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  "body": "\n## Summary\n\nLet $S$ be a graded ring.\nSerre's twisting sheaf $\\mathcal{O}(1)$ on $\\operatorname{Proj} S$ is the sheaf attached to the graded module $S$ shifted by one degree.\nOn projective space $\\mathbb{P}^n_k$ it is the line bundle whose global sections are the linear forms in the homogeneous coordinates.\nIts powers $\\mathcal{O}(m)$ turn graded modules into sheaves and sheaves back into graded modules, and they control embeddings into projective space.\n\n## Explanations\n\n### L1 human\n\nA homogeneous polynomial of degree $m$ is not a function on projective space, since its value changes when one rescales the coordinates.\nIt is a section of a line bundle instead, and this line bundle is $\\mathcal{O}(m)$.\nFor example, on the projective line, the linear forms $x_0$ and $x_1$ are sections of $\\mathcal{O}(1)$, and $x_0^2 + x_1^2$ is a section of $\\mathcal{O}(2)$.\nThe zeros of such a section are well defined, and they form a hypersurface of projective space.\n",
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  "name": "Serre's twisting sheaf",
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