Determinantal schemes and buchsbaum-rim sheaves

Let ø be a generically surjective morphism between direct sums of line bundles on ℙⁿ and assume that the degeneracy locus, X, of ø has the expected codimension. We call B_ø = ker ø a (first) Buchsbaum-Rim sheaf and we call X a standard determinantal scheme. Viewing ø as a matrix (after choosing bases), we say that X is good if one can delete a generalized row from ø and have the maximal minors of the resulting submatrix define a scheme of the expected codimension. In this paper we give several characterizations of good determinantal schemes. In particular, it is shown that being a good determinantal scheme of codimension r+1 is equivalent to being the zero-locus of a regular section of the dual of a first Buchsbaum-Rim sheaf of rank r + 1. It is also equivalent to being standard determinantal and locally a complete intersection outside a subscheme Y ⊂X of codimension r + 2. Furthermore, for any good determinantal subscheme X of codimension r + 1 there is a good determinantal subscheme S codimension r such that X sits in S in a nice way. This leads to several generalizations of a theorem of Kreuzer. For example, we show that for a zeroscheme X in ℙ³, being good determinantal is equivalent to the existence of an arithmetically Cohen-Macaulay curve S, which is a local complete intersection, such that X is a subcanonical Cartier divisor on S.