If appointed a public defender by the court, can a DA notify the PD that the case has been dropped and then put out a bench warrant because defendant failed to appear court?

There is always the possibility that if the accused in a criminal matter has a public defender representing him or her, the deputy district attorney can advise the public defender that the case has been dismissed and then a bench warrant could be issued by the court for the arrest of the accused due to the failure of him or her to appear in court.

If this is the situation with respct to you, I suggest that you speak with the public defender who was representing you in order to get the matter resolved with respect to the possible recall of the bench warrant. From what you have written, it seems as though there may have been some miscommunication between the public defender's office, the district attorney's office and the court if supposedly the case against you was being or was dismissed and then a bench warrant was issued for your arrest due to your non-appearance at a scheduled hearing.