Abstract
In algebraic geometry, divisors are codimension 1 subvarieties, whose significance is to serve as the locus of zeros and poles of rational functions. The chapter discusses Weil divisors and locally principal divisors (or Cartier divisors), and linear equivalence between them. A divisor gives rise to a linear system, that one visualises as a family of codimension 1 subvarieties parametrised by a vector space. Algebraic groups are varieties with a group law given by regular maps. This includes linear algebraic groups (or matrix groups), but also elliptic curves and their higher dimensional generalisations, the Abelian varieties. Differential forms are dual to tangent fields, and have many applications. A principal aim is to discuss the canonical class of a variety. The chapter includes a complete discussion and proof of the Riemann–Roch theorem for curves.
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Notes
- 1.
The current literature is inconsistent, some authors using \(\mathop{\mathrm{Div}}\nolimits X\) for the group of “ordinary” divisors ∑k i C i (Weil divisors) described here, some for locally principal divisors (Cartier divisors) (see Section 1.2). In case of ambiguity, one can write \(\operatorname{WDiv}\) or \(\operatorname{CDiv}\).
- 2.
The divisor \(\operatorname{div} f\) is also traditionally denoted by (f) in the literature.
- 3.
For a modern introduction to algebraic groups, including proofs of Theorems A–B, see Humphreys [40].
- 4.
The so-called “Principle of conservation of number” (roughly, algebraically equivalent cycles have the same numerical properties) is discussed in Fulton [29, Chapter 10], although the proof given there is very high-powered and abstract. Section 19.3 of the same book also contains a condensed discussion of the Néron–Severi theorem over \(\mathbb{C}\).
- 5.
Elements of Ω n[U] are called canonical differentials, following a suggestion of Mumford.
- 6.
κ=−1 also occurs in the literature. The invariant κ is usually called the Kodaira dimension, although it was introduced in different contexts by the Shafarevich seminar [69] and by Iitaka.
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Shafarevich, I.R. (2013). Divisors and Differential Forms. In: Basic Algebraic Geometry 1. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-37956-7_3
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