Hilbert Function

Let $\Gamma = \{p_1, \ldots, p_m\}\subset\Bbb{P}^2$ be a collection of $m$ distinct points. Then the number of conditions imposed by $\Gamma$ on forms of degree $d$ is called the Hilbert function $h_\Gamma$ of $\Gamma$. If curves $X_1$ and $X_2$ of degrees $d$ and $e$ meet in a collection $\Gamma$ of $d\cdot e$ points, then for any $k$, the number $h_\Gamma(k)$ of conditions imposed by $\Gamma$ on forms of degree $k$ is independent of $X_1$ and $X_2$ and is given by

$$h_\Gamma(k)={k+2\choose 2}-{k-d+2\choose 2}-{k-e+2\choose 2}+{k-d-e+2\choose 2},$$

where the Binomial Coefficient ${a\choose 2}$ is taken as 0 if $a<2$ (Cayley 1843).

References

Eisenbud, D.; Green, M.; and Harris, J. ``Cayley-Bacharach Theorems and Conjectures.'' Bull. Amer. Math. Soc. 33, 295-324, 1996.