---- Let $X$ be a [[scheme]]. > [!definition] Definition. ([[effective divisor]]) > > > A [[prime divisor in a scheme|Weil divisor]] $\sum_{Y \subset X \text{ p.d.}} a_{Y} Y$ is **effective** if $a_{Y} \geq 0$ for all [[prime divisor in a scheme|prime divisors]] $Y \subset X$. > > A [[Cartier divisor]] $D \in \Gamma\left( X, \frac{\mathcal{K}_{X}^{*}}{\mathcal{O}_{X}^{*}} \right)$ is **effective** if it is [[quotient sheaf|represented by]] $\{ (U_{i}, f_{i}) \}$ with $f_{i}$ regular: $f_{i} \in \Gamma(U_{i}, \mathcal{O}_{X})$ for all $i$. ---- #### ---- #### 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 > ```