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.
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Notes
- 1.
Quite generally, \(\mathcal{O}_{X,Y}\) is a subring of the direct product of function fields k(X i ) of the irreducible components X i of X that meet Y, that is,
$$\mathcal{O}_{X,Y}\subset\prod_{Y\cap X_i\ne\emptyset}k(X_i). $$We can also view it as a quotient of the local ring \(\mathcal{O}_{\mathbb{A}^{n},Y}\) of rational functions on the ambient space regular on a dense open set of Y, modulo the ideal \(\mathfrak{a}_{X}\mathcal{O}_{\mathbb{A}^{n},Y}\) of functions vanishing on X. In discussing rational maps and rational functions as in Chapter 1, a point to grasp is that rational functions are defined as fractions, and the locus where they are regular is determined subsequently; otherwise you have to worry about when two functions or maps with different domains are equal (for example, is the function z/z with a removable singularity equal to 1?).
- 2.
The term smooth is used interchangeably with nonsingular in the current literature. The first English edition of this book used the archaic term simple, which goes back to Zariski, and is a literal translation of the Russian, but is not in current use.
- 3.
This notion appears in the literature under many other names: σ-process, monoidal transformation, dilation, quadratic transformation, etc.
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Shafarevich, I.R. (2013). Local Properties. In: Basic Algebraic Geometry 1. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-37956-7_2
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