Determinantal schemes and Buchsbaum-Rim sheaves

Let phi be a generically surjective morphism between direct sums of line bu ndles on P-n and assume that the degeneracy locus, X, of phi has the expect ed codimension. We call B-phi = ker phi a (first) Buchsbaum-Rim sheaf and w e call X a standard determinantal scheme. Viewing phi as a matrix (after ch oosing bases), we say that X is good if one can delete a generalized row fr om phi and have the maximal miners of the resulting submatrix define a sche me of the expected codimension. In this paper we give several characterizat ions 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 lo cally a complete intersection outside a subscheme Y subset of X of codimens ion r + 2. Furthermore, for any good determinantal subscheme X of codimensi on 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 theor em of Kreuzer. For example, we show that for a zeroscheme X in P-3, being g ood 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. (C) 2000 Elsevier Science B.V. All righ ts reserved.