Small schemes and varieties of minimal degree

We prove that if X ⊂ P^r is any 2-regular scheme (in the sense of Castelnuovo-Mumford) then X is small. This means that if L is a linear space and Y := L ∩ X is finite, then Y is 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 ⊂ P^r 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.