Consider the space of all homogeneous degree $d$ polynomials in three variables $[X,Y,Z]$, i.e. $$ f[X,Y,Z] = f_{300} X^d + f_{210} X^{d-1}Y + \ldots .$$ 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.

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.

Consider the space of all homogeneous degree $d$ polynomials in three variables $[X,Y,Z]$, i.e. $$ f[X,Y,Z] = f_{300} X^3 + f_{210} X^2Y + f_{111} X Y Z + \ldots .$$ 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.

Consider the space of all homogeneous degree $d$ polynomials in three variables $[X,Y,Z]$, i.e. $$ f[X,Y,Z] = f_{300} X^d + f_{210} X^{d-1}Y + \ldots .$$ 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.