Timeline for The 2-sphere and $\mathbb{CP}^1$
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 21, 2009 at 21:21 | vote | accept | Aston Smythe | ||
| Nov 21, 2009 at 20:50 | comment | added | Greg Kuperberg | 1. $\vec{z}$ is a vector in $\mathbb{C}^{n+1}$, and complex vectors have lengths (en.wikipedia.org/wiki/Inner_product). Normalization means rescaling to unit length. 2. A list of $d$ global, real functions on a set $X$ can be viewed as a function from $X$ to $\mathbb{R}^d$. That is how I am using real and imaginary parts of $x_{jk}$. In this case the image of $X$ happens to be an affine variety. 3. Google finds some references, but they are too advanced to help here, or they don't fit. | |
| Nov 21, 2009 at 20:09 | comment | added | Aston Smythe | ... or perhaps just a reference, I seem to be missing more background than is fair for you to explain. | |
| Nov 21, 2009 at 19:00 | comment | added | Aston Smythe | It's beginning to make sense, just two last concrete questions: (1) What exactly do you mean by the normalisation of $z_i$? (2) How, given global coordinates, can a projective variety be expressed as an affine variety? | |
| Nov 21, 2009 at 15:47 | comment | added | Greg Kuperberg | That's exactly the point: The projective coordinate $z_j$ is not globally defined, but the product $x_{jk}$ is globally defined, provided that the vector of projective coordinates $\vec{z}$ is rescaled to have length 1. If a variety is described by global coordinates, then it is an affine variety. | |
| Nov 21, 2009 at 11:00 | comment | added | Aston Smythe | Thanks for your answer, it seems to be what I'm looking for, but could you please explain the first part of your comment a little more (I'm very new to AG). For me a complex algebraic variety is the set of points in $\mathbb{C}^n$ that vanish for all elements of an ideal $I$, whereas a projective variety is the set of points in $\mathbb{CP}^n$ that vanish for all elements of an ideal $J$. I don't see why multiplying together coordinates (which as far as I understand are not even globally defined - hence I think your line bundle comment) changes a projective variety into an algebraic one. | |
| Nov 21, 2009 at 0:28 | history | answered | Greg Kuperberg | CC BY-SA 2.5 |