Let \(X\) be a projective variety and \(L\) a globally generated line bundle on \(X\). Using the notion of multigraded Castelnuovo-Mumford regularity, the authors give a cohomological criterion for \(L\) to satisfy the property \(N_p\). Several corollaries are obtained in the case when \(X\) is a toric variety. For instance, it is shown that if \(L\) is an ample line bundle on \(X\), then \(L^d\) satisfies \(N_p\) for any \(d \geq \dim X - 1 + p\). As another corollary, if \(X = {\mathbb P}^{n_1} \times \dots \times {\mathbb P}^{n_l}\), then the Veronese-Segre embedding \(\mathcal{O}_X (d_1,\dots,d_l)\) satisfies \(N_p\) for \(p \leq \text{min} \{ d_i : d \neq 0 \}\).
As the authors point out, their results are not sharp in general. For example, it is known that \(\mathcal{O}_{{\mathbb P}^2} (d)\) satisfies \(N_{3d-3}\) for \(d \geq 3\) and it is an open conjecture that the same should hold for any \({\mathbb P}^n\), \(n \geq 2\) [see D. Eisenbud, M. Green, K. Hulek and S. Popescu, Compos. Math. 141, No. 6, 1460–1478 (2005; Zbl 1086.14044)]. Motivated by L. Ein and R. Lazarsfeld [Invent. Math. 111, No. 1, 51–67 (1993; Zbl 0814.14040)], the authors also study the \(N_p\) property for adjoint bundles on a Gorenstein toric variety.