Donīt ever use alum on a "real" cut in your face!
I am not able to explain the exact chemical progress but I can tell you that using alum on a fresh cut leaves some heavy scar-tissue (guess where Iīve got my Nickname from?).

When using cold water from a bowl to rinse off you face after shaving, you can just put the alum block into the water for a few minutes.
That way itīs effect is a little less concentrated.