In $\mathbb P_{n+1}$ we consider $d$ varieties $V_1,\ldots V_d$, each of them is defined by $d-1$ equations, as follows: we have $d-1$ polynomials $f_2(X_0,\ldots,X_n),\ldots,f_d(X_0,\ldots,X_n)$ and $d\cdot(d-1)$ nonzero coefficents $l_{ij}$ ($i=2,\ldots,d$, $j=1,\ldots,d$). The variety $V_j$ is then defined by the polynomials $f_i(X_0,\ldots,X_n)-l_{ij}X_{n+1}^i$ for $i=2,\ldots,d$. Finally, we consider the projection $\pi:\mathbb P_{n+1}\rightarrow\mathbb P_n$ onto the first $n+1$ coordinates.
My questions are: how do I compute the dimension of $\bigcap_{j=1}^d\pi(V_j)$? How can I relate this dimension to other quantities which could be easier to calculate, such as the dimension of the $V_j$'s?
Answers with assumptions on the given data are as welcome as more general ones.