2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/70999We 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 Fröberg.27 pages, Plain TeX + diagrams.tex, minor revisions, updated abstract + titleAlgebraic GeometryCommutative AlgebraCombinatorics14N05, 14N25, 13D02, 14M20text