Projective Algebraic Geometry - Exercise 4(a) Cox et al - Section 8.1

[Math Amateur]

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:https://www.physicsforums.com/attachments/5738Can someone please help me with Exercise 4(a) ... ... indeed, what is actually involved in (rigorously) showing that the equation \(\displaystyle x^2 - y^2 = z^2\) is a well-defined curve in \(\displaystyle \mathbb{P}^2 ( \mathbb{R} )\) ... but I am very unsure of exactly how this works ... ... Presumably, what is involved is not only (rigorously) showing that the equation \(\displaystyle x^2 - y^2 = z^2\) is a well-defined curve in \(\displaystyle \mathbb{P}^2 ( \mathbb{R} )\) but showing that \(\displaystyle x^2 - y^2 = z^2\) is the representation in \(\displaystyle \mathbb{P}^2 ( \mathbb{R} )\) of the curve \(\displaystyle x^2 - y^2 = 1\) in \(\displaystyle \mathbb{R}^2\) ... ... ?Indeed whatever the meaning of the question, I would like to be able to show that the curve \(\displaystyle x^2 - y^2 = 1\) in \(\displaystyle \mathbb{R}^2\) becomes the curve \(\displaystyle x^2 - y^2 = z^2\) in \(\displaystyle \mathbb{P}^2 ( \mathbb{R} )\) ... indeed I think this is true ... BUT ... how do you rigorously show this ...

Further I would like to understand the general approach for taking an algebraic curve in \(\displaystyle \mathbb{R}^2\) and finding the corresponding curve in \(\displaystyle \mathbb{P}^2 ( \mathbb{R} )\) ... ... BUT ... how is this done ...Hope someone can help ... ...Peter

======================================================================To give readers of the above post some idea of the context of the exercise and also the notation I am providing some relevant text from Cox et al ... ... as follows:https://www.physicsforums.com/attachments/5739
https://www.physicsforums.com/attachments/5740
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[GJA]

Hi Peter,

Can someone please help me with Exercise 4(a) ... ... indeed, what is actually involved in (rigorously) showing that the equation \(\displaystyle x^2 - y^2 = z^2\) is a well-defined curve in \(\displaystyle \mathbb{P}^2 ( \mathbb{R} )\) ... but I am very unsure of exactly how this works ... ...

Whenever you see "well-defined" in some sort of quotient space it means that you must check that a particular choice of representative element for a member of the quotient is irrelevant at the courser level detail offered by the quotient space.

In exercise 4(a) you must show that if $(x,y,z)$ and $(x',y',z')$ are homogeneous coordinates for the curve $C,$ then they actually described the same curve $C$ in projective space. This is done in a similar fashion to what the authors outline after Definition 3.

Indeed whatever the meaning of the question, I would like to be able to show that the curve \(\displaystyle x^2 - y^2 = 1\) in \(\displaystyle \mathbb{R}^2\) becomes the curve \(\displaystyle x^2 - y^2 = z^2\) in \(\displaystyle \mathbb{P}^2 ( \mathbb{R} )\) ... indeed I think this is true ... BUT ... how do you rigorously show this ...

Further I would like to understand the general approach for taking an algebraic curve in \(\displaystyle \mathbb{R}^2\) and finding the corresponding curve in \(\displaystyle \mathbb{P}^2 ( \mathbb{R} )\) ... ... BUT ... how is this done ...

I believe you can find the answer to your question in the "Plane Projective Curves" section of this Wikipedia entry: https://en.wikipedia.org/wiki/Algebraic_curve

 

[Math Amateur]

Hi Peter,
Whenever you see "well-defined" in some sort of quotient space it means that you must check that a particular choice of representative element for a member of the quotient is irrelevant at the courser level detail offered by the quotient space.

In exercise 4(a) you must show that if $(x,y,z)$ and $(x',y',z')$ are homogeneous coordinates for the curve $C,$ then they actually described the same curve $C$ in projective space. This is done in a similar fashion to what the authors outline after Definition 3. I believe you can find the answer to your question in the "Plane Projective Curves" section of this Wikipedia entry: https://en.wikipedia.org/wiki/Algebraic_curve

Thanks GJA ... most helpful ...

Re-reading the relevant part of Cox et al and studying the Wikipedia entry ...

Thanks again,

Peter