(Hilbert polynomial)
For \(X \subseteq \PP^n\) closed with homogeneous coordinate ring \(S(X) = k[x_0,\ldots,x_n]/I(X)\), the Hilbert function is \[ h_X(r) \da \dim_k S(X)_r . \] For \(r \gg 0\) it agrees with a polynomial \(P_X(r)\), the Hilbert polynomial of \(X\).
For a coherent sheaf \(\mcf\) on a closed subscheme \(X \subseteq \PP^n_k\), the Hilbert function of \(\mcf\) is \(h_\mcf(m) \da h^0(X, \mcf(m))\).
For \(m \gg 0\), \(h_\mcf(m) = \chi(X, \mcf(m)) = P_\mcf(m)\), the value of the Hilbert polynomial of \(\mcf\) (D-COHEULER), and \(P_X = P_{\OO_X}\). [@Har10a, Exercise III.5.2]
(Reading off the coefficients)
Write \(\dim X = d\). Then \(\deg P_X = d\), and
the leading coefficient is \(\deg(X)/d!\); this defines the degree of \(X\) [@Har10a, §I.7];
the constant term is \(P_X(0) = \chi(\OO_X)\), and the arithmetic genus is \(p_a(X) = (-1)^d \qty(P_X(0) - 1)\), which is \(1 - \chi(\OO_X)\) for a curve.
The leading coefficient, and hence the degree, depends on the embedding: \(\PP^1\) embedded by \(\OO(3)\) has degree \(3\). The constant term \(\chi(X,\OO_X)\) and the arithmetic genus do not depend on the embedding [@Har10a, Exercise III.5.2].
(Hilbert polynomials of $\PP^n$ and the twisted cubic)
\(P_{\PP^n}(r) = \binom{r+n}{n}\), of degree \(n\) and leading coefficient \(1/n!\), so \(\deg \PP^n = 1\) and \(p_a(\PP^n) = 0\).
On \(\PP^1\), one has \(h^0(\PP^1,\OO(m))=m+1\) for \(m\ge0\) and zero for \(m<0\) [@Har10a, Proposition II.5.13]. Taking \(m=3r\) for \(r\ge0\) gives the Hilbert polynomial of the twisted cubic, \(P(r)=3r+1\), with degree \(3\) and arithmetic genus \(0\).
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Backlinks
PR-JR7TSA hypersurface of equation-degree $d$ has degree $d$usesD-VARDEGDegree as an intersection countusesD-HILBSCHEMEThe Hilbert functor and the Hilbert schemeusesD-G1AEHThe genus in its several sensesuses