Equations of the multi-Rees algebra of fattened coordinate subspaces

In this paper we describe the equations defining multi-Rees algebra $k[x_1,\dots,x_n][I_1^{a_1}t_1,\dots,I_r^{a_r}t_r]$, where ideals are generated by subsets of $x_1,\dots,x_n$. We also show that a family binomials whose leading terms squrefree, form Gr\"{o}bner basis for with lexicographic order. if remove include $x$'s, then remaining toric ideal associated to multi-fiber ring. However binomials, including in not necessarily corresponding symmetric algebra. Despite fact, is type.