clifford dimension \(\mathrm{Cliff}(L)\) equal to the Clifford index \(\mathrm{Cliff} (C)\) of \(C\) (it implies \(h^1(L)\leq 2)\). If \(h^1(L)=0\), then \(L\) embeds \(C\) as a projectively normal curve [\textit{E. Ballico} and \textit{Ph. Ellia}, Math. Z. 188, 215--223 (1985; Zbl 0631.14029)]. Here the author proves the cases \(h^1(L) =1, 2\). The result seems to be covered by \textit{E. Ballico} and \textit{C. Fontanari} [J. Pure Appl. Algebra 214, No. 6, 837--840 (2010; Zbl 1184.14039); J. Pure Appl. Algebra 214, No. 8, 1450--1455 (2010; Zbl 1185.14028)], but the proof is new and interesting. It uses deformation theory (at all orders, not just a tangent space computation) taking as \(C\) a smoothing of a general curve of compact type \(X\cup Y\) with two components. The line bundle on \(X\cup Y\) has bad postulation, but its badness is killed at a finite order of the deformation.