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

  1. Jul 19, 2016 #1
    1. The problem statement, all variables and given/known data

    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:

    ?temp_hash=188679c2af2003c8b689ae9e31bef7ba.png

    Can someone please help me to get started on Exercises 5(a) and 5(b) shown above ...

    2. Relevant equations


    The definitions 1, 2 and 3 in Cox et al Section 8.1 may be relevant ... see the text provided below ...
    ...

    3. The attempt at a solution

    I am very unsure how to start on this exercise ... but i suspect that the following transformation or map as given in Cox et al Section 8.1 directly after Definition 3 is involved:

    ##\mathbb{R}^2 \longrightarrow \mathbb{P}^2 ( \mathbb{R} ) ##

    where ##(x, y) \in \mathbb{R}^2## is sent to the point ##p \in \mathbb{P}^2 ( \mathbb{R} )## whose homogeneous coordinates are ##(x, y, 1)##


    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:


    ?temp_hash=a85b650a1e3c7face457e0bf39da7bc0.png
    ?temp_hash=a85b650a1e3c7face457e0bf39da7bc0.png
    ?temp_hash=a85b650a1e3c7face457e0bf39da7bc0.png
    ?temp_hash=a85b650a1e3c7face457e0bf39da7bc0.png
    ?temp_hash=a85b650a1e3c7face457e0bf39da7bc0.png
    ?temp_hash=a85b650a1e3c7face457e0bf39da7bc0.png
     
    Last edited: Jul 19, 2016
  2. jcsd
  3. Jul 19, 2016 #2

    andrewkirk

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    I can have a go at (a).
    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)).

    So far so good. I haven't thought about c or d yet.
     
  4. Jul 19, 2016 #3



    Thanks for the help Andrew ... but ... your solution seems to have been achieved with some good insight ...

    Is there a method or process for transforming curves in ##\mathbb{R}^2## to curves in ##\mathbb{P}^2 ( \mathbb{R} )##?

    Thanks again ...

    Peter

    P.S. if you have any ideas about parts (c) and (d) please let me know ...
     
  5. Jul 19, 2016 #4

    andrewkirk

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    I don't know of any general method but, for curves whose equations are polynomial in x and y, the method is easy enough. You just write out the equation and find the degree of each term, which is the exponent for x plus the exponent for y. Let M be the highest such degree in the equation. Then, to make the equation homogeneous, we multiply each term by ##z^{M-d}## where ##d## is the degree of the term. We then have an equation that is homogeneous - hence making a well-defined curve in the projective plane - and which also involves ##z##.

    There are a couple of examples of this technique in the author's discussion following Proposition 4.

    (d) is easy. Just set ##x## equal to any nonzero constant, such as 1. The equation then becomes ##yz=1##, which is the classic hyperbola equation.

    To do (c) we'd need to understand what the author means by 'tangent to' in the context of a projective plane. Do you know what he means by that?
     
  6. Jul 22, 2016 #5

    Thanks Andrew ...

    By the way, your approach seems to be a general method to extend algebraic curves in the Euclidean plane to the Projective plane ... the book: "Conics and Cubics: A Concrete Introduction to Algebraic Curves"by Robert Bix outlines the method as follows:


    ?temp_hash=a87a0a0b6a65645f99589b298392e9af.png
    ?temp_hash=a87a0a0b6a65645f99589b298392e9af.png
    ?temp_hash=a87a0a0b6a65645f99589b298392e9af.png
     

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