The oily pen scratches on the diamond surface can be wiped off with the European rag (like pure wool carpet) that is usually used to wipe cups.

In fact, it can be erased, but it doesn't have to use any cloth. The key is not to scratch with metal. It's too soft. It's useless. Including nails, can be removed.

Then there is ultrasonic cleaning, which is available in jewelry stores and optical shops. It's usually put there for free and made casually.