On syzygies of Segre embeddings

We study the syzygies of the ideals of the Segre embeddings. Let $d \in {\mathbf N}$, $d \geq 3$; we prove that the line bundle ${\mathcal O}(1,...,1)$on the $P^1 \times ... \times P^1$ ($d$ copies) satisfies Property $N_p$ of Green-Lazarsfeld if and only if $p \leq 3$. Besides we prove that if we have a projective variety not satisfying Property $N_p$ for some $p$, then the product of it with any other projective variety does not satisfy Property $N_p$. From this we also deduce other corollaries about syzygies of Segre embeddings.