Question

Consider the space of all homogeneous degree $d$ polynomials in three variables $[X,Y,Z]$, i.e. $$ f[X,Y,Z] = f_{d00} X^d + f_{d10} X^{d-1}Y + \ldots .$$ This can be thought of as a section of the line bundle

$$ \gamma_{\mathbb{P}^2} ^{* d} \rightarrow \mathbb{P}^2 $$

Upto scaling this is the projective space $\mathbb{P}^{\delta_{d}}$, where $\delta_{d} = \frac{d(d+3)}{2}.$
Note that ``passing through a point '' imposes a linear condition on the coefficients $f_{ijk}$. Fix any number $k \leq \delta_{d}$. Is there an abstract way to show that there exist $k$ points $[X_1, Y_1, Z_1]\ldots, [X_k, Y_k, Z_k]$ such that passing through those $k$ points imposes linearly independent conditions on the coefficients? More precisely, the conditions will not be linearly independent implies that a certain determinant involving the $[X_i,Y_i, Z_i]$ is zero. How do I know that this determinant is not $\textit{always}$ zero?
One can explicitly write out the matrix whose determinant should not be zero and try to produce some explicit choice of points for which the determinant is not zero. But is there a way to avoid doing that? Does the Bertini theorem (or some modification of that) imply that there exists $k$ points so that passing through $k$ points imposes linearly independent conditions? In any case what is the simplest way to prove this seemingly obvious statement? Everything is over the complex numbers.

Answer 1

It is quite easy to give $k$ points imposing independent conditions. Just choose the points $p_1,\dots p_k$ one at a time as follows:

After choosing the first $j$ points for some $j<k$, take any curve $C$ passing through these $j$ points. Pick any point not lying on $C$. Call this point $p_{j+1}$ and continue. The effect is that the sets of curves passing through the first $j$ points are strictly decreasing sets as $j$ increases, which is all you that you want to achieve.

Answer 2

The simplest solution of the problem is very-well known in Approx. Theory.The desired construction is called Berzolari-Radon: One takes d+1 lines - l_0,...,l_d. Take any d+1 points on l_0. Next take d points on l_1 not lying on l_0, take d-1 points on l_2 not lying on l_0 and l_1, so on, finally take 1 point on L_d not lying on any other previous lines. Thus we get maximal number of d-independent points. Of course any its subset (of k points) is independent as well.