 Homogeneous coordinate ring

For $X=V_{+}(I)\subseteq {\mathbb{P}}^n$, the homogeneous coordinate ring is the graded ring $k[x_0,\dots ,x_n]/I$.

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 \(k\).