The definition of projective variety is equivalent to the locus of zero set of homogeneous polynomials that generate a prime ideal in the algebraically closed field they are a subset of (polynomial ring).
Now, what about the variety generated by the locus of zero set of homogeneous polynomials that do not necessarily generate a prime ideal, but are a groebner basis for the ideal they generate? Are these varieties specially studied? Can these be decomposed to projective varieties? Thanks beforehand.