For given $n,m \geq 1$, consider the set of $n$-tuples of quadratic forms $(Q_1, \cdots, Q_n)$ with $Q_j \in \mathbb{C}[x_0, \cdots, x_5]$ for $j = 1, \cdots, n$, which vanish at a fixed set of points $\{\mathbf{y}^{(1)}, \cdots, \mathbf{y}^{(m)}\} \subset \mathbb{P}^4(\mathbb{C})$. That is, for each $1 \leq i \leq n$ and $1 \leq j \leq m$, we have $Q_i(\mathbf{y}^{(j)}) = 0$.
For $(n,m) = (9,6)$ and $(14, 6)$, how do we characterize the set of $n$-tuples $(Q_1, \cdots, Q_n)$ vanishing at a prescribed set of $m$ points?