The "Varieties" part is is the Del Pezzo-Bertini classification of nondegenerate varieties of degree one more than their codimension. A convenient reference is my paper from the Bowdoin conference, just 20 years ago, with Joe Harris:
On varieties of minimal degree (a centennial account). Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), 3--13, Proc. Sympos. Pure Math., 46, Part 1, Amer. Math. Soc., Providence, RI, 1987.
The "sets" part is work with Green, Hulek and Popescu on the arXiv http://front.math.ucdavis.edu/math.AG/0404517 (to appear in the American Journal of Mathematics). The "schemes" part is as I write this still mostly conjectures. I'll survey the state of the art as I know it.