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

In summary, the conversation is about projective algebraic geometry and the projective plane, specifically chapter 8, section 1. The request for help is for exercises 5(a) and 5(b) from 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. The exercise involves finding an equation that satisfies both the equation y=x^2 and is homogeneous in x, y, and z. A possible solution is yz=x^2, which also includes two points at infinity on the curve. The process for extending algebraic curves from the Euclidean plane to the projective plane
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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
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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 \(\displaystyle y=x^2\) and we also want it to give a well-defined curve, which means it must be homogeneous in \(\displaystyle x,y\) and \(\displaystyle z\).A simple equation that satisfies both those is \(\displaystyle yz=x^2\). Then for \(\displaystyle z=1\) this gives the original equation. Any point in \(\displaystyle \mathbb R^2\) with nonzero \(\displaystyle z\) is the same as a point with \(\displaystyle z=1\). The only other points are those with \(\displaystyle z=0\), which are at infinity. For such points we will also have, courtesy of the equation, \(\displaystyle x=0\). So the set of points on the curve at infinity are those on the \(\displaystyle y\) axis in \(\displaystyle \mathbb R^2\). 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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1. What is projective algebraic geometry?

Projective algebraic geometry is a branch of mathematics that studies geometric objects and their properties using algebraic techniques. It focuses on the study of objects called projective varieties, which are sets of points defined by polynomial equations.

2. What is Cox et al Section 8.1?

Cox et al Section 8.1 refers to the eighth chapter, section 1 of the book "Ideals, Varieties, and Algorithms" written by David Cox, John Little, and Donal O'Shea. This section discusses the basics of projective algebraic geometry, including projective varieties, projective morphisms, and projective space.

3. What are Exs 5(a) & 5(b) in Cox et al Section 8.1?

Exs 5(a) & 5(b) refer to Exercises 5(a) and 5(b) in Section 8.1 of Cox et al's book. These exercises provide practice problems for understanding and applying the concepts of projective algebraic geometry discussed in the section.

4. How do I approach solving Exs 5(a) & 5(b)?

The best way to approach solving these exercises is to first review the concepts discussed in Section 8.1, and then carefully read and understand the problem statements. Next, try to break down the problem into smaller, more manageable steps and use the techniques and formulas discussed in the section to solve them. If you are still having trouble, you can consult the solutions manual or seek help from a professor or fellow mathematician.

5. Why is understanding projective algebraic geometry important?

Projective algebraic geometry has many applications in mathematics, physics, and engineering. It is also closely related to other areas of mathematics such as algebraic geometry, commutative algebra, and topology. Understanding projective algebraic geometry can also help in solving real-world problems in areas such as computer vision, robotics, and cryptography.

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