Vector bundles on algebraic curves

Vector bundles on algebraic curves are important concepts in algebraic geometry and have applications in various fields, including number theory, representation theory, and mathematical physics. Here's an overview of what vector bundles are in this context: ### Basic Definitions 1. **Algebraic Curve**: An algebraic curve is a one-dimensional algebraic variety. It can be viewed over an algebraically closed field (like the complex numbers) or more generally over other fields.