Is there a way to not show some fields of a content type on the add form?
The Field Permissions module sounds exactly like what you're after.


  The Field Permissions module allows site administrators to set
  field-level permissions to edit, view and create fields on any entity.
  
  Features
  
  Enable field permissions on any entity, not just nodes. Role-based
  field permissions allowing different viewing patterned based on what
  access the user has. Author-level permissions allow viewing and
  editing of fields based on who the entity owner is. Permissions for
  each field are not enabled by default. Instead, administrators can
  enable these permissions explicitly for the fields where this feature
  is needed. Field permissions overview


Simply don't give anonymous users permission to add/edit these fields.