Page "Algebraic variety" Paragraph 29

worrying whether the resulting object can be put into some projective space.

This also leads to difficulties since one can introduce somewhat pathological objects, e. g. an affine line with zero doubled.

Such objects are usually not considered varieties, and are eliminated by requiring the schemes underlying a variety to be separated.

( Strictly speaking, there is also a third condition, namely, that one needs only finitely many affine patches in the definition above.