I am trying to define line bundles on $\mathbb{P}^n$ as a triplet $E \xrightarrow{p} \mathbb{P}^n$. It follows the idea.
Let $m \in \mathbb{Z}$ a fixed integer. Take $E$ the quotient of $\mathbb{C}^{n+1} \times \mathbb{C}$ by the action of $\mathbb{C}^\times$ given by
$$ t \cdot ((z_0,\dots,z_n),\lambda)=(t(z_0,\dots,z_n),t^m\lambda)$$
and let me denote it by $E=\mathbb{C}^{n+1} \times_{\mathbb{C}^\times} \mathbb{C}$. We have a natural $p$ which is the projection onto the first factor:
$$ E=\mathbb{C}^{n+1} \times_{\mathbb{C}^\times} \mathbb{C} \rightarrow \mathbb{P}^n$$
$$ (z,l) \mapsto [z] $$
I'm wondering if it is $\mathcal{O}(m)$ or $\mathcal{O}(-m)$. Where $\mathcal{O}(m)$ is defined in terms of sheaves as the sheaf of "holomorphic maps of degree $m$".
Best Answer
Question: "I'm wondering if it is O(m) or O(−m). Where O(m) is defined in terms of sheaves as the sheaf of "holomorphic maps of degree m"."
Answer: You may use schemes since the classification of holomorphic line bundles on $\mathbb{P}^n$ viewed as a complex manifold is "the same" as the classification of invertible sheaves on $\mathbb{P}^n$ viewed as a scheme. If you are a PhD student it is a good exercise to understand the relation between sheaves of sections and vector bundles mentioned in the comments.
Let $G:=Spec(k[t,1/t])$ be the multiplicative group scheme with $G(k)\cong k^*$ the units in $k$, and let $S:=Spec(k[x_0,x_1])$ and $T:=Spec(k[z])$. Define the coaction
$$G\times S \times T \rightarrow S\times T$$
by
$$\sigma: k[x_0,x_1,z] \rightarrow k[x_0,x_1,z]\otimes_k k[t,,1/t]$$
$$\sigma(x_0):=tx_0, \sigma(x_1):=tx_1$$
and
$$\sigma(z):=t^mz.$$
Let $U:=S-\{(0,0\}$ and let $V:=U \times T$ and consider the induced action $u: G\times U \rightarrow U$. You get a canonical projection map
$$\pi: E:=V/G \rightarrow U/G \cong \mathbb{P}^1$$
and it follows $\pi^{-1}(D(x_i)):=Spec(A_i)$ with
$$A_0:=k[\frac{x_1}{x_0}, (\frac{1}{x_0})^mz]$$
and
$$A_1:=k[\frac{x_0}{x_1},(\frac{1}{x_1})^mz].$$
If you consider the sheaf $A:=Sym^*_{\mathcal{O}_{\mathbb{P}^1}}(\mathcal{O}(-m))$ on $\mathbb{P}^1$ it follows there is an isomorphism of line bundles
$$ E \cong \mathbb{V}(\mathcal{O}(-m)):=Spec(A).$$
This holds in general: Your vector bundle $E$ satisfies $E \cong \mathbb{V}(\mathcal{O}(-m))\cong \mathbb{V}(\mathcal{O}(m)^*)$ where $\mathcal{O}(-m)$ is the sheaf corresponding to the graded module $k[x_0,..,x_n](-m)$ in the sense of Hartshornes book, Chapter I and II. Here $E:=V/G$ and $U/G$ are "geometric quotients" in the sense of geometric invariant theory. The global sections of the projection map
$$\pi: \mathbb{V}(\mathcal{O}(-m)) \rightarrow \mathbb{P}^n$$
are in 1-1 correspondence with $H^0(\mathbb{P}^n,\mathcal{O}(m))$, which is the vector space of homogeneous polynomials $f(x_0,..,x_n)$ of degree $m$. Hence the sheaf of sections of $\pi$ is the invertible sheaf $\mathcal{O}(m)$.
Why is the projection map $\mathbb{A}^{n+1}_k\setminus \{0\} \to \mathbb{P}^n_k$ a morphism of schemes?
In general if $L$ is an invertible sheaf on $X$, it follows $L$ is the sheaf of sections of the canonical map $\pi: \mathbb{V}(L^*) \rightarrow X$, where $\mathbb{V}(L^*):=Spec(B)$ with $B:=Sym_{\mathcal{O}_X}^*(L^*)$. Hence the map $\pi$ has $H^0(X,L)$ as global sections.
More generally: Whenever you have a linear algebraic group $G \subseteq GL_k(V)$ and a closed subgroup $H \subseteq G$ and a left $H$-module $\rho: H \rightarrow GL_k(W)$, it follows there is a quotient scheme $G/H$ that is regular of finite type and quasi projective over $k$. The $H$-module $(W,\rho)$ gives a finite rank algebraic vector bundle $\pi: E(\rho) \rightarrow G/H$. The construction is "similar" to the above example.
Example: You may construct $\mathbb{P}^n$ as follows: Let $dim_k(V)=n+1$ and let $l \subseteq V$ be a line. Let $P \subseteq SL_k(V)$ be the subgroup fixing $l$. It follows there is an isomorphism $\mathbb{P}^n \cong SL_k(V)/P$. You may choose the projection map
$$\pi: SL_k(V) \rightarrow \mathbb{P}^n$$
to map a "matrix" $g \in SL_k(V)(k)$ to its "first column".
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