Determinantal Schemes and Buchsbaum-Rim Sheaves

A natural and efficient method for producing numerous examples of interesting schemes is to consider the vanishing locus of the minors of a homogeneous polynomial matrix. If the matrix satisfies certain genericity conditions then the resulting schemes have a number of well described properties. These objects have been studied in both a classical context and a modern context and go by the name of determinantal schemes. Some of the classical schemes that can be constructed in this manner are the Segre varieties, the rational normal scrolls, and the Veronese varieties. In fact, it can be shown (cf. [10]) that any projective variety is isomorphic to a determinantal variety arising from a matrix with linear entries! Due to their important role in algebraic geometry and commutative algebra, determinantal schemes and their associated rings have both merited and received considerable attention in the literature. Groundbreaking work has been carried out by a number of different authors; we direct the reader to the two excellent sources [1] and [8] for background, history, and a list of important papers.