Actually, using a mini DisplayPort to DVI ou HDMI adapter and connecting the External Monitor through it, is working fine. and you get full 1080p resolution.
While HDMI port does not on DELL XPS 15, as it is optimus based.
Installing Bumblebee project or Ironhide enbales Optimus and the use of the Nvidia card but HDMI support is still in the laboratory.
One can enable the HDMI video output under a native resolution (supposing that one installed bumblebee), on terminal:
$ optirun nvidia-settings -c :8
$ DISPLAY=:8 firefox
Check this to enable the HDMI port with native resolution:
UPDATE: Check the new support for Optimus in Ubuntu 13.04, named REVERSE PRIME (Only Nouveau)