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Varieties

In: Basic Algebraic Geometry 2

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  • Igor R. Shafarevich

    (Steklov Mathematical Institute of the Russian Academy of Sciences, Algebra Section)

Abstract

Scheme theory provides the modern definition of a variety, and a convenient language for all the constructions of algebraic geometry, ancient and modern. A variety over an algebraically closed field k is a separated reduced scheme of finite type over k. The general properties of quasiprojective varieties from Volume 1 of the book are reinterpreted in this intrinsic framework. There follows a comparison between varieties and projective varieties, including an intrinsic treatment of blowups, Chow’s lemma that any variety has a blowup that is quasiprojective, and a brief discussion of different criteria for a variety to be projective. On the other hand, an example is given (and illustrated on the front cover of Volume 2) of a complete variety that cannot be embedded in any projective space. The chapter also discusses in some detail two other circles of ideas: sheaves of modules, including locally free sheaves and coherent sheaves, and the idea of a scheme representing a functor, that plays an central role in the modern theory of moduli.

Suggested Citation

  • Igor R. Shafarevich, 2013. "Varieties," Springer Books, in: Basic Algebraic Geometry 2, edition 3, chapter 0, pages 49-111, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-38010-5_2
    DOI: 10.1007/978-3-642-38010-5_2
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