Determinantal schemes and Buchsbaum-Rim sheaves
Abstract
Let
ϕ
be a generically surjective morphism between direct sums of line bundles on $\proj{n}$ and assume that the degeneracy locus,
X
, of
ϕ
has the expected codimension. We call
B
ϕ
=kerϕ
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 $\proj{3}$, 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
.