Thanks for this - I know I am not alone here, and I have made some progress. I have now tested the set-up with two different monitors, both Samsung and both with lower resolutions. What worked well with the newer of the two was to find out the resolution (1366 x 768), set the SP to that resolution and then connect the monitor. The surface then sets itself to its proper resolution (2160 x 1440) and both screens look right and work well together.
When I tried this with my LG monitor it still didn't work. This suggests to me that this is a resolution problem so I will go back to the MS chat site and see if I can make more progress under that heading.
Any further comments would be appreciated.