Why do lightning rods have a sharp point at the top?

Suppose there is a charged cloud floating over your conductor. Then making your lightning conductor pointy at the edge would facilitate better discharge by setting up a high electric field.

We will take a spherical approximation of the pointed end, then ${\sigma}=\frac{q}{4\pi r^2}$ is the surface charge density of the end. It has a very high surface charge density due to its small radius. Hence, in this case, the electric field over that small part will be $E=\frac{\sigma}{\epsilon_0}$ which is also very high.

Then, for a pointy metal rod, the electric field set up at pointy ends is high. Now for some reason, if the discharge of the cloud occurs, the charge will be easily passed through the lightning conductor and conducted to the ground. Your artifact which you are trying to save is ultimately protected from damage.

The point of the point is to increase the electric field near the point. Small radius curves will have a higher local electric field, eventually creating a localize area where the field is greater than the dielectric strength of the air. This results in what I refer to as "micro-lightning." This microlightning discharges the air (or cloud) before the charge difference between the cloud and ground builds to the point where a very long path of breakdown is formed. The main idea is to prevent big lightning by have near-continual (during storms) microlightning.

You can demonstrate this with a small Tesla coil or classroom Van de Graaff generator. Set up a situation with the coil or generator causing long (>10 cm) sparks. Then get a pointy object like a key or a nail, ground it, and bring it near the discharge. The spark will stop, but if you listen carefully, you can hear a crackle near the pointy object. You won't get a large spark around the pointy object until you get close to the coil tip or generator sphere. Then remove the pointy object and the long sparks will start again.