On the projective normality of complete linear series on an algebraic curve (Q1076747)

Main result of this beautiful paper: Let L be a very ample line bundle on a smooth curve X of genus g, \(h^ 0(L)=r+1\), \(\deg (L)=2g+1-k\) with \(2k+1\leq g\) and let \(Y\subset {\mathbb{P}}^ r\) be the embedding of X determined by L; \((a)\quad L\) is not normally generated if and only if there is n, \(1\leq n\leq r-2\), and an effective divisor D on Y, \(\deg (D)\geq 2n+2\), such that \(H^ 1(Y,L^ 2(-D))=0\) and D spans an n-plane in which D fails to impose independent conditions on quadrics; \((b)\quad if\) \(\deg (L)\geq 2g+1-2h^ 1(L)-Cliff(X)\), then L is normally generated. Furthermore if \(g>>Cliff(X)\), the authors classify the line bundles with \(\deg (L)=2g-2h^ 1(L)-Cliff(X)\) and not normally generated. They discuss several conjectures on syzygies of projective curves and the Clifford index of curves. [Conjecture 3.8 is now proved by the reviewer; see Proc. Am. Math. Soc. 97, 217-218 (1986; Zbl 0591.14020).]