Homogeneous coordinate ring

In algebraic geometry, the homogeneous coordinate ring R of an algebraic variety V given as a subvariety of projective space of a given dimension N is by definition the quotient ring

R = K[X₀, X₁, X₂, ..., X_N] / I

where I is the homogeneous ideal defining V, K is the algebraically closed field over which V is defined, and

K[X₀, X₁, X₂, ..., X_N]

is the polynomial ring in N + 1 variables X_i. The polynomial ring is therefore the homogeneous coordinate ring of the projective space itself, and the variables are the homogeneous coordinates, for a given choice of basis (in the vector space underlying the projective space). The choice of basis means this definition is not intrinsic, but it can be made so by using the symmetric algebra.