Projective Algebraic Geometry - the Projective Plane ....

[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 with Exercise 1 ... Exercise 1 in Section 8.1 reads as follows:
View attachment 5710Can someone please help me to get a start on this exercise ... I am somewhat lost ...Help will be appreciated ...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 5711
View attachment 5712
View attachment 5713
https://www.physicsforums.com/attachments/5714
https://www.physicsforums.com/attachments/5715
View attachment 5716

 

[GJA]

Hi Peter,

I am currently focused on Chapter 8, Section 1: The Projective Plane ... ... and need help with Exercise 1 ...

According to the authors' Definition 1 $\mathbb{P}^{2}(\mathbb{R})$ is $\mathbb{R}^{2}$ together with all of the various "points at infinity." Using this to attack Exercise 1(a), do you think you could list what the three cases are that they mention in the problem? This would be a good question to answer in order to get the ball rolling.

Let me know what you think. Good luck!

 

[Math Amateur]

Hi Peter,
According to the authors' Definition 1 $\mathbb{P}^{2}(\mathbb{R})$ is $\mathbb{R}^{2}$ together with all of the various "points at infinity." Using this to attack Exercise 1(a), do you think you could list what the three cases are that they mention in the problem? This would be a good question to answer in order to get the ball rolling.

Let me know what you think. Good luck!

Hi GJA,

Thanks ... but how to list the three points ... say in coordinates ... is my first problem ...

can you give some more explicit help ...

Peter

 

[GJA]

Hi Peter,

The three, two-distinct point cases are:

1. Both come from $\mathbb{R}^{2}$: $p_{1}=(x_{1},y_{1})\in\mathbb{R}^{2}$ and $p_{2}=(x_{2},y_{2})\in\mathbb{R}^{2}$
2. One comes from $\mathbb{R}^{2}$ and the other is a point at infinity for some line $L$: $p=(x,y)\in\mathbb{R}^{2}$ and $[L]_{\infty}$
3. Both are points at infinity for two lines, say $L_{1}$ and $L_{2},$ that are not parallel: $[L_{1}]_{\infty}$ and $[L_{2}]_{\infty}$

What's left to be shown is that each case determines a unique projective line from $\mathbb{P}^{2}(\mathbb{R})$.

Hopefully that helps get things going for you. Good luck!

 

[Math Amateur]

Hi Peter,

The three, two-distinct point cases are:

1. Both come from $\mathbb{R}^{2}$: $p_{1}=(x_{1},y_{1})\in\mathbb{R}^{2}$ and $p_{2}=(x_{2},y_{2})\in\mathbb{R}^{2}$
2. One comes from $\mathbb{R}^{2}$ and the other is a point at infinity for some line $L$: $p=(x,y)\in\mathbb{R}^{2}$ and $[L]_{\infty}$
3. Both are points at infinity for two lines, say $L_{1}$ and $L_{2},$ that are not parallel: $[L_{1}]_{\infty}$ and $[L_{2}]_{\infty}$

What's left to be shown is that each case determines a unique projective line from $\mathbb{P}^{2}(\mathbb{R})$.

Hopefully that helps get things going for you. Good luck!

Thanks GJA ...

Now I think a proof of Case 1 would go something like the following: (please feel free to critique the proof)\(\displaystyle (x_1, y_1)\) and \(\displaystyle (x_2, y_2)\) being distinct points in \(\displaystyle \mathbb{R}^2\), determine a line, say \(\displaystyle L\), in \(\displaystyle \mathbb{R}^2\).Let \(\displaystyle [L]_{\infty}\) be the common point at infinity of all lines parallel to \(\displaystyle L\).

Then the set \(\displaystyle \overline{L} = L \cup [L]_{\infty}\) is the projective line corresponding to, or determined by the two distinct points \(\displaystyle (x_1, y_1)\) and \(\displaystyle (x_2, y_2)\) in \(\displaystyle \mathbb{R}^2\).
Does the above constitute a rigorous proof that two distinct points in \(\displaystyle \mathbb{R}^2\) determine a projective line in \(\displaystyle \mathbb{P}^2 ( \mathbb{R} )\) ?I am not sure how to formulate a riigorous proof of the other two cases ... can you help ...

Peter

 

[Math Amateur]

Hi Peter,

The three, two-distinct point cases are:

1. Both come from $\mathbb{R}^{2}$: $p_{1}=(x_{1},y_{1})\in\mathbb{R}^{2}$ and $p_{2}=(x_{2},y_{2})\in\mathbb{R}^{2}$
2. One comes from $\mathbb{R}^{2}$ and the other is a point at infinity for some line $L$: $p=(x,y)\in\mathbb{R}^{2}$ and $[L]_{\infty}$
3. Both are points at infinity for two lines, say $L_{1}$ and $L_{2},$ that are not parallel: $[L_{1}]_{\infty}$ and $[L_{2}]_{\infty}$

What's left to be shown is that each case determines a unique projective line from $\mathbb{P}^{2}(\mathbb{R})$.

Hopefully that helps get things going for you. Good luck!

Hi again GJA ... just had an idea regarding Case 2 ...Maybe, informally, a basis for a proof is as follows:

Case 2: A point at infinity essentially determines a slope for a set of parallel lines in \(\displaystyle \mathbb{R}^2\) and the point \(\displaystyle (x, y)\) together with the slope in \(\displaystyle \mathbb{R}^2\) determines a line \(\displaystyle L\) in \(\displaystyle \mathbb{R}^2\) ... the line \(\displaystyle L\) together with the point at infinity together determine a line in \(\displaystyle \mathbb{P}^2 ( \mathbb{R} )\) ...

Is that correct ... ?

How would I formalise the above into a valid proof ... ?Still not sure of Case 3 ... unless we have that two points at infinity determine \(\displaystyle H_{\infty}\), the line at infinity ...
Can you help further ... Peter

 

[GJA]

Hi Peter,

Maybe, informally, a basis for a proof is as follows:

Case 2: A point at infinity essentially determines a slope for a set of parallel lines in \(\displaystyle \mathbb{R}^2\) and the point \(\displaystyle (x, y)\) together with the slope in \(\displaystyle \mathbb{R}^2\) determines a line \(\displaystyle L\) in \(\displaystyle \mathbb{R}^2\) ... the line \(\displaystyle L\) together with the point at infinity together determine a line in \(\displaystyle \mathbb{P}^2 ( \mathbb{R} )\) ...

Is that correct ... ?

How would I formalise the above into a valid proof ... ?Still not sure of Case 3 ... unless we have that two points at infinity determine \(\displaystyle H_{\infty}\), the line at infinity ...

As strange as it may sound, what constitutes a "rigorous" proof can be quite subjective. Personally, the arguments you've provided (the one regarding Case 3 being correct, by the way) are perfectly clear and, to me, complete the exercise. Some other reader may prefer that you use some sort of proof by contradiction here. If there was some truly critical consideration that was overlooked using a direct proof, I would agree that a proof by contradiction would be needed, but, again, I don't believe that's the case here, though I would not argue that the other reader's opinion is "wrong."

Tangential Rant: My personal belief is that rigour is a critical element of mathematical thinking, but it can sometimes be overdone and, at times, obscure what might be relatively simple idea otherwise. I would never tell anyone serious about math/hard science that reading/writing clear, complete, detailed proofs is something they shouldn't take seriously because it is an essential part of the "job." Furthermore, it is often the case that when you are working out the details you come to appreciate ideas and techniques you perhaps overlooked previously. That said, it is my belief that developing a feel (which takes lots of time and, admittedly, slogging through proofs) for how to link various ideas through analogy and less rigid terminology than what's used in a hardcore proof is equally important and that we should provide students with as much training for the intuition behind theorems as is possible/practical (though, again, I admit that this cannot always be done). I could continue to ramble, so I think I'll stop here. Take everything with a grain of salt, though, because I am certainly not the arbiter of mathematical rigour and I am known to have made a mistake or two along the way when it comes to getting the details right.

 

[Deveno]

I'd like to add at this point my thoughts about GJA's tangential rant.

Math is, in my opinion, not about "mathematical objects" (whatever you might conceive those to be) but rather a discourse about ideas. The formalism, which is sometimes unavoidable, is an attempt to be unambiguous about what is being expressed about those ideas.

Just as we have isomorphisms between two structures of the same type, "formalization" is a sort of "meta-isomorphism" between two types of description.

With the real projective plane, you can attempt to describe it in various ways. This is analogous to being able to describe the rational number "$\dfrac{1}{2}$" as:

"a half (of something)"

the ratio $1:2$ (which is also the ratio $k:2k$ for any integer $k$)

"the inverse in the field of fractions of $\Bbb Z$ of the integer $2$"

$[([(\{\emptyset\},\emptyset)]_G,[(\{\emptyset,\{\emptyset\}\},\emptyset)]_G)]_Q$

where the brackets represent equivalence classes of the Grothendieck groupification (sub $G$), and the equivalence classses (the sub $Q$) of the localization of the groupification ring (considering the group so obtained naturally as a $\Bbb Z$-algebra) at 0.

The last description is clearly "the most formal", but is almost unintelligible. The first description (the English one) is intuitively obvious, and yet mathematically imprecise. The other two are somewhere in-between.

So where does "proof" fit into all this? Well, simply put: proof is in the mind of the beholder. If you are trying to convince a computer, your proof may well be unreadable to mortal beings. If you are trying to convince another being who is "at your level", a few words may suffice.

Personal understanding is a hard thing to quantify. One benchmark I like to use is "I can recognize a thing no matter what description is used." For example whether or not you are using a description like:

$\{z \in \Bbb C: |z| = r, r \in \Bbb R^+\}$

$\{(x,y) \in \Bbb R^2:x^2 + y^2 = r^2\}$

$\text{im }f, f: \Bbb R \to \Bbb R^2, f(t) = (r\cos(t), r\sin(t))$

or simply $rS^1$, you should recognize that all of these refer to a circle of radius $r$.

Topologically, the projective plane is a "twisted plane" where the "upper infinite semi-circle" is sewed to the "lower infinite semi-circle" in the opposite direction. Just as the "twisted cylinder" (the Moebius strip) requires more than two dimensions to "create", the projective plane requires more than three dimensions to do so (we can't create a three-dimensional model without some self-intersection). One such "immersion" (which isn't perfect) is this figure (a "cross-cap"):

View attachment 5736

A slightly better one is afforded by Boy's surface, which looks like this:

View attachment 5737

I suspect what you are "intended" to see in Cox et. al.'s text is that the different ways they present to "view" the real projective plane all "act" the same.

 

[Math Amateur]

Hi Peter,
As strange as it may sound, what constitutes a "rigorous" proof can be quite subjective. Personally, the arguments you've provided (the one regarding Case 3 being correct, by the way) are perfectly clear and, to me, complete the exercise. Some other reader may prefer that you use some sort of proof by contradiction here. If there was some truly critical consideration that was overlooked using a direct proof, I would agree that a proof by contradiction would be needed, but, again, I don't believe that's the case here, though I would not argue that the other reader's opinion is "wrong."

Tangential Rant: My personal belief is that rigour is a critical element of mathematical thinking, but it can sometimes be overdone and, at times, obscure what might be relatively simple idea otherwise. I would never tell anyone serious about math/hard science that reading/writing clear, complete, detailed proofs is something they shouldn't take seriously because it is an essential part of the "job." Furthermore, it is often the case that when you are working out the details you come to appreciate ideas and techniques you perhaps overlooked previously. That said, it is my belief that developing a feel (which takes lots of time and, admittedly, slogging through proofs) for how to link various ideas through analogy and less rigid terminology than what's used in a hardcore proof is equally important and that we should provide students with as much training for the intuition behind theorems as is possible/practical (though, again, I admit that this cannot always be done). I could continue to ramble, so I think I'll stop here. Take everything with a grain of salt, though, because I am certainly not the arbiter of mathematical rigour and I am known to have made a mistake or two along the way when it comes to getting the details right.

Generally agree with you GJA ... most interesting as well ...

I very definitely agree with you when you write:

"... ... it is often the case that when you are working out the details you come to appreciate ideas and techniques you perhaps overlooked previously. That said, it is my belief that developing a feel (which takes lots of time and, admittedly, slogging through proofs) for how to link various ideas through analogy and less rigid terminology than what's used in a hardcore proof is equally important and that we should provide students with as much training for the intuition behind theorems as is possible/practical ... ... I definitely strive to gain an intuitive feel for mathematical notions and ideas ... and yes, slogging through proofs and some exercises is definitely a part of achieving this ...

Thanks for the interesting post ...

Peter

- - - Updated - - -

I'd like to add at this point my thoughts about GJA's tangential rant.

Math is, in my opinion, not about "mathematical objects" (whatever you might conceive those to be) but rather a discourse about ideas. The formalism, which is sometimes unavoidable, is an attempt to be unambiguous about what is being expressed about those ideas.

Just as we have isomorphisms between two structures of the same type, "formalization" is a sort of "meta-isomorphism" between two types of description.

With the real projective plane, you can attempt to describe it in various ways. This is analogous to being able to describe the rational number "$\dfrac{1}{2}$" as:

"a half (of something)"

the ratio $1:2$ (which is also the ratio $k:2k$ for any integer $k$)

"the inverse in the field of fractions of $\Bbb Z$ of the integer $2$"

$[([(\{\emptyset\},\emptyset)]_G,[(\{\emptyset,\{\emptyset\}\},\emptyset)]_G)]_Q$

where the brackets represent equivalence classes of the Grothendieck groupification (sub $G$), and the equivalence classses (the sub $Q$) of the localization of the groupification ring (considering the group so obtained naturally as a $\Bbb Z$-algebra) at 0.

The last description is clearly "the most formal", but is almost unintelligible. The first description (the English one) is intuitively obvious, and yet mathematically imprecise. The other two are somewhere in-between.

So where does "proof" fit into all this? Well, simply put: proof is in the mind of the beholder. If you are trying to convince a computer, your proof may well be unreadable to mortal beings. If you are trying to convince another being who is "at your level", a few words may suffice.

Personal understanding is a hard thing to quantify. One benchmark I like to use is "I can recognize a thing no matter what description is used." For example whether or not you are using a description like:

$\{z \in \Bbb C: |z| = r, r \in \Bbb R^+\}$

$\{(x,y) \in \Bbb R^2:x^2 + y^2 = r^2\}$

$\text{im }f, f: \Bbb R \to \Bbb R^2, f(t) = (r\cos(t), r\sin(t))$

or simply $rS^1$, you should recognize that all of these refer to a circle of radius $r$.

Topologically, the projective plane is a "twisted plane" where the "upper infinite semi-circle" is sewed to the "lower infinite semi-circle" in the opposite direction. Just as the "twisted cylinder" (the Moebius strip) requires more than two dimensions to "create", the projective plane requires more than three dimensions to do so (we can't create a three-dimensional model without some self-intersection). One such "immersion" (which isn't perfect) is this figure (a "cross-cap"):
A slightly better one is afforded by Boy's surface, which looks like this:
I suspect what you are "intended" to see in Cox et. al.'s text is that the different ways they present to "view" the real projective plane all "act" the same.

Thanks for an informative and interesting post, Deveno ... ...

Peter