Conormal Geometry of Maximal Minors

Abstract

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) N is 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 Thom's A f -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 authors' generalization to modules of Böger's 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 .

Associated points and integral closure of modules

“…3.1 we see that we can replace the rank assumption on M and N with the hypothesis that the two modules are equal at the generic point of each irreducible component of X. Finally, in [KT00] and [KT01] it's proved that b −1 Y is of codimension one, whereas we prove that b −1 i Y j is of codimension one in an irreducible component of B for each irreducible component Y j of Y .…”

“…In this sense, our approach is similar in spirit to that of Simis, Ulrich and Vasconcelos [SUV01], who analyze the integrality of R(N ) over R(M) by reducing it to a local problem at codimension one primes of R(M). However, in contrast to [SUV01], [KT94] and [KT00], our approach allows us to work in a more general setting: we don't require X to be equidimensional, and we allow the ranks of M and N to vary across the generic points of the irreducible components of X. Finally, we strengthen the Kleiman-Thorup theorem by showing that the inverse image in Proj(R(M)) of each irreducible component of the locus in X where N is not integral over M is of codimension one in an irreducible component of Proj(R(M)).…”

“…The original version of Thm. 6.1 is stated in [KT94] (see [KT00] and [KT01] for subsequent versions with slightly different hypothesis). In [KT00] it's assumed that X is irreducible universally catenary scheme, N is torsion-free and M and N are generically equal.…”

The Af condition and relative conormal spaces for functions with nonvanishing derivative

“…Sct. 1.5 in [KT00]). Instead of working with conormal spaces, we work below with Projans of Rees algebras of modules.…”

An intrinsic definition of the Rees algebra of a module

“…There are more examples of these kinds of generalizations, see e.g. [KT00], [SUV01], [Liu98], [GK99], [Kat95], [Kod95], [Ree87] and [Vas94].…”