---- > [!definition] Definition. ([[divisor of zeros of a global line bundle section]]) > > - Let $X$ be an *[[integral scheme|integral]]* [[scheme]] and $\mathcal{L}$ a [[locally free sheaf|line bundle]] on $X$. > - Let $s \in \Gamma(X, \mathcal{L})$ be a nonzero global section of $\mathcal{L}$. > - Let $\{ U_{i} \}$ be a trivializing [[cover]] of $X$ with [[morphism of sheaves of modules|isomorphisms]] $\varphi_{i}:\mathcal{L} |_{U_{i}} \to \mathcal{O}_{U_{i}}$. > > Then we get a [[Cartier divisor]], called the **divisor of zeros of $s$**, as[^1] $(s)_{0}:= \{ \big(U_{i}, \varphi_{i}(s)\big) \}.$ > As $\varphi_{i}(s) \in \mathcal{O}_{X}(U_{i})$ by definition, $(s)_{0}$ is manifestly [[effective divisor|effective]]. > > ![[CleanShot 2025-03-16 at [email protected]]] ---- #### [^1]: To see this is indeed a [[Cartier divisor]], need to check $\varphi_{i}(s)$ is a [[unit]] [[sheaf of rational functions|in]] $\mathcal{K}_{X}(U_{i})$ for all $i$, and that on overlaps $U_{i} \cap U_{j}$ the difference $\frac{\varphi_{i}(s)}{\varphi_{j}(s)}$ lives in $\mathcal{O}_{X}^{*}(U_{i} \cap U_{j})$. The former is true because $\varphi_{i}(s) \neq 0$ (since $s \neq 0$) and hence is invertible in the [[field of fractions]] $K(X)$. The latter is true because $\frac{\varphi_{i}(s)}{\varphi_{j}(s)}$ is precisely the [[lines bundles and transition functions|transition function]] $g_{ij} \in \mathcal{O}_{X}^{*}(U_{i} \cap U_{j})$ between $U_{i}$ and $U_{j}$. ---- #### 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 > ```