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In Hartshorne's book "Algebraic geometry", page XV ("Terminology") says: "all varieties in Chapter 1 are quasi-projective". Could anyone give me a detailed explanation why is this true?

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That assertion is not a theorem, it is just informing you that those are the only varieties consideredin Chapter I and that's what you should understand when you read the word variety within that chapter. If you look at Chapter I you will see that he defines (right after I.3.1.1) a variety to be "any affine, quasi-affine, projective or quasi-projective variety". Abstract varieties are introduced in IV.2.

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