"
a nasty scar
Got a long list of ex-lovers
They'll tell
She knows that this boyfriend has a high chance to leave her, so "I've got a blank space, baby" means that she has a list of ex-boyfriends and doesn't care about how long the list of them can/will go on for, and "And I'll write your name" means she isn't afraid of if he breaks up with her as much as someone else may be.