Connection between polynomials and groups

[cduston]

Hey Everyone,
I'm reading a paper by Claude LeBrun about exotic smoothness on manifolds and he is talking about a connection between polynomials and groups that I am not familiar with (or at least I think that's what he's talking about). He's creating a line bundle (which happens to be O(2)) over the complex projective space CP^2 and he states that the "elements n of the line bundle are degree 2" (not actually a direct quote, but I'm almost positive that's what he's trying to say). So what this says to me is that there is some connection between the degree 2 polynomials (like p(x)=a+b*x + c*x^2) and the group of all 2x2 orthogonal matricies. I've been running this around in my head for a few days and I can't come up with a good explanation, so could someone help me out?

Thanks so much!

 

[Hurkyl]

Are you sure O(2) is denoting an orthogonal group, and is not instead supposed to be the twisted sheaf \mathcal{O}(2)? This latter interpretation sounds better to me, because if my understanding of algebraic geometry is correct, \mathcal{O}(2) on the projective plane is supposed to be (roughly speaking) the set of all homogenous degree 2 polynomials in three variables.

 

[cduston]

Oh no...I think you are right, that's what the Os look like in the paper. I also had a conversation with my advisor about this but it was over e-mail, so I'm sure she wrote O(2) when she really meant (that crazy italic O)(2). Ok, it's back to Hatcher to learn about twisted sheafs! If anyone else has any insights they would be appreciated, but I think Hurkyl has the right idea. Thanks!

 

[matt grime]

This is a quickie, and possibly wrong since it is what I think of but I'm no geometer.

The projective plane is just two copies of the affine plane glued together. You can think of the coords on each plane as, say,

k[x,y] and k[u,v]

We glue them, roughly speaking by sending x to u, and y to v^-n, you think of this as twisting n times (n can be positive or negative here), and this is O(n). (The font is mathcal, I think).