Connecting the PC to the TV with HDMI showed a good amount of overscan: the viewable resolution was something like 1820x1030 pixels when the native resolution of the TV is 1900x1080, causing the borders of the PC's desktop to be cut, and even worse, causing the display to be blurry because of some image resizing happening somewhere.
After fiddling with many of the nVidia control panel settings (the PC still runs Vista until I manage to properly install the nVidia Linux drivers), and scouting the web where many people report similar problems, I finally found out the solution was a non obvious setting on the TV side. Go to "Settings / Set-up / Screen settings / Display Area" and select the "Full pixel" option. In French, this is in "Réglages / Réglage / Réglage écran / Zone d'affichage" and the value is "Tous les pixels".
Basically we tell the TV to show all the pixels that come from the HDMI input, and once done the display is perfect! But I can't understand why we would like to do otherwise, i.e. not display everything and have the TV resize a perfectly clean and properly sized digital input... Overscan was understandable with analog transmission when the end of scan lines could be noisy or cut, but with a digital signal, I'm quite puzzled by why we would like to do that, and even more why this is the default setting on the TV.