Is it possible to sequence the centromere region of a specific chromosome with precision? Are there differences in this region between different chromosomes?
The sequences of centromeres are different among different chromosomes. The diversity of centromere sequences have been reviewed in a number of articles and is actually focus of a lot of studies. Regarding the sequencing part, most genome sequences available do have the sequences of centromeres in them. If you need to sequence some specific centromeres, it might be difficult but not impossible. One could take out a specific chromosome DNA after separating using PFGE and then sequence using Illumina, PacBio sequencing etc. Overall, it will not be straight forward. If you could explain your question in a bit more detail, it will be easy to answer.
Sorry for the delay in answering as I was traveling. You could refer some of these listed reviews for centromere diversity:
1. Article Building centromeres: Home sweet home or a nomadic existence?
2. Article Diversity in Requirement of Genetic and Epigenetic Factors f...
You may find a decent amount of information. I can tell you some more on this topic, if necessary.
Regarding the other part, I am not sure about how one can sort using FACS. May be if you have centromere sequence specific probes, you could do that. But again you need the sequence information for the same. Similar analysis could be possible with magnetic beads as well. But you would need centromere sequence information in that case and make sure that the probe sequence does not show similarity with any other sequence in the genome.