Abstract
Let V be a vector space of dimension (n + 1) over a field \(\mathbb{k}\). Besides the (n + 1)-dimensional affine space \(\mathbb{A}^{n+1} = \mathbb{A}(V )\), associated with V is the n- dimensional projective space \(\mathbb{P}_{n} = \mathbb{P}(V )\), called the projectivization of V. By definition, points of \(\mathbb{P}(V )\) are 1-dimensional vector subspaces in V, or equivalently, lines in \(\mathbb{A}(V )\) passing through the origin. To observe such points as usual “dots,” we have to use a screen, that is, an n-dimensional affine hyperplane in \(\mathbb{A}(V )\) that does not pass through the origin (see Fig. 11.1).
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- 1.
See Sect. 6.5 on p. 142.
- 2.
See Sect. 6.5 on p. 142.
- 3.
Or just the infinity.
- 4.
See Sect. 6.5.2 on p. 143.
- 5.
These orientations will coincide when we overlap the planes by a continuous movement along the surface of the sphere.
- 6.
It is infinite dimensional as soon V is infinite as a set.
- 7.
Dotted things in (11.6) are not changed.
- 8.
That is, two distinct points of \(\mathbb{P}(V )\).
- 9.
Move x 1 2 to the right-hand side and divide both sides by x 2 + x 1.
- 10.
See Proposition 3.5 on p. 50.
- 11.
In Fig. 11.6, the tangent line through p crosses L at the point (0: 1: 0) lying at infinity.
- 12.
See formula (11.17) on p. 268.
- 13.
That is, all numbers p 1, p 2, p 3, p 4 are finite.
- 14.
See Example 11.5 on p. 266.
- 15.
That is, a statement about the annihilators of points and lines from Pappus’s theorem that holds in \(\mathbb{P}_{2}^{\times }\). It could start thus: “Given two distinct points and two triples of concurrent lines intersecting in these points….”
- 16.
This is, lie in the same pencil.
- 17.
See Example 11.6 on p. 266.
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Gorodentsev, A.L. (2016). Projective Spaces. In: Algebra I. Springer, Cham. https://doi.org/10.1007/978-3-319-45285-2_11
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