linear dependencies. This is done by using the local dependencies to construct rows of a design matrix $M$, showing that its rank is high and then arguing that the dimension of the original set is small since it must lie in the kernel of $M$.

Suppose we have a configuration of points with many $r$-rich lines. Clearly, $r \geq 3$ collinear points are also linearly dependent. However, this conclusion does not use the fact that $r$ may be larger than 3. To use this information, we observe that a certain map, called the Veronese embedding, takes $r$-collinear points to $r$ linearly dependent points in a larger dimensional space (see Section \[sec-veronese-embedding\]). Thus we can create a design matrix using these linear dependencies similarly to the constructions of [@BDWY11; @DSW12] to get an upper bound on the dimension of the [*image*]{} of the original set, under the Veronese embedding. We use this upper bound to conclude that there is a polynomial of degree $r-2$ which contains all the points in our original configuration. We then proceed in a way similar to the proof of the Joints conjecture by Guth and Katz [@GK10b] to conclude that there is a hyperplane which contains 