On syzygies of projective varieties
Let \(X\) be a projective variety over an algebraically closed field of characteristic zero. A line bundle \({\mathcal L}\) on \(X\) is said to have property \(N_p\) for some \(p>0\), if the ideal defining the embedding of \(X\) in the complete linear system of \({\mathcal L}\) is quadratically generated and has linear syzygies up to the \(p\)-th step. The author proves that for ample \({\mathcal L}\) and any \(p>0\), a sufficiently high power \({\mathcal L}^n\) has property \(N_p\), thereby generalizing the analogous result by \textit{L. Ein} and \textit{R. Lazarsfeld} in the smooth case [cf. Invent. Math. 111, No. 1, 51-67 (1993; Zbl 0814.14040)]. He expresses the hope that the explicit nature of the vanishing conditions used in his proof will lead to effective bounds for the required size of \(n\).
scientific article; zbMATH DE number 1054232