I've looked through a lot of the other "black screen on reboot" problems and none of them have seemed to work for me.
I've done too many fresh installs of Ubuntu 13.10 onto my HP mini 1000 to try and narrow this down any further.
Every install, it works fine, I can do anything I want.
But after update the Ubuntu Base under the security updates tab of the Software Updater application, and then reboot, I get a blank screen. No flashing cursor, no nothing.
I have no idea if I have nvidia, as some of the other answers have addressed, but if I look at the packages that the Ubuntu Base is trying to install, I don't see nvidia on there anywhere.
If I cd .. as far as possible and then "ls -R | grep nvidia" I do get a bunch of "cannot open" in the directories ./ect ./lib and ./proc as well one in ./sys and one in ./usr. I also get some python files and .desktop files with nvidia in the names so I'm guessing I do have nvidia? But the Ubuntu Base doesn't have any nvidia packages in its update so it is confusing.
I would rather install the updates than not, but if there is no way around it, then I'm fine avoiding them every time they pop up, I would just like to know what is the route I should take.