Subschemes of special determinantal and Pfaffian projective schemes
By the well-known result of Hilbert-Burch resp. Buchsbaum-Eisenbud an arithmetically Cohen-Macaulay codimension 2 resp. an arithmetically Gorenstein codimension 3 subscheme is defined by the maximal minors of an \(r \times (r+1)\)-matrix resp. by the submaximal Pfaffians of a skew-symmetric matrix. In the paper under review the authors consider particular cases of these kind of varieties where the corresponding matrices have ``too many'' zeros on the lower right part. The precise definition in the paper is called \(Z_{\rho}\)-type property. It turns out that this special form of the structure matrices forces the schemes to have two ``natural'' subschemes completely determined by the matrix. As applications of their technique the authors recover a result of \textit{E. D. Davis} [Rend. Sem. Mat., Torino 42, No. 2, 25--28 (1984; Zbl 0598.14047)], and some of their own [Commun. Algebra 29, No. 1, 303--318 (2001; Zbl 1027.14022)].
codimension 2 arithmetically Cohen-Macaulay subscheme
codimension 3 arithmetically Gorenstein subscheme
scientific article; zbMATH DE number 5627207