Vanishing of weight one syzygies of projective varieties

In this article we study conditions under which weight one Koszul cohomology vanishes on projective varieties. As corollary of more general results, we obtain statements on the so-called property (M_q) reflecting on the higher syzygies of minimal surfaces and higher dimensional projective varieties. By considering both properties (M_q) and (N_p), we gain a significantly deeper understanding of the minimal free resolution.