Turning down drill bit shank

How to reduce the diameter of the dill bit cylindrical part to fit into a small chuck?

It can easily be done with a lathe and a tool that is harder than the drill bit material. A drill bit is typically made in HSS (high speed steel, which means that it can be used at high cutting speeds) and can be machined with any harder material, for example carbide  tool. All those cheap tool sets with brazed on insert are carbide.

Make sure the drill bit is well centred before turning it down. You can check it with a dial indicator. If the chuck jaws push in the void of the helix, the drill bit isn’t supported and it won’t be centered. To avoid this, wrap a brass shim all around ONLY the cylindrical part of the shank. It also prevents marks on the jaws (which are hardened, but also the drill bit is hard, so is good to have some soft metal in between). Obviously if you have a collect chuck is better.