1997/08/26 by Martin Kreuzer, Kreuzer, M., Juan Migliore +5
Mathematics · Pharmacology, Toxicology and Pharmaceutics · #13C40 #14C20 #14F05 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Alkaloids: synthesis and pharmacology #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Primary 14M12 #Secondary 13D02
paper · pdf · doi:10.48550/arxiv.alg-geom/9708021
openalex publication_date 1997/08/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let ϕ be a generically surjective morphism between direct sums of line bundles on \projn and assume that the degeneracy locus, X, of ϕ has the expected codimension. We call Bϕ = 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 \proj3, 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.