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Homework Statement
I am reading the undergraduate introduction to algebraic geometry entitled "Ideals, Varieties and Algorithms: An introduction to Computational Algebraic Geometry and Commutative Algebra (Third Edition) by David Cox, John Little and Donal O'Shea ... ...
I am currently focused on Chapter 8, Section 1: The Projective Plane ... ... and need help getting started with Exercise 4(a) ...Exercise 4 in Section 8.1 reads as follows:
Can someone please help me with Exercise 4(a) ... ... indeed, what is actually involved in (rigorously) showing that the equation ##x^2 - y^2 = z^2## is a well-defined curve in ##\mathbb{P}^2 ( \mathbb{R} )## ... but I am very unsure of exactly how this works ... ...
Presumably, what is involved is not only (rigorously) showing that the equation ##x^2 - y^2 = z^2## is a well-defined curve in ##\mathbb{P}^2 ( \mathbb{R} )## but showing that ##x^2 - y^2 = z^2## is the representation in ##\mathbb{P}^2 ( \mathbb{R} )## of the curve ##x^2 - y^2 = 1## in ##\mathbb{R}^2## ... ... ?
Hope someone can help ... ...
Homework Equations
Definitions 1, 2, and 3 of Cox et al Section 8.1: The Projective Plane are relevant (see text below for Cox et al Section 8.1)
The Attempt at a Solution
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I am very uncertain about what is required in this example and so it is hard to make any substantial progress ... but I think the following map would be central to answering the question:
##\mathbb{R}^2 \longrightarrow \mathbb{P}^2 ( \mathbb{R} )##
which is defined by sending ##(x, y) \in \mathbb{R}^2## to the point ##p \in \mathbb{P}^2 ( \mathbb{R} )## whose homogeneous coordinates are ##(x, y, 1)## ... ...
BUT ... how do we get the variable ##z## explicitly in the equation when ##(x, y)## is sent to ##(x, y, 1)## ... ... ?
Hope someone can help ... ...
Peter======================================================================To give readers of the above post some idea of the context of the exercise, the relevant definitions and propositions, and also the notation I am providing some relevant text from Cox et al ... ... as follows:
Attachments
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Cox et al ... Exercise 4, Section 8.1 ... ....png20.8 KB · Views: 633
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Cox et al ...- 1 - The Projective Plane - page 1 ... ....png54.6 KB · Views: 676
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Cox et al ...- 2 - The Projective Plane - page 2 ... ....png23.8 KB · Views: 648
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Cox et al ...- 3 - The Projective Plane - page 3 ... ....png42.7 KB · Views: 685
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Cox et al ...- 4 - The Projective Plane - page 4 ... ....png37.2 KB · Views: 632
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Cox et al ...- 5 - The Projective Plane - page 5 ... ....png43.6 KB · Views: 677
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Cox et al ...- 6 - The Projective Plane - page 6 ... ....png44 KB · Views: 576
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