Lagmental Vicfred

Degree Is the Leading Coefficient of the Hilbert Polynomial by Vicfred

tags: mathematics, projective geometry, Proj, Hilbert polynomials

For a projective variety of dimension r, its degree appears after multiplying the leading coefficient by r factorial. The example is deliberately concrete; it is a test of the statement, not a substitute for it.

Objects and notation

A graded ring \(S=\bigoplus_{d\ge0}S_d\) produces \(\operatorname{Proj}S\), whose points are homogeneous primes avoiding the irrelevant ideal \(S_+\). Projective space is the basic example.

$$ P_X(t)=\chi(X,\mathcal O_X(t)) $$

There are two layers here: the object \(\mathsf D\) and the law \(\mathsf C\). Writing them separately makes the direction of \(\Longrightarrow\) visible and keeps an accidental converse from slipping in.

$$ P_X(t)=\frac{\deg X}{r!}t^r+O(t^{r-1}) $$

Push the symbols

The middle display is intentionally dense: it is where signs, bounds, multiplicities, or normalising factors are most likely to be lost.

$$ X=V(F)\subset\mathbf P^n,\quad\deg F=d\quad\Longrightarrow\quad P_X(t)=\binom{t+n}{n}-\binom{t+n-d}{n} $$

Structural reading

A good test for understanding is to change the presentation while keeping the invariant fixed. The aligned form makes that comparison unusually easy.

$$ \begin{aligned} \mathsf{D}\;&:\quad P_X(t)=\chi(X,\mathcal O_X(t)),\\[5pt] \mathsf{C}\;&:\quad P_X(t)=\frac{\deg X}{r!}t^r+O(t^{r-1}). \end{aligned} $$

A hypothesis worth keeping

Homogeneity is essential: a nonhomogeneous equation is not well defined under rescaling of projective coordinates.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] P_X(t)=\frac{\deg X}{r!}t^r+O(t^{r-1}) \end{gathered}} $$

I would use the boxed line as a reference later, while returning to the full display whenever a hypothesis becomes uncertain. That division keeps compression from becoming ambiguity.

This article was posted on Wed 13 March 2013. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.