 Homogeneous coordinate ring

For an embedded projective variety $X\subseteq {\mathbb{P}}_k^n$, let $I(X)$ be its homogeneous vanishing ideal. Its homogeneous coordinate ring is the graded ringThe grading records polynomial degree, and $\mathrm{Proj}S(X)$ recovers the embedded projective variety. The affine scheme $\mathrm{Spec}S(X)$ is its affine cone.

The chosen homogeneous ideal matters: different graded quotients can have the same Proj construction. For the usual degree-one polynomial grading, saturating an ideal by the irrelevant ideal recovers the ideal determined by the associated closed closed subscheme. Thus an arbitrary presentation $X=V_{+}(I)$ should not silently identify $I$ with $I(X)$.^[ref]

The ring also depends on the projective embedding. A projective line in its standard embedding has homogeneous coordinate ring $k[s,t]$, while its degree-two Veronese map image has homogeneous coordinate ring $k[X,Y,Z]/(XZ-Y^2)$.

The homogeneous coordinate ring is a mathematical construct used primarily in algebraic geometry and projective geometry. It provides a way to systematically handle projective space and the geometric objects that reside within it, such as points, lines, and higher-dimensional varieties. ### Definition Consider projective space \(\mathbb{P}^n\) over a field \(k\).