UKC

Scenario: You are on the decent of a steep Alpine climb. for whatever reason you need to rappel down about 20/30 meters to continue your decent. 

All climbers rappel to the bottom, but the rope is left stuck at the anchor at the top.

What are ways to retrieve the rope and equipment left at the top?

Probably more easily understood from watching a youtube. You wouldn't fix the rope to your top anchor, you'd feed it through and descend from two strands. So leaving the middle of the rope at the anchor rather than a fixed end. 

Like Whitlew said, rather than tying a single rope to and anchor and abseiling off that, you feed it over/through/behind something at the middle of the rope, toss both ends down the face you're abseiling, abseil down using both sides then pull on one end of the rope to pull it down.  This is one of the reasons ropes often have middle markers.

Of course this means that, unless you're using some slightly advanced techniques, you'll need a rope that is double the length of the longest abseil you'll make. 

I see - I did think think of this - but then I guess that the anchor(s) are just lost completely then? 

Yes, hence the term abseil tat (cord + metalwork left behind).

You may also find someone else’s abseil tat and use that after careful checking of integrity.

Or there may already be abseil anchors in place.

Sometimes yes, but this depends what sort of alpine climb you're talking about. A popular route in the Alps would have bolts installed so you just thread directly through those, no need to add your own gear. Sometimes you can put the rope directly around a block or tree with no extra gear, or through a V thread, but in these cases you need to take care that you will still be able to pull the rope down. Otherwise you leave some inexpensive cord and maybe nuts or pitons, but I avoid this if I think it's safe to do so, as it's tantamount to littering.

Or the propeller of a crashed bomber....

Is that where the term “bomber placement“ comes from….?!

Very good!