macmini1,1 + 12.04 + 1080p display = black screen

I dual boot an old macmini1,1 (intel core duo 1.83 GHz) between Mac OS and Ubuntu 12.04. The video card in it is a GMA 950 (intel 945GM x86/MMX/SSE2). I have tried hooking up a Sony 17" monitor (1280 x 1024) via the DVI port and both OS's detect the display and work fine. A 50" Panasonic display (1280 x 720) also works in both OS's.

I have a 54" Panasonic display (1920 x 1080) that I access via a DVI -> HDMI converter (apple branded). The Mac OS detects the display and drives it fine, but Ubuntu shows only a black screen. When I first boot the computer, refit throws up a screen where I choose Ubuntu. The little penguin icon slowly fades away, and then the screen goes black before the GRUB screen comes up.

I would love to be able to debug this, but I'm not sure how to proceed. I would appreciate any pointers.