Hi IvanSaiz,
Thanks for being involved in Apple Communities.
I'm sorry to hear you haven't been able to use Illustrator since you upgraded to macOS Catalina. If you're still looking for a solution, Apple Communities will help you find your way out.
In this case, it is possible that your app isn't designed for a 64-bit use. Apple has been working with developers to transition their apps, and in 2018 Apple informed them that macOS Mojave would be the last version of macOS to run 32-bit apps. You can easily check what version your app is running: Finder > Applications: if it has a prohibitory sign, it is not compatible with the system you're using: 32-bit app compatibility with macOS High Sierra 10.13.4 and later
Instead of downgrading the system (where you aso downgrade your security features) I would advise you to get in touch with the app developper, either from the App Store (if you downloaded the App from the App Store) or from the developper's website.
I hope this helps. Please let us know whether you managed to solve the issue. I'm sure this will help a lot of us.
Kind regards.