Equations of the multi-Rees algebra of fattened coordinate subspaces

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