Hi Mike. Sounds like your battery needs replacing to me. Although, nominally, they are 12V, each of the six cells that make the battery are 2.1 - 2.2V so, fully charged, the battery should read 12.6 - 12.8V. Anything less than 12.4V and you'll probably have trouble starting, especially with a Ducati.
To check, disconnect the battery, fully charge it for 8 hours then measure the voltage after an hour (this allows the 'surface charge' to dissipate for an accurate reading). Let the battery stand for another 24 hours and see what the voltage is - if it shows 12.4V or lower you've got a problem. As you say yours is only showing 12.1V just an hour after charging, I think it's Motobatt time!
As an example, the battery on my Gilera dropped to 12.3V when it failed to start. After I charged it up it read 12.8V and held that charge for a week while I waited for the new R/R to arrive.
Hope that helps.
PS I found this table online, which is quite useful. You can see that, although your battery has only dropped maybe half a volt, it's actually holding less than 50% of the charge of a healthy battery.
