RESUMEN

We prove that if $X\subset {\Bbb P}^r$ is any 2-regular scheme (in the sense of Castelnuovo-Mumford) then $X$ is {\it small}. This means that if $L$ is a linear space and $Y:= L\cap X$ is finite, then $Y$ is {\it linearly independent\/} in the sense that the dimension of the linear span of $Y$ is $\deg Y+1$. 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 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 $X\subset {\Bbb P}^r$ is 2-regular, then so is $X_{\rm red}$, 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.