Appendix C: The Gotzmann Theorems and the Hilbert scheme
by A. Iarrobino and Steven L. Kleiman.

Appendix C was written by the first author and Steven L. Kleiman, and it holds a particular place in the book. Section C1 contains what is used in the body: Macaulay's characterization of Hilbert functions of graded quotients of the polynomial ring, using $O$-sequences, and the particular case of graded quotients of $k[x_1,x_2]$ (Corollary C6. The rest of the appendix treats various aspects of the theory of the Hilbert scheme of $\Hilb^P(\mathbb P^n)$, which parametrizes subschemes of $\mathbb P^n$ with Hilbert polynomial $P$. The last Section C5 proves some new results about annihilating schemes of homogeneous polynomials and the varieties $\Gor(T)$.

In Section C2 we define Macaulay and Gotzmann polynomials, and give some of their properties; in Section C3 we state the Persistence and Regularity Theorems of G. Gotzmann \cite{Go}.
Next, in Section C4 we report on the relation of these results to G.~Gotzmann's identification of the Hilbert scheme $\Hilb^P(\mathbb P^n)$ as a simply described subscheme in a product of two Grassmanians. Gotzmann's description is based on Grothendieck construction and on his own improvement of Mumford's effective bound for the Castelnuovo--Mumford regularity degree. Here we also establish D. Bayer's related conjecture that the Hilbert scheme is a determinantal subscheme of a Grassmannian.

The last Section C5 of the Appendix applies the Gotzmann Hilbert Scheme Theorem and a related result \cite{Go4} to certain schemes $\Gor(T)$, in a manner similar to some applications in an article of A. Bigatti, A.~Geramita, and J. Migliore \cite{BGM}. Thus we obtain another example, in addition to those of M. Boij \cite{Bj2}, of a scheme $\Gor(T)$ for $r=4$ having several irreducible components.

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