A note on Property Np

Let M be a very ample line bundle on a smooth complex projective variety Y and let ϕ _M :Y→P(H ⁰(Y, M)^*) be the map associated to M; we are concerned with the problem to see whether the syzygies of ϕ _M (Y) give information on the syzygies of ϕ _M ^s (Y). In particular we prove that if Y is a smooth complex projective variety and M is a line bundle on Y satisfying Property N _p , then M ^s satisfies Property N _p if s≥p. Received: 11 June 1999 / Revised version: 22 November 1999