They are just the seed pods of the potatoe, Tom, nothing to worry about. They are same family as tomatoe but DO NOT try to eat these as they are poisonous!
Desiree, an excellent all round potatoe, is a main crop and should be ready for lifting at the end of August.

If you grow the seeds, you might wind up with your own variety, like Luther Burbank! I wonder how you would breed for blight resistance?

To breed a plant for a particular "strength" either blight resistance, scab or any other of the many problems that can affect a plant you first find two plants that seem to survive in these conditions, pollinate these plants together and then collect the seeds. Grow these plants on and induce the disease. If they survive then you keep improving the strain until after about ten years and hundreds if not thousands of pounds later you will have a blight resistant potatoe. Then you have to grow seed potatoes every year from the seed pods!

Hmm. I thought (right or wrong) that the usual seed potatoes were just the runts of the litter from a crop that was carefully maintained virus-free.

Ooooh, T! Don't let a Scottish farmer hear you say that. There is a huge amount of work goes into producing seed potatoes for the market

Sorry! Sorry! My ignorance is showing! I knew that there was a lot of work involved in ensuring that they were virus and disease free, but the actual mechanics escaped me. As far as I knew, practically all modern potato varieties are still produced asexually. So, if not by seed, then by tissue culture, then?

I am no expert, Tugbrethil. I believe, that seed sized potatoes are collected from the previous years crop and planted as if for shop sized potatoes but they are harvested early to be used as next years seed potatoes. In between a lot of work goes on to ensure that they are disease free and pure. I think that it would make sense for farmer in area 'A' to sell all his potatoes and buy his next years seed potatoea from a farmer in area 'S'. I suppose that this could be described as "tissue culture" - I had never thought of it in these terms.

Tissue culture is where the propagator cuts out tiny plugs of the dividing cells (called meristem in plants), separates the cells, and grows them in rotating flasks of nutrient solution. When they have enough cells, they are spread thinly over nutrient spiked agar in glass saucers, allowed to grow for a while, and (sometimes) treated with hormones to force the resultant blobs of green tissue to form roots and stems. It's a lot of rigmarole, but it's the most certain way of getting exact clones of a plant, without virus included. Practically all of the named varieties of orchid on the market, and some hostas, are produced this way. I have no idea if seed potato growers do this, too, but I would think that would be expensive. It actually sounds like seed potatoes are grown like new potatoes, but maybe treated differently to make them mature while small? So much to learn!