Normally generated line bundles on general curves, II

Let C be a general curve of genus g≥3. Here we prove that there is a normally generated L∈Picd(C) such that h0(C,L)=r+1≥4 (i.e. a very ample line bundle which embeds C in Pr as a projectively normal curve) if and only if (r+1)h1≤g≤r(r−1)/2+2h1, where h1≔g+r−d=h1(C,L).