MHB Help with Projective Algebraic Geometry - Cox et al Section 8.1, Exs 5(a) & 5(b)

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Projective Algebraic Geometry - the Projective Plane ... Cox et al - Section 8.1, Exs 5(a) & 5(b)

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 Exercises 5(a) and 5(b) ... ...Exercise 5 in Section 8.1 reads as follows:View attachment 5745
Can someone please help me to get started on Exercises 5(a) and 5(b) shown above ...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:
View attachment 5746
https://www.physicsforums.com/attachments/5747
https://www.physicsforums.com/attachments/5748
View attachment 5749
View attachment 5750
View attachment 5751
 
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Re: Projective Algebraic Geometry - the Projective Plane ... Cox et al - Section 8.1, Exs 5(a) & 5(b

Peter said:
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 Exercises 5(a) and 5(b) ... ...Exercise 5 in Section 8.1 reads as follows:
Can someone please help me to get started on Exercises 5(a) and 5(b) shown above ...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:
Just reporting to MHB members that I have had the following help from Andrew Kirk on the Physics Forums:"... ... ... We want the equation to be compatible with the equation $$y=x^2$$ and we also want it to give a well-defined curve, which means it must be homogeneous in $$x,y$$ and $$z$$.A simple equation that satisfies both those is $$yz=x^2$$. Then for $$z=1$$ this gives the original equation. Any point in $$\mathbb R^2$$ with nonzero $$z$$ is the same as a point with $$z=1$$. The only other points are those with $$z=0$$, which are at infinity. For such points we will also have, courtesy of the equation, $$x=0$$. So the set of points on the curve at infinity are those on the $$y$$ axis in $$\mathbb R^2$$. This comprises two equivalence classes: [(0,0,0)] and [(0,1,0)]. So there are two points at infinity, which sounds like what we would want for a parabola (which answers part (b)). ... ... "I have also found a description of the process for extending algebraic curves from the Euclidean plane to the Projective plane in Robert Bix' book: "Conics and Cubics: A Concrete Introduction to Algebraic Curves" ... ... as follows:View attachment 5769
https://www.physicsforums.com/attachments/5770
https://www.physicsforums.com/attachments/5771
Peter
 
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