Projective Algebraic Geometry - Exercise 6 in Section 8.1, Cox et al

[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 6 ... Exercise 6 in Section 8.1 reads as follows:https://www.physicsforums.com/attachments/5752
Can someone please help me with Exercise 6 ... ...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:
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[jvicens]

Exercise 6 in Section 8.1 of "Ideals, Varieties and Algorithms" is focused on understanding the concept of a projective closure of an affine variety. This exercise involves a specific example of a projective closure, where the affine variety is a circle given by the equation x^2 + y^2 = 1 and the projective closure is given by the equation x^2 + y^2 = z^2.

To start this exercise, you will need to first understand the concept of projective coordinates and how they relate to affine coordinates. Projective coordinates are denoted as (x:y:z) and are defined as the equivalence classes of (x,y,z) under the relation (x,y,z) ~ (λx,λy,λz) for any nonzero scalar λ. This means that (x:y:z) represents all points on the same line through the origin as (x,y,z).

Next, you will need to understand the concept of homogenization, which is the process of taking an affine equation and making it homogeneous by adding an extra variable. In this case, homogenizing the equation x^2 + y^2 = 1 will give us the projective equation x^2 + y^2 = z^2.

To complete this exercise, you will need to use the concepts of projective coordinates and homogenization to show that the projective closure of the affine variety x^2 + y^2 = 1 is indeed x^2 + y^2 = z^2. This will involve manipulating equations and understanding the properties of projective coordinates.

I hope this explanation helps you get started with Exercise 6 in Section 8.1. If you have any further questions, please feel free to ask. Good luck!