One example is the close relationship between homogeneous polynomials and projective varieties.

A cubic surface is a projective variety studied in algebraic geometry.

This is because only the constants are globally regular functions on a projective variety.

Some properties of a projective variety follow from completeness.

For example, if X is a projective variety over k, then .

Next, one can define projective and quasi-projective varieties in a similar way.

Every projective variety is complete, but not vice versa.

So, for the purposes of birational classification, we can work only with projective varieties, and this is usually the most convenient setting.

This is important to know, but it can still be very hard to determine whether two smooth projective varieties are birational.

This is a property shared by sufficiently positive embeddings of any projective variety.