Consider the sheaves $\mathcal{O}_{\mathbb{P}^1}(d)$ on $\mathbb{P}^1$, where $\mathcal{O}_{\mathbb{P}^1}(d)(U)$ are holomorphic functions on $\pi^{-1}(U)$ which are homogeneus of degree $d$ and where $\pi:\mathbb{C}^2\setminus\lbrace 0 \rbrace\longrightarrow \mathbb{P}^1$ is the quotient map. How I can show that the sheaf $\mathcal{O}_{\mathbb{P}^1}(d)$ is isomorphic to the sheaf of the global sections of the line bundle $L(d)$ on $\mathbb{P}^1$.
$\mathcal{O}_{\mathbb{P}^1}(d)\cong L(d)$
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algebraic-geometry
sheaf-theory
line-bundles
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0What is $L(d)$? – 2017-01-10
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0With $L(d)$ I denote the tensor product (d times) of $L(1)=\mathbb{P}^2-\lbrace (0:0:1)\rbrace $. – 2017-01-10
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1I guess the map $L(1) \to \mathbb P^1$ is the projection from $(0:0:1)$. First you can try to show that $L(1) \cong O(1)$ by computing sections. Then you can show that $O(a) \otimes O(b) \cong O(a+b)$ for any $a,b$. – 2017-01-13