Let $k$ be a field, $R=k[x_1,\dots,x_r]$ the polynomial ring in $r$ indeterminates and $I_1,I_2\dots,I_d$ homogeneous ideals of $R$ generated by linear forms. Define $J = I_1 I_2 \cdots I_d$ and suppose that $\dim R/J = 0$.
Question: Is it true that $(R/J)_d = 0$? If yes, how can we see that?
Since the $I_i$ are generated by homogeneous linear forms, they are prime ideals. Since $J \subset I_i$, you have that $$\mathrm{ht}~J \leq \mathrm{ht}~I_i$$ for all $i$. Since you claim that $J$ has codimension $r$, it must be the case that each $I_i$ has height $r$, which in turn means it is equal to the homogenous maximal ideal $M =(x_1,\ldots,x_r)$. Now the result follows since $(R/M^d)_d =0$.