---- Let $k$ be a [[field]], and denote by $\mathbb{P}^{n}_{k}$ the [[projective space|projective]] $n$-space over $k$, i.e., the [[scheme]] $\mathbb{P}^{n}_{k}:=\text{Proj }k[X_{0},\dots X_{n}],$ where $\text{Proj }$denotes the [[proj construction]] and the [[polynomial 4|polynomial ring]] $k[X_{0},\dots,X_{n}]$ is taken with its usual [[graded ring|grading]] (elements of $k$ have degree zero). > [!theorem] Theorem. ([[the imperfect dictionary to divisors]]) > Let > - $X$ be an [[integral scheme|integral]] [[subscheme|closed subscheme]] of $\mathbb{P}^{n}_{k}$; > - $k=\overline{k}$ be [[algebraically closed]]. > > Let $D_{0}$ be a [[Cartier divisor]] of $X$ and $\mathcal{L} \cong \mathcal{O}_{X}(D_{0})$ the [[line bundle associated to a Cartier divisor|associated]] [[locally free sheaf|line bundle]]. Then: > > 1. For every nonzero global section $s \in \Gamma(X, \mathcal{L})$, the [[divisor of zeros of a global line bundle section|divisor of zeros]] $(s)_{0}$ is an [[effective divisor]] linearly equivalent to $D_{0}$: $(s)_{0} \sim D_{0}$; > 2. Every [[effective divisor]] linearly equivalent to $D_{0}$ is of the form $(s)_{0}$ for some global section $s \in \Gamma(X, \mathcal{L})$; > 3. Two global sections $s,s' \in \Gamma(X, \mathcal{L})$ have $(s)_{0} =(s')_{0}$ if and only if there exists $\lambda \in k^{*}$ with $s = \lambda s'$. > This builds us a dictionary between sections of line bundles and effective Cartier divisors. > [!basicexample] Example. (For $\mathbb{P}^{n}_{k}$ itself) > [[class group of projective space|We know]] $\text{Cl }\mathbb{P}^{n}_{k}=\mathbb{Z} \langle [H] \rangle$, i.e., the [[divisor (of zeros and poles)|class group]] of $\mathbb{P}^{n}_{k}$ is generated by the class of a hyperplane $H=V(\langle X_{0} \rangle)$. $\mathbb{P}^{n}_{k}$ is [[regular scheme|nonsingular]] and [[integral scheme|integral]], and so satiasfies the isomorphism conditions respectively in [[relating Weil and Cartier divisors]] and [[relating Cartier divisors and line bundles]], thus: $\mathbb{Z} \langle [H] \rangle = \mathbb{Z} \cong\text{Cl }\mathbb{P}^{n}_{k} \cong \text{CaCl }\mathbb{P}^{n}_{k} \cong \text{Pic }\mathbb{P}^{n}_{k}.$ > Can check that the [[line bundle associated to a Cartier divisor|line bundle associated to]] $H$ (viewed as a [[Cartier divisor]]), $\mathcal{O}_{\mathbb{P}^{n}_{k}}(H)$, is isomorphic to [[universal line bundle on projective space|the generator]] $\mathcal{O}_{\mathbb{P}^{n}_{k}}(1)$ of the [[Picard group]] $\text{Pic }\mathbb{P}^{n}_{k}$. > > So now we know all [[locally free sheaf|line bundles]] on $\mathbb{P}^{n}_{k}$. They are ($d \in \mathbb{Z}$) > $\begin{align} > \mathcal{O}_{\mathbb{P}^{n}_{k}} (d)& := \mathcal{O}_{\mathbb{P}^{n}_{k}}(1)^{\otimes d} & d > 0 \\ > \mathcal{O}_{\mathbb{P}^{n}_{k}} (-d)& := \mathcal{O}_{\mathbb{P}^{n}_{k}} (d) ^{\vee} & d < 0 > \end{align}$ > > > Check that $\mathcal{O}_{\mathbb{P}^{n}_{k}}(dH)\cong \mathcal{O}_{\mathbb{P}^{n}_{k}}(d)$. [[on the sheaf cohomology of line bundles over projective space|We will soon prove]] that if $S=k[X_{0},\dots,X_{n}]$ is such that $\mathbb{P}^{n}_{k}=\text{Proj }S$, then $\Gamma \big(\mathbb{P}^{n}_{k}, \mathcal{O}_{\mathbb{P}^{n}_{k}}(d)\big) \cong S_{d}$. > [!proof]- Proof. ([[the imperfect dictionary to divisors]]) > Note that since $X$ is [[integral scheme|integral]], $\mathcal{K}_{X}=\underline{K(X)}=(U \text{ nonempty} \mapsto K(X))$, where $K(X)$ is the [[generic point of an integral scheme|function field]] of $X$ and we have noted the nice form a [[constant sheaf]] on an [[irreducible topological space|irreducible space]] takes. > **Setup.** $\mathcal{L}=\mathcal{O}_{X}(D_{0})$ is a [[subsheaf of modules generated by local sections|subsheaf]] of $\mathcal{K}_{X}=\underline{K(X)}$, so $s \in \Gamma(X, \mathcal{L})$ corresponds to rational function $f \in K(X)$. Supposing $D_{0}$ is represented as $\{ (U_{i}, f_{i}) \}$, $\mathcal{O}_{X}(D_{0})$ by construction has as local trivializations the maps $\begin{align}\mathcal{O}_{X}(D_{0}) |_{U_{i}} &\xrightarrow{\varphi_{i}} \mathcal{O}_{U_{i}}\\ t & \mapsto t f_{i} \end{align}$ > **1.** > The definition $(s)_{0}$ is the [[Cartier divisor]] represented by $\{ \big(U_{i}, \varphi_{i}(f |_{U_{i}})\big) \}=\{ \big( U_{i}, f f_i \big) \}=D_{0}+(f)$, where we have switched to additive language and used that principal divisor $(f)$ is represented by $\{ (X,f) \}$. Thus $(s)_{0} \sim D_{0}$ (why did this get cut off?) ---- #### ----- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```