rights issue question?

under what circumstances does a company end up with "rump" stock, that is not taken up in a rights issue? If Im a shareholder and I sell the nill paid rights instead of taking up the rights, doesn;t somoen else buy the new stock armed with the rights ?

is unbought new stock as a result of stockholders that just do nothing ?

also, aren;t the underwriters of the issue obliged to buy up whatever new stock isn;t taken up?

thanks

sorry, so my overall question is why does a share price rally if there is a large take up of the rights issue? is it not largely irrelevant ?