Conormal geometry of maximal minors. (English) Zbl 0974.13010
Let \(A\) be a Noetherian domain, \(N\) a finite torsion-free \(A\)-module and \(M\) a proper submodule such that \(\text{rank }M=\text{rank }N\). The authors consider a graded domain \(S=A[N]\) where this notation just means that \(A_1=N\) generates \(S\). Moreover, let \(A[M]\) be the subalgebra of \(S\) generated by \(M\), and let \(W\) be the subset of those \({\mathfrak p}\in\text{Spec}(A)\) for which \(A[N]_{\mathfrak p}\) is not a finite \(A[M]_{\mathfrak p}\)-module. Finally, \(E\) is the preimage of \(W\) in \(C=\text{Proj}(A[M])\) and \(r=\dim C\).
The main result is that \(\dim E=r-1\) if
(a) \(N\) is free and \(A[N]\) the symmetric algebra of \(N\), or
(b) \(W\) is non-empty and \(A\) is universally catenary.
Furthermore \(E\) is equidimensional if (a) holds and \(A\) is universally catenary.
As a consequence the authors derive (a generalized version of) the Eagon-Northcott bound for heights for ideals of maximal minors and a generalization of Böger’s criterion for the integral dependence of ideals. A further application is made for the geometry of the dual variety of a projective variety.
The main result is that \(\dim E=r-1\) if
(a) \(N\) is free and \(A[N]\) the symmetric algebra of \(N\), or
(b) \(W\) is non-empty and \(A\) is universally catenary.
Furthermore \(E\) is equidimensional if (a) holds and \(A\) is universally catenary.
As a consequence the authors derive (a generalized version of) the Eagon-Northcott bound for heights for ideals of maximal minors and a generalization of Böger’s criterion for the integral dependence of ideals. A further application is made for the geometry of the dual variety of a projective variety.
Reviewer: W.Bruns (Osnabrück)
MSC:
| 13C15 | Dimension theory, depth, related commutative rings (catenary, etc.) |
| 13H15 | Multiplicity theory and related topics |
| 13A02 | Graded rings |
| 13G05 | Integral domains |
Keywords:
Noetherian domain; graded domain; Eagon-Northcott bound; heights for ideals; dependence of idealsReferences:
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