Suppose that $H_1$,...,$H_d$ are hyperplanes in $\mathbb P^n$ (over some field -- you can pick). For $k \geq n$, let $t_k$ denote the number of points through which there pass exactly $k$ hyperplanes.
I'd to know some inequalities among the $t_k$. They should say, roughly, that points with very large numbers of hyperplanes through them are relatively uncommon, and that most of the intersection points have few. I am flexible about the specific form of the inequalities, and not picky that they be very sharp.
I'm aware of a result of Hirzebruch in the case $n=2$, which says that (over $\mathbb C$), we always have $t_2 + \frac34 t_3 \geq d + \sum_{k=5}^{d-2} (2k-9) t_k$ (assuming $t_d = t_{d-1} = 0$). The proof, however, uses some highly nontrivial algebraic geometry specific to dimension $2$, and it is not apparent to me how to extend it to higher dimensions.
Are there similar constraints in higher dimensions? What are the keywords I should be looking for in the literature?
