Lots of people seem to be having this same issue.
I have the problem on my PC. I log in, my internet disconnects. It's not just the game disconnecting, my whole router disconnects. I get the message.. you have been logged out of SL...
I have an Ethernet connection from a TalkTalk Router. I run a laptop via an Ethernet connection from the same router and that is working fine, which makes me think it's a problem with my PC.
The problem has just started and I can't pin it down to anything in particular, like a viewer update. I tried a clean install wiping SL off my PC completely (as far as I can tell), as one suggestion was that the cache was corrupted. That made no difference.
I'm not very technical, so I guess that unless somebody cleverer than me can give me a simple solution, I'm locked out of SL.