Lagmental Vicfred

Homogenization Adds the Hyperplane at Infinity by Vicfred

Last updated: Wed 06 November 2019

tags: mathematics, projective geometry, Proj, Hilbert polynomials

A polynomial of degree d becomes homogeneous after inserting powers of a new coordinate. The example is deliberately concrete; it is a test of the statement, not a substitute for it.

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.

$$ f(x_1,\ldots,x_n)=\sum_{|\alpha|\le d}c_\alpha x^\alpha $$

The typography mirrors the proof: first declare \(\mathsf D\), then state \(\mathsf C\). The symbol \(\Longrightarrow\) below is a logical dependency, not extra mathematical structure.

$$ f^h(x_0,\ldots,x_n)=\sum_\alpha c_\alpha x_0^{d-|\alpha|}x^\alpha $$

Stress the formula

The computation below is not a second theorem. It is a checksum for the definitions and a place to inspect the difficult LaTeX at full size.

$$ y-x^2=0\quad\leadsto\quad YZ-X^2=0,\qquad Z=0\Longrightarrow X=0 $$

Interpretation

What survives the example is not its particular numbers but the relation encoded by the two rows below. That relation is the part worth transporting to a new setting.

$$ \begin{aligned} \mathsf{D}\;&:\quad f(x_1,\ldots,x_n)=\sum_{|\alpha|\le d}c_\alpha x^\alpha,\\[5pt] \mathsf{C}\;&:\quad f^h(x_0,\ldots,x_n)=\sum_\alpha c_\alpha x_0^{d-|\alpha|}x^\alpha. \end{aligned} $$

Limit of the argument

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] f^h(x_0,\ldots,x_n)=\sum_\alpha c_\alpha x_0^{d-|\alpha|}x^\alpha \end{gathered}} $$

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 Thu 25 May 2017. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.