Relations between the $${\mathcal {}}$$-ultrafilters

Under CH we show the following results:

1. (1)
   There is a discrete ultrafilter which is not a \({\mathcal {Z}}_{0}\)-ultrafilter.
    
2. (2)
   There is a \(\sigma \)-compact ultrafilter which is not a \({\mathcal {Z}}_{0}\)-ultrafilter.
    
3. (3)
   There is a \({\mathcal {J}}_{\omega ^{3}}\)-ultrafilter which is not a \({\mathcal {Z}}_{0}\)-ultrafilter.