Dvir–Gopi rich-lines conjecture

Let $r\geq 2$, and let $\mathcal P\subset\mathbb{C}^d$ be a set of $n$ points. Write $\mathcal{L}_r(\mathcal P)$ for the set of lines incident to at least $r$ points of $\mathcal P$. Suppose that

$$|\mathcal{L}_r(\mathcal P)|\gg_d \frac{n^2}{r^{d+1}}+\frac{n}{r}.$$

Then there exists an integer $1<t<d$ and a subset $\mathcal P'\subset\mathcal P$ of size $\gtrsim_d n/r^{d-t}$ contained in a $t$-flat. Dvir–Gopi conjecture. The number of rich lines in a point set in $\mathbb{C}^d$ should force many points to lie in a lower-dimensional affine flat. The conjecture would sharpen the stated Dvir–Gopi theorem, whose bounds are not believed to be tight, and is presented as a tight bound in the source.