Determinantal schemes and Buchsbaum–Rim sheaves

Abstract

Let φ be a generically surjective morphism between direct sums of line bundles on P n and assume that the degeneracy locus, X, of φ has the expected codimension. We call B φ= ker φ a ( first) Buchsbaum–Rim sheaf and we call X a standard determinantal scheme. Viewing φ as a matrix (after choosing bases), we say that X is good if one can delete a generalized row from φ 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 P 3 , 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.