The PC version of Final Fantasy XIII is locked at a maximum resolution of 720p, according to numerous online reports.
A user-created patch from Dark Souls saviour Durante, however, allows users to increase the resolution up to 3840x2160, albeit with unwanted side effects. Current bugs introduced by the patch include crashes when taking screenshots and issues with scaling.
Still, if you've given Final Fantasy XIII a download and want to try running it at a higher resolution, head through here for more details on the patch and instructions on how to install it.
Source: reddit.com, neogaf.com, blog.metaclassofnil.com