How to create a cachable contextual custom block based on URL

Hi,

I have several custom blocks on my site that display contents based on arguments from the URL, for example the id of the taxonomy on the taxonomy page would be $tid = arg(2);

Unfortunately this method does not work when using authcache_ajax or authcache_esi (after enabling personalization for this blocks, user based), most probably because the URL is now something like /fragment/block...

What is the correct way to deal with that?

Thanks.