Syzygy algebras of the Segre embeddings

We will talk about syzygy algebras of projectivizations of highest weight orbits. In the paper (http://arxiv.org/abs/math/0602316) A.L.Gorodentsev, A.S. Khoroshkin, A.N.Rudakov find syzygy algebras of the Plucker embeddings of the Grassmannians $Gr(2,n)$. We will shortly discuss this result and in a slightly different way will find syzygy algebras of the Segre embeddings (http://arxiv.org/abs/1108.3733).
The sheaf $i_*O_X$ for an embedding $i\colon X\to\mathbb{P}^n$ of projective variety to projective space may be expressed by the sheaves $O(i)$, $-n\leqslant i\leqslant 0$ and syzygy spaces. In the same way we will find $GL(U)\times GL(V)$-equivariant resolutions for the sheaves $O(a,b)$ for the Segre embedding $\mathbb{P}(U)\times\mathbb{P}(V)\to\mathbb{P}(U\otimes V)$ and $a,b\geqslant0$.