Conormal Geometry of Maximal Minors

Let A be a Noetherian local domain, N be a finitely generated torsion-free module, and M a proper submodule that is generically equal to N. Let A[N] be an arbitrary graded overdomain of A generated as an A-algebra by N placed in degree 1. Let A[M] be the subalgebra generated by M. Set C Proj(A[M]) and r dim C. Form the (closed) subset W of Spec(A) of primes p where A[N]p is not a finitely generated module over A[M]p, and denote the preimage of W in C by E. We prove this: (1) dim E = r − 1 if either (a) Nis free and A[N] is the symmetric algebra, or (b) W is nonempty and A is universally catenary, and (2) E is equidimensional if (a) holds and A is universally catenary. Our proof was inspired by some recent work of Gaffney and Massey, which we sketch; they proved (2) when A is the ring of germs of a complex-analytic variety, and applied it to improve a characterization of Thomʹs Af-condition in equisingularity theory. From (1), we recover, with new proofs, the usual height inequality for maximal minors and an extension of it obtained by the authors in 1992. From the latter, we recover the authorsʹ generalization to modules of Bögerʹs criterion for integral dependence of ideals. Finally, we introduce an application of (1), being made by Thorup, to the geometry of the dual variety of a projective variety, and use it to obtain an interesting example where the conclusion of (1) fails and A[N] is a finitely generated module over A[M].