  We prove that if X is any 2-regular projective scheme (in the sense of
Castelnuovo-Mumford) then X is "small". This means that if L is a linear space
and Y:= L\cap X is finite, then Y is "linearly independent" in the sense that
the dimension of the linear span of Y is 1+deg Y. The converse is true and
well-known for finite schemes, but false in general. The main result of this
paper is that the converse, "small implies 2-regular", is also true for reduced
projective schemes (algebraic sets). This is proven by means of a delicate
geometric analysis, leading to a complete classification: we show that the
components of a small algebraic set are varieties of minimal degree, meeting in
a particularly simple way. From the classification one can show that if X is
2-regular, then so is X_{red}, and so also is the projection of X from any
point of X.
  Our results extend the Del Pezzo-Bertini classification of varieties of
minimal degree, the characterization of these as the varieties of regularity 2
by Eisenbud-Goto, and the construction of 2-regular square-free monomial ideals
by Fr\"oberg.
