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I'm reading my Algebraic geometry textbook and wanted to know if they interchange the words varieties and affine varieties. Just beginning my studies and didn't want to make any assumptions.

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Which book? It depends very much on local conventions. But any serious book on the subject will want to talk about projective varieties, which are not affine outside of trivial cases. More generally one will want objects that look locally like affine varieties, but this usually waits until schemes have been introduced. –  Dylan Moreland Jul 23 '12 at 23:07
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Usually a variety is defined as something locally an affine variety. For example, projective space is a variety, but it is not an affine variety. (but the answer to this question really depends upon the convenctions of your textboox) –  Fredrik Meyer Jul 23 '12 at 23:08

1 Answer 1

It does depend on the definition of variety given in your textbook. Following the conventions of Hartshorne's book Algebraic Geometry (wich is the canonical reference on the subject) he defines over an algebraic closed set $k$ a variety over $k$ as any affine, quasi-affine, projective or quasi-projective variety. Where we say that a variety is

  1. Affine if is an irreducible closed subset of $\mathbb{A}^n$ with the Zariski topology.
  2. Quasi-affine if is an open subset of an affine variety.
  3. Projective if is an irreducible subset of $\mathbb{P}^n$ with the Zariski topology.
  4. Quasi-projective if is an open set of a projective variety.

So you cannot interchange those terms freely. Every time you start reading an Algebraic Geometry book you should be careful checking what notation and conventions they are following, because they change a lot between authors.

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