I have two sort of answers:

1. Really portably, I don't know. On debian /etc/alternatives/www-browser is a symlink to the configured system browser. It gives no hints on what arguments to provide, though most browsers will probably accept a plain url. You may want to check out Selenium which has methods for starting most browsers on a lot of systems. It's in Java, but you may be able to learn some tricks. See also WWW::Selenium for the perl interface.

2. What do you mean by "check an url once a while if a given calculation is performed"? If you just want to check a url periodically for changes, you can use sleep, or - on unix - use a crontab entry to run the program periodically. You may also want to check WWW::Mechanize which will give you a nice interface for programmatic web scraping (it's much easier to install than WWW::Selenium but it does not support javascript and some other things that may or may not be relevant to you).