Hilbert Polynomial at a point? One of the review problems in my final review is the following:

Let $X\subset\mathbb P_{\mathbb C}^n$ be a hypersurface, and $P\in X$ a singular point. Let $L$ be a line not contained in $X$ that intersects $X$ at $P$.
Prove that $h_{X\cap L}(P)\geq2$, the intersection multiplicity of $X$ and $L$ at $P$, where $h_{X\cap L}(P)$ refers to the hilbert polynomial of $X\cap L$ at $P$.

From my readings, it doesn't seem like the hilbert polynomial accepts a point as input, but rather a natural number. So what does this even mean?

The following is a community wiki answer recording the discussion in the comments so that this question might be marked as answered (once this answer is upvoted or accepted).

Hm yeah I am not sure that this makes sense? Not an expert though. Could they mean like intersection number of $X\cap L$ at $P$? – user113102
Oh it even says that in the question lol. Yeah I think the stuff after the semicolon is just a typo or they forgot to delete it or something. – user113102

