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2017 | OriginalPaper | Buchkapitel

12. Algebraic Manifolds

verfasst von : Alexey L. Gorodentsev

Erschienen in: Algebra II

Verlag: Springer International Publishing

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Everywhere in this chapter we assume by default that the ground field \(\mathbb{k}\) is algebraically closed.

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Vorheriges Kapitel Affine Algebraic Geometry

Nächstes Kapitel Algebraic Field Extensions

Recall that \(\mathcal{O}_{Z}(V ) =\{\, f \in \mathbb{k}(Z)\mid V \subset \mathop{\mathrm{Dom}}\nolimits (\,f)\}\) denotes the algebra of rational functions, regular in V ⊂ Z, on an affine algebraic variety Z (see Sect. 11.4.1 on p. 254 for details).

Without the epithet “affine.”

See Chap. 11 of Algebra I.

See Example 11.2 of Algebra I.

The first index i is the order number of the chart, while the second index numbers the coordinates within the ith chart and takes n values \(0\leqslant \nu \leqslant n\), ν ≠ i.

See Sect. 2.6.4 on p. 49.

See Example 9.8 on p. 194.

The first formula relates 2n affine coordinates (x ₁, …, , x _n, y ₁, … , y _n) in \(\mathbb{A}^{n} \times \mathbb{A}^{n} = \mathbb{A}^{2n}\), whereas the second deals with two collections of homogeneous coordinates (x ₀: x ₁: ⋯ : x _n), (y ₀: y ₁: ⋯ : y _n) on \(\mathbb{P}_{n} \times \mathbb{P}_{n}\) (note that they cannot be combined into one collection). We will see in Exercise 12.12 that the latter equations actually determine a closed submanifold of \(\mathbb{P}_{n} \times \mathbb{P}_{n}\) in the sense of Sect. 12.1.2.

Given an irreducible algebraic manifold X, a (Weil) divisor on X is an element of the free abelian group generated by all closed irreducible submanifolds of codimension 1 in X (the dimensions of algebraic varieties will be discussed in Sect. 12.5 on p. 281).

Compare with Sect. 11.3.2 of Algebra I.

See Example 11.6 of Algebra I, especially formula (11.14) there.

That is, to the points of the “hypersurface” \(Z(A) \subset \mathbb{P}_{1}\), some of which may be multiple.

In Example 12.9 on p. 288, we will see that the same holds for every system of homogeneous polynomial equations such that the number of equations equals the number of unknowns.

This means that both binary forms A, B do not vanish at the point (0: 1).

See Example 12.4 on p. 271.

That is, indecomposable into a disjoint union of two nonempty closed subsets.

Possibly after appropriate renumbering of the coordinates x ₁, x ₂, …, x _n. Note that this holds over every infinite field \(\mathbb{k}\), not necessarily algebraically closed.

In particular, this implies that \(\mathop{\mathrm{tr}}\nolimits \deg \mathbb{k}[x_{1},x_{2},\ldots,x_{n}]/(\,f) = n - 1\).

In honor of Emmy Noether, who proved a version of this claim in 1926.

See Sect. 10.4 on p. 236.

For i = 1, this means that f ₁ is not a zero divisor in \(\mathbb{k}[X]\). A sequence of functions possessing this property is called a regular sequence, and the corresponding subvariety V ( f ₁,f ₂,…,f _m) ⊂ X is called a complete intersection.

See Sect. 12.3 on p. 274.

Compare with Problem 17.20 of Algebra I.

See Sect. 3.5.4 of Algebra I.

See Exercise 12.9 on p. 271.

That is, without singular points; see Sect. 2.5.5 on p. 40.

That is, an irreducible variety of dimension one not contained in a hyperplane.

[DK]

Danilov, V.I., Koshevoy, G.A.: Arrays and the Combinatorics of Young Tableaux, Russian Math. Surveys 60:2 (2005), 269–334.MathSciNetCrossRef

[Fu]

Fulton, W.: Young Tableaux with Applications to Representation Theory and Geometry. Cambridge University Press, 1997.MATH

[FH]

Fulton, W., Harris, J.: Representation Theory: A First Course, Graduate Texts in Mathematics. Cambridge University Press, 1997.MATH

[Mo]

Morris, S. A.: Pontryagin Duality and the Structure of Locally Compact Abelian Groups, London Math. Society LNS 29. Cambridge University Press, 1977.

Titel: Algebraic Manifolds
verfasst von: Alexey L. Gorodentsev
Verlag: Springer International Publishing
Buch: Algebra II
Print ISBN: 978-3-319-50852-8

Electronic ISBN: 978-3-319-50853-5

Copyright-Jahr: 2017
DOI: https://doi.org/10.1007/978-3-319-50853-5_12

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