I have a question about the second part of Exercise 5.13 in Harris' Algebraic Geometry: A First Course: Given $n \leq 2d + 1$ points in $\mathbb{P}^2$, characterize the subset of $(\mathbb{P}^2)^n$ for which the points do not pose independent conditions on polynomials of degree $d$ (where we say that the points pose independent conditions, if the space of homogeneous polynomials vanishing at the points has codimension $n$ in the space of homogeneous polynomials of degree $d$). The answer given in the Hints-section is that this is the case if and only if $d+2$ of the points are colinear.
I see, that if $d + 2$ of them are colinear, they don't pose independent conditions: The line through the $d+2$ points and a polynomial of degree $d-1$ passing through the remaining $n - d -2$ points will satisfy the conditions, thus the space of polynomials vanishing at the points has dimension at least $d(d + 1)/2 - (n - d - 2) = (d^2 + 3d + 4)/2 - n$, which is bigger than the dimension we would get for independent conditions, namely: $(d+1)(d + 2)/2 - n = (d^2 + 3d + 2)/2 - n$.
In the other direction, if I use the same argument as above and the bound on $n$, I see why the conditions imposed can not be dependent due to a curve of degree greater than one in the base locus of the linear system. But this does not cover all the cases, how does one show in general, that if the conditions are dependent, $d + 2$ of the points are colinear?