How do US Ph.D. applicants without research experience convince admissions committees that they are interested in a particular field?
I would have thought that in general the time allotted for a PhD is too short (3 - 4 years full time) to spend too much of that learning the basics of your field to the level of being able to produce original research.  Not forgetting that most PhD students start publishing heavily during their candidature, so that really you would be expected to be producing publishable papers within the first 18 months to two years.  I can't imagine any mathematical academic taking on a PhD candidate who didn't already show some expertise in the field - maybe a few papers already; certainly previous studies in the discipline.  (There may be some exceptions, but I imagine they'd be rare.)  If you had read a little about algebraic geometry and wanted to research into it, you would in many cases be recommended to enrol in a few postgraduate courses first - pretty much to test your ability and expertise - before enrolling in a PhD.  Also, supervisors aren't going to teach you the area - although you may be recommended some background reading - they are just wanting to encourage your research, and also of course to get their name on your papers.