They do have to be sharp. As an exercise, consider a metallic cone with a charge of 1 and look at the distribution of the charge that minimizes the electrostati

They do have to be sharp. As an exercise, consider a metallic cone with a charge of 1 and look at the distribution of the charge that minimizes the electrostatic energy. You will find that the charge collects near the tip of the cone. In general, charges will collect where the curvature is high. It's easier to push electrons out of a pointy tip (or inside of a pointy tip) because they are denser there.