TY - JOUR

T1 - Determinantal schemes and buchsbaum-rim sheaves

AU - Kreuzer, Martin

AU - Migliore, Juan C.

AU - Peterson, Chris

AU - Nagel, Uwe

PY - 2000/6/26

Y1 - 2000/6/26

N2 - 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 Bø = 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.

AB - 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 Bø = 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.

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U2 - 10.1016/S0022-4049(99)00046-8

DO - 10.1016/S0022-4049(99)00046-8

M3 - Article

AN - SCOPUS:0034717360

SN - 0022-4049

VL - 150

SP - 155

EP - 174

JO - Journal of Pure and Applied Algebra

JF - Journal of Pure and Applied Algebra

IS - 2

ER -