Syzygies of adjoint linear series on projective varieties

Syzygies of algebraic varieties has long been a topic of intense interest among algebraists and geometers alike. After the pioneering work of Mark Green on curves, numerous attempts have been made to extend these results to higher dimensions. It has been proposed that the syzygies of adjoint linear series L=K+mA, with A ample is the natural analog for higher dimensions. The very ampleness of adjoint linear series is not known even for threefolds. So the problem that has been open for 30 years is the following (problem): If A is ample and basepoint-free, does L satisfy property Np for m≥n+1+p? Ein and Lazarsfeld proved this when A is very ample in 1993. We give a positive answer to the problem above.