I have found out that doll bases (what I have called the thing that holds up my dolls) and doll stands are completely different. I found this out by doing a google search.
Doll bases are an unfinished doll form- a blank- to make your own doll on. Most of the ones I found by googling were computer generated- lots of naked, hairless barbie-type girls and also a full range of anime type figures, aliens, or probably anything anyone would want. See some here and here.