Consider the Hilbert polynomial for a projective scheme.
The degree, dimension and arithmetic genus extract information from the lowest term and the highest term in the polynomial. What about all other terms? It would seem they encode some more info about our scheme. I could not find any reference to these coefficients, though.
So my question is what else can we learn about our scheme from the Hilbert polynomial?


Thanks



What information Hilbert polynomial encodes other than dimension, degree and arithmetic genus?
For a curve, that's all of course, since the polynomial is linear.
Now let's say $X$ is a smooth surface with an ample divisor $H$ and canonical
divisor $K$, we have the Hilbert polynomial
$$\chi(\mathcal{O}\_X(nH))= \frac{1}{2}nH(nH-K) + \chi(\mathcal{O}\_X)$$
by Riemann-Roch. So the linear coefficient gives you the degree of the canonical
divisor. In higher dimensions, the more general form of Riemann-Roch
$$\chi(\mathcal{O}\_X(nH)) = \int\_X ch(\mathcal{O}(nH))td(X)$$
tells you that you're basically getting certain Chern numbers in $X$ and $H$ as coefficients.


What is perhaps simpler, is to use the recurrence formula
$$\chi(\mathcal{O}\_{H\cap Y}(nH)) =\chi(\mathcal{O}\_Y(nH))- \chi(\mathcal{O}\_Y((n-1)H))$$
to see that the Hilbert polynomial determines and is determined by the arithmetic genera of $X$ and
 the complete intersections $H$, $H\_1\cap H\_2$ etc.


