Q: When I connect my Surface RT to HDMI the resolution drops to 720 and I can't run Windows Store applications. What can I do?
A: The Surface RT has a resolution of 1366x768. However, if you connect it via HDMI and select to duplicate the display, you might find the resolution suddenly drops to 720 on the RT (to match the HDMI) so you can no longer run Windows Store applications.
One solution would be to select to extend the display instead of duplicating it. This would allow the RT device to maintain its native resolution and use up to 1080 on the projector.
Alternatively, if the projector can support 1366x768 resolution, then ensure the projector sends back the EDID signal that lets RT know the projector can use 1366x768, and it will work fine. It might be necessary to actually edit the EDID table in the projector connected via HDMI to add support for 1366x768, but this might not always be possible on some projectors.
Essentially the Surface RT can't override the resolutions sent back from the displays connected via HDMI and so the resolutions available will depend on the display connected and the capabilities it exposes via the EDID.