Subject: Why is cerium oxide used to polish quartz?

I'm wondering why tin oxide (Moh's: 6-7) and especially cerium oxide 
(Moh's: 6) are usually used to polish hard minerals like quartz (Moh's: 7) 
when a soft material (lower Moh value) cannot scratch a harder material no 
matter how hard you try.  As I understand polishing, the polish eliminates 
visible scratches and replaces them with smaller scratches which are <= 1 
um and are invisible to the naked eye so the surface "looks" mirror 
smooth.  Why aren't Al2O3 or silicon carbide used when these have a 
hardness= 9?

Re: Why is cerium oxide used to polish quartz?