Picard group ($\mathrm{Pic}(X)$)

The Picard group, often denoted as \(\text{Pic}(X)\), is an important concept in algebraic geometry and algebraic topology. It is primarily associated with the study of line bundles (or invertible sheaves) over a scheme or a topological space \(X\). ### Definition 1. **Line Bundles:** A line bundle over a space \(X\) is a locally free sheaf of rank 1.