# ag.algebraic geometry – Linearly independent quadratic forms vanishing on a finite set of points

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 $$(n,m,k) = (4,9,6)$$.