The question I am interested in can be summed up as follows: given positive integers $n,m,k$, how do we write down $m$ linearly independent quadratic forms $Q_1, cdots, Q_m in mathbb{C}(x_0, cdots, x_n)$ such that the common-zero locus

$$V = V_{Q_1, cdots, Q_m} = {mathbf{x} in mathbb{P}^n(mathbb{C}) : Q_i(mathbf{x}) = 0 text{ for all } 1 leq j leq m}$$

has size exactly equal to $k$?

Clearly, we require $m geq n$ in order for there to be any hope, since $V$ is a variety of co-dimension at most $m$ so when $m < n$ then $V$ has positive dimension and hence cannot be a finite set.

The two base cases I am considering are when $(n,m,k) = (2, 2, 4)$ and $(3,5,5)$. In the former case, any pair of linearly independent ternary quadratic forms will do: indeed, the intersection $V_{Q_1, Q_2}$ is a complete intersection in $mathbb{P}^2$ and consists of four points. When $(n,m,k) = (3,5,5)$ a beautiful construction gives that such a quintuple $(Q_1, cdots, Q_5)$ is given by the set of $4 times 4$ pfaffians of a $5 times 5$ skew-symmetric matrix of linear forms in $x_0, cdots, x_3$.

Are there general ways to parametrize such tuples for any other values of $(n,m,k)$? Specifically, I want to know whether such a parametrization exists when $(n,m,k) = (4,9,6)$.