What are the legal rights of a new mom when the father was not named on the birth certificate. The mom and dad split before pregnancy was known. The dad is an abusive alcoholic. Mom wants to protect the baby, but would like to let the dad know about the baby.

A father as the same rights of a mother. But if the father is not named then he has to fight for his rights by going to court and establishing paternity. Once paternity is established then he can file for joint custody and visitation.

As far as him being an abusive alcoholic, that apparently didn't stop you from having sex with him. That fact will be taken in by the court.

Agreed, a father has all the rights, married or not, or named on the birth certificate or not,

He merely has to prove his paternity in court, if he is not listed on the Birth certificate.

So it is best if you file for custody, ask for child support and see if he even asks for visits though the court.