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Local Properties

In: Basic Algebraic Geometry 1

Author

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  • Igor R. Shafarevich

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

Abstract

A variety has a local ring at a point. It also has a tangent space, that is determined from the local ring. Singular and nonsingular points are characterised in terms of the tangent space. A nonsingular variety is locally factorial, which means that a codimension 1 subvariety is locally given by one equation. Next comes a discussion of birational maps and birational equivalence, exemplified by the idea of a blowup. Normal varieties are defined by the algebraic idea that the affine coordinate rings are integrally closed. This implies that the singular locus has codimension at least 2, so that every codimension 1 subvariety defines a valuation of the field of fractions. Finally, the notion of singularity and ramification are extended to regular maps between varieties.

Suggested Citation

  • Igor R. Shafarevich, 2013. "Local Properties," Springer Books, in: Basic Algebraic Geometry 1, edition 3, chapter 0, pages 83-146, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-37956-7_2
    DOI: 10.1007/978-3-642-37956-7_2
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