Abstract
Everywhere in this chapter we assume by default that the ground field \(\mathbb{k}\) is algebraically closed.
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Notes
- 1.
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).
- 2.
Without the epithet “affine.”
- 3.
See Chap. 11 of Algebra I.
- 4.
See Example 11.2 of Algebra I.
- 5.
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.
- 6.
See Sect. 2.6.4 on p. 49.
- 7.
See Example 9.8 on p. 194.
- 8.
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.
- 9.
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).
- 10.
Compare with Sect. 11.3.2 of Algebra I.
- 11.
See Example 11.6 of Algebra I, especially formula (11.14) there.
- 12.
That is, to the points of the “hypersurface” \(Z(A) \subset \mathbb{P}_{1}\), some of which may be multiple.
- 13.
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.
- 14.
This means that both binary forms A, B do not vanish at the point (0: 1).
- 15.
See Example 12.4 on p. 271.
- 16.
That is, indecomposable into a disjoint union of two nonempty closed subsets.
- 17.
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.
- 18.
In particular, this implies that \(\mathop{\mathrm{tr}}\nolimits \deg \mathbb{k}[x_{1},x_{2},\ldots,x_{n}]/(\,f) = n - 1\).
- 19.
In honor of Emmy Noether, who proved a version of this claim in 1926.
- 20.
See Sect. 10.4 on p. 236.
- 21.
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.
- 22.
See Sect. 12.3 on p. 274.
- 23.
Compare with Problem 17.20 of Algebra I.
- 24.
See Sect. 3.5.4 of Algebra I.
- 25.
See Exercise 12.9 on p. 271.
- 26.
That is, without singular points; see Sect. 2.5.5 on p. 40.
- 27.
That is, an irreducible variety of dimension one not contained in a hyperplane.
References
Danilov, V.I., Koshevoy, G.A.: Arrays and the Combinatorics of Young Tableaux, Russian Math. Surveys 60:2 (2005), 269–334.
Fulton, W.: Young Tableaux with Applications to Representation Theory and Geometry. Cambridge University Press, 1997.
Fulton, W., Harris, J.: Representation Theory: A First Course, Graduate Texts in Mathematics. Cambridge University Press, 1997.
Morris, S. A.: Pontryagin Duality and the Structure of Locally Compact Abelian Groups, London Math. Society LNS 29. Cambridge University Press, 1977.
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Gorodentsev, A.L. (2017). Algebraic Manifolds. In: Algebra II. Springer, Cham. https://doi.org/10.1007/978-3-319-50853-5_12
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