Normal generation of line bundles of degree 2p-2 on curves
Let L be a very ample line bundle on a genus \(g\) curve C. A theorem of Green and Lazarsfeld asserts that L is normally generated if \(\deg (L)\geq 2g+1-2h^ 1(L)-\text{Cliff}(C)\) [\textit{M. Green} and \textit{R. Lazarsfeld}, Invent. Math. 83, 73-90 (1986; Zbl 0594.14010)]. Now assume that L is a general line bundle of degree \(2g-2\). The main theorem of this article shows that if C is nonhyperelliptic and \(g\geq 22\), then L is again normally generated. Using an earlier result of the authors [J. Reine Angew. Math. 356, 1-18 (1985; Zbl 0561.14009)] the main theorem is reduced to the case when C is double elliptic. Suppose that \(f: C\to E\) is the degree two mapping from C to an elliptic curve E. Then \(f_*L\) is a degree \(g-1\) vector bundle on E. Suppose that M is a subline bundle of maximal degree of \(f_*L\). Let \(N=f_*L/M\). It is shown that \(\deg M\geq (g/2)-1\) and \(\deg N\geq 3\). Since M and N are normally generated, it follows that L is also normally generated.
normal generation of very ample line bundle on curve
scientific article; zbMATH DE number 3983352