Different authors seem to have different conventions when they define the term affine variety (similarly projective variety). For the purposes of this question let us stick with the affine case, and let us work over an algebraically closed field. For example:
- In Harris's Algebraic Geometry: A First Course, an affine variety is the zero set in the affine space, of a collection of polynomials. So, it is just a closed subset of the affine space under the Zariski topology. He calls an irreducible closed subset, an irreducible affine variety. (A similar convention is used in the book by Cox, Little and O'Shea)
- In Hartshorne's Algebraic Geometry, a closed subset of the affine space is called an affine algebraic set, and an irreducible closed subset is called an affine variety.
- In the recent book Algebraic Geometry I: Schemes With Examples by Goertz and Wedhorn, the authors use the terms affine algebraic set, and irreducible affine algebraic set for closed and irreducible closed subsets of the affine space respectively. They reserve the term affine variety for a space with functions that is isomorphic to a space with functions associated to an irreducible affine algebraic set (so, this is more in the spirit of Hartshorne).
While it is usually clear from the context, what the authors of a particular book mean, when they use the terms above in bold, why are there different terminologies? Is there a consensus among mathematicians today, as to what they mean when they use the term affine algebraic variety?