I am not sure what you mean by "core name".
Most motherboards come with the lowest level of BIOS listed on the support page. The exception generally are boards marked "Max" such as the Tomahawk I mentioned. The standard model is set for a 2xxx CPU where the Max model is already capable of 3xxx CPU out of the box.
The single easiest thing is to browse the top listed features. It will say stuff like, ready for Ryzen 3, ready for Ryzen 5, etc. meaning the model # (3xxx/5xxx). THEN go look at the CPU compat list and see if the same BIOS revision as the lowest offered supports the CPU you have on hand.
For instance it could say:
Ryzen 5 2600 and than have some string of number/letter for the BIOS revision. For the sake of ease, let us say that BIOS rev 1 works on the 2600. But then below you might see Ryzen 5 3600 and BIOS rev 2. This would insinuate that you have to have a 2600 running to then update the BIOS in order for it to work with the 3600. The Max version come out of the box on rev 2. (this is a for instance, I pulled those rev # from my butt)
Some new motherboards offer a feature where you can update the BIOS before setup via a special USB slot on back and a button. I do not suggest going this way unless you are familiar, know what you are doing, or know someone that is.