It can be any problem on the PC, the motherboard and all associated components. It can be the power Supply. It can even be the display itself. There is no way to narrow the problem without a lot of spare parts.
Thanks, Techdude --
The display is fine, it works with my other machine. I am assuming the PSU is ok too, because the HDD powers up and fans are running. Also, I do not see any suspiciously looking capacitors on the mainboard. I have also popped in a spare CPU, but nothing. Have tried a good set of spare ram memory - nothing. So I figure: CPU ok, Ram ok, PSU ok, monitor ok. As far as my modest expertise goes, it is the mobo or some component of it, such as maybe the onboard video module?Replacing the mobo is not worth it to me, but maybe it is something repairable....