FiOS, 1080i, and 1080p

[Doug]

Hello:

FiOS is in the house now, for phone, TV, and the internet.  All three
work, so far so good.

I also upgraded the computer, adding a big 28" monitor.  That is fun!
It does 1080p.  The monitor was also relatively cheap ($500, but I
have seen it online now for $450).  If I take the HDMI wire from FiOS,
and connect it to the monitor, I see a program on the screen for a few
seconds, then the screen goes gray.  I can still hear the sound fine.
FiOS puts out signals as 1080i, not 1080p.  Looks like there is a unit
out there that will convert on the fly a 1080i signal to a 1080p for
$1500, which would be an utter waste of cash.

FiOS also serves up a 720p signal.  My standard definition overhead
projector can display the signal - what ever it is - without a
problem.  The monitor works for a moment, then goes gray.  Any idea
how I might get the monitor to show what is on FiOS, hi-def or not?
The behavior is the same on digital as well as analog channels (low
numbers, >500).

Doug