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

In summary, Peter is seeking help with Exercise 4(a) in Section 8.1 of the book "Ideals, Varieties and Algorithms: An introduction to Computational Algebraic Geometry and Commutative Algebra (Third Edition)" by David Cox, John Little and Donal O'Shea. He is unsure of how to rigorously show that the equation x^2 - y^2 = z^2 is a well-defined curve in projective space and how to find the corresponding curve in projective space for a given algebraic curve in \mathbb{R}^2. He has received some helpful explanations and references from other users in response to his question.
  • #1
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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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  • #2
Hi Peter,

Peter said:
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.
Peter said:
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
 
  • #3
GJA said:
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
 

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

1. What is Projective Algebraic Geometry?

Projective Algebraic Geometry is a branch of mathematics that studies the properties and structures of algebraic varieties in projective space. It deals with the study of polynomial equations and their solutions in multiple dimensions.

2. Who are the authors of "Projective Algebraic Geometry - Exercise 4(a) Cox et al - Section 8.1"?

The authors of "Projective Algebraic Geometry - Exercise 4(a) Cox et al - Section 8.1" are David Cox, John Little, and Donal O'Shea.

3. What is the main topic of Exercise 4(a) in Section 8.1?

The main topic of Exercise 4(a) in Section 8.1 is the study of projective spaces and their properties, such as dimension, singularities, and rational maps.

4. What are some practical applications of Projective Algebraic Geometry?

Projective Algebraic Geometry has many practical applications in fields such as computer vision, engineering, and physics. It is used to solve systems of polynomial equations, model real-world objects and shapes, and analyze projective transformations.

5. How can I improve my understanding of Projective Algebraic Geometry?

To improve your understanding of Projective Algebraic Geometry, it is recommended to practice solving problems and exercises, read textbooks and research papers, and attend lectures or workshops on the topic. Collaborating with other mathematicians and discussing concepts can also be helpful.

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