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We knew that the projective space $\mathbb P^1$ is the $1$-dim subspace of $\mathbb A^2$ (affine space).

image

From Fig 1 and 2 we have, \begin{align} \phi_x:\mathbb A^1_x&\rightarrow\mathbb P^1\\ x&\mapsto\text{line through }(x,1)\\\\ \phi_y:\mathbb A^1_y&\rightarrow\mathbb P^1\\ y&\mapsto\text{line through }(1,y) \end{align} And $\phi_x$ and $\phi_y$ are injective. Let $U_x=\operatorname{Im}\phi_x=$ subsets of lines not parallel to $y=1$ and $U_y=\operatorname{Im}\phi_y=$ subsets of lines not parallel to $x=1$. In order to be chart compatible we can show that, $$x\in\mathbb A^1_x\setminus\{0\}\quad\text{with}\quad y=\frac1x\in\mathbb A^1_y\setminus\{0\}\quad\text{and}\quad\mathbb P^1=U_x\cup U_y$$

That's all I knew about $\mathbb P^1$. I was trying to understand different line bundle over complex projective space $\mathbb {CP}^1$ and their sections from topological perspective (I guess similar term exist in algebraic geometry). Internet have bunch of resources but I can't visualize them and most of the resources based on algebraic geometry which I want to skip right now. I got the following book which compile several line bundles. On page 38 from A Beginner’s Guide to Holomorphic Manifolds introduced three different line bundle: (i) holomorphic line bundle of degree $k$, (ii) tautological bundle and (iii) hyperplane bundle (here the author use different terminology like Holomorphic instead of complex, page 3). I knew that a local section $$s_i: U_i\to \pi^{-1}(U_i)\overset{\psi_i}\to U_i\times {\mathbb C}$$ should satisfy $$s_i(-) = g_{ij}(-) s_j(-)$$
where $g_{ij}$ are the transition fuctions.

The holomorphic line bundle of degree $k \in \mathbf{Z}$ over $\mathbf{P}^1$ is constructed explicitly as follows. Using the standard atlas, take two "trivial" families over $U_0$ and $U_1$ with coordinates $\left(z^0, \zeta^0\right)$ and $\left(z^1, \zeta^1\right)-z^i$ is the base coordinate and $\zeta^i$ is the fibre coordinate over $U_i$-and "glue them together" over the set $\mathbf{C}^{\times}=\mathbf{P}^1 \backslash\{0, \infty\}$ by the identification $$ z^0=\frac{1}{z^1}, \quad \zeta^0=\frac{\zeta^1}{\left(z^1\right)^k}.\tag1 $$ This line bundle is usually denoted $\mathcal{O}_{\mathbf{P}^1}(k)$, or simply $\mathcal{O}(k)$ for brevity.

Question 1: I didn't understand the meaning of the term "degree" here? and how the identification come from for fiber coordinates in $(1)$ with degree $k$?

Exercise 3.6: Let $\left[Z^0: Z^1: Z^2\right]$ be homogeneous coordinates on $\mathbf{P}^2$. The projection map $p:\left[Z^0: Z^1: Z^2\right] \mapsto\left[Z^0: Z^1: 0\right]$ is defined everywhere except at the point $[0: 0: 1]$, and the image is the $\mathbf{P}^1$ in $\mathbf{P}^2$ with equation $Z^2=0$ (see also Exercise 1.1). Let $H=\mathbf{P}^2 \backslash\left[\begin{array}{ll}0: 0: 1\end{array}\right]$. Show that $p: H \rightarrow \mathbf{P}^1$ is a holomorphic line bundle, and find the corresponding value of $k$. Show that the space of holomorphic sections is the set of linear functions $\left\{a_0 Z^0+a_1 Z^1 \mid a_i \in \mathbf{C}\right\}$.

Question 2: How to visualize the point $[0: 0: 1]$ in $\mathbb{CP}^2$? How to know that the sections for this hyperplane bundle is the set of linear functions $\left\{a_0 Z^0+a_1 Z^1 \mid a_i \in \mathbf{C}\right\}$?

In shorts, I guess I couldn't understand those bundle intuitively which is the main reason I couldn't come up with the sections of those bundles.

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Question 1 is just terminology: The exponent $k$ in the denominator is by definition the degree.

For Question 2, the geometric picture is Cartesian three-space with projection along the third axis. The third axis itself maps to $(0, 0, 0)$. Since this does not represent a point of the projective plane, projection is not defined on the third axis. But as a point of projective space, the third axis is $[0 : 0 : 1]$. Does that help with writing a formal argument?

To find the degree, we look at a trivializing cover:

  • In the set $U_{0}$ where $Z^{1} \neq 0$ [sic], we have affine coordinate $z^{0} = Z^{0}/Z^{1}$. Since $$ [Z^{0} : Z^{1} : Z^{2}] = [Z^{0}/Z^{1} : 1 : Z^{2}/Z^{1}] = [z^{0} : 1 : Z^{2}/Z^{1}], $$ we may take fibre coordinate $\zeta^{0} = Z^{2}/Z^{1}$.
  • In the set $U_{1}$ where $Z^{0} \neq 0$, we have affine coordinate $z^{1} = Z^{1}/Z^{0} = 1/z^{0}$ and fibre coordinate $\zeta^{1} = Z^{2}/Z^{0}$.

In the intersection $U_{0} \cap U_{1}$ (where $Z^{0}$ and $Z^{1}$ are both non-zero), we have $$ \zeta^{0} = \frac{Z^{2}}{Z^{1}} = \frac{Z^{2}/Z^{0}}{Z^{1}/Z^{0}} = \frac{\zeta^{1}}{z^{1}}, $$ from which we read off $k = 1$.

A fairly standard argument to find the space of sections (that generalizes to arbitrary integers $k$, and may be found in Wells' Differential Analysis on Complex Manifolds, and surely elsewhere) is to write a holomorphic section as a pair of entire functions $f_{0}$ in $U_{0}$ and $f_{1}$ in $U_{1}$ satisfying $$ f_{1}(z^{1}) = g_{01}(z^{0}) f_{0}(z^{0}) = \frac{f_{0}(z^{0})}{z^{0}}, $$ or more simply, $f_{1}(1/z) = f_{0}(z)/z$.

Expand $f_{1}(w) = a_{0} + a_{1}w + a_{2}w^{2} + \cdots$ as a power series. Set $w = 1/z$ and equate coefficients in the transition formula. I leave details to you as needed, but qualitatively the left-hand side has only terms of non-positive degree, while the right-hand side has only terms of degree at least $-1$. There are two "overlapping" degrees, so each section is identified with a graph $$ Z^{2}/Z^{0} = \zeta^{1} = f(w) = a_{0} + a_{1}w = a_{0} + a_{1}Z^{1}/Z^{0}, $$ i.e., a linear function $Z^{2} = a_{1}Z^{0} + a_{1}Z^{1}$.

Finally, here is a geometric, after-the-fact way to see that linear expressions in homogeneous coordinates are sections of this bundle: Since the total space is a punctured projective plane, the image of a section is a projective line not passing through $[0 : 0 : 1]$, i.e., the graph of a linear function $Z^{2} = a_{0}Z^{0} + a_{1}Z^{1}$. Inversely, any such projective line maps bijectively to the projective line $Z^{2} = 0$ under projection, so is the image of a section.

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