What is 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

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

is the polynomial ring in N + 1 variables Xi. 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. The choice of basis means this definition is not intrinsic, but it can be made so by using the symmetric algebra.