Algebraic variety

Projective varieties are also equipped with the Zariski topology by declaring all algebraic sets to be closed.

Hilbert's Nullstellensatz says that closed subvarieties of an affine or projective variety are in one-to-one correspondence with the prime ideals or homogeneous prime ideals of the coordinate ring of the variety.

in the 2-dimensional affine space (over a field of characteristic not two). It has the associated cubic homogeneous polynomial equation: