How to know my Arduino core version and its location on my Ubuntu laptop?
The ancient version installed by apt lives in
/usr/share/arduino/hardware/arduino/cores/arduino.

If you download and unpack a recent version, it will stay wherever you
left it. For example, I have an Arduino core in
/home/edgar/Downloads/arduino-1.8.10/hardware/arduino/avr/cores/arduino.
The installation script doesn't really install those files. It does
install some icons and desktop shortcuts, but neither the core nor the
Arduino IDE. Instead, it assumes the downloaded package will stay
wherever it was when you ran the script.

If you want a somewhat cleaner installation, I suggest you move the
package to /opt, recursively chown it to root, and run the setup script
there. The core will then be in
/opt/arduino-&lt;version&gt;/hardware/arduino/avr/cores/arduino.