Properties of 3-Codimensional Gorenstein Schemes

Subschemes of Special Determinantal and Pfaffian Projective Schemes

“…It turns out that this special form of the structure matrix forces the schemes to have two "natural" subschemes which are determined by the matrix. This approach will permit us to recover an old result of Davis in [4], about Hilbert functions with a "flat part" (see also [10]) and to recover Corollary 3.4 in [11]. From this result we are able to produce an easy way to construct 3-codimensional Gorenstein schemes containing Gorenstein subschemes with less generators than the given scheme, and a way to construct 4-codimensional Gorenstein schemes.…”

“…Note that the above corollary gives a complete description of the subschemes that were found in Corollary 3.4 in [11]. …”

“…After ordering the d i 's in a non-decreasing way, we call X. It is well known that b is the Betti sequence of a 3-codimensional aG scheme of P n (briefly aG Betti sequence) iff it satisfies the following Diesel conditions (see [5], Proposition 3.1 and [11], Proposition 1.1):…”

“…It was observed in [11] Theorem 3.1 that an aG scheme with such a Hilbert function has an associated Buchsbaum-Eisenbud alternating matrix of Z ρ -type.…”

“…One can observe that particular Hilbert functions or graded Betti numbers of the scheme force the defining structure matrices to take a particular form (roughly, there can be "too many" zeros on the lower right part), see for instance [4,7,11,12]. On the other hand there are many schemes which have a defining structure matrix of this special type.…”