It should be a basic question, but somehow, I doubt if it is doable by elementary methods. Fix a vector space $V$ over a field $k$ with $\dim V=r+1$. Consider the moduli space parametrizing $(V_1, \cdots,V_n)$ where $V_i\subset V$ and $\Sigma_{i=1}^n\dim(V_i)=(n-1)r+n$, with $\dim V_i=m_i$ also fixed for each $i$ (i.e. each $m_i$ is a fixed number).
We know if I pose $\dim(V_1\cap\cdots\cap V_n) < p+1$, then this condition cuts an open locus of this moduli space. Now my question is: what is the minimum $p$ such that this open condition cuts an non empty locus of this moduli space?
At first, I thought this might be done by basic linear algebra, but when $n\geq 3$, it is really messy if we just use basic linear algebra. And also note that even for $n=3$ $\dim(\text{span}(V_1, V_2, V_3))\neq \Sigma_{i=1}^3\dim(V_i)-\Sigma_{i\neq j}\dim{V_i\cap V_j}+\dim(V_1\cap V_2\cap V_3)$. This inequality makes basic linear algebra technique "almost" impossible. So I am wondering if there is some other approach which might be helpful.
