[Paper Review] Conormal Geometry of Maximal Minors

Steven L. Kleiman, Anders Thorup|ArXiv.org|Aug 22, 1997

Commutative Algebra and Its Applications5 references4 citations

TL;DR

This paper investigates the conormal geometry of maximal minors in algebraic geometry, focusing on the dimension and equidimensionality of the exceptional locus E in Proj(A[M]) where A[N] fails to be finitely generated over A[M]. Using techniques from commutative algebra and algebraic geometry, the authors prove that dim E = r−1 under certain conditions, including when N is free and A[N] is the symmetric algebra, or when A is universally catenary and W is nonempty. The result recovers and re-proves classical height inequalities for maximal minors and extends criteria for integral dependence of ideals.

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 perfect 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 B"oger's criterion for integral dependence of ideals. Finally, we introduce an application of (1), being made by the second author, 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].

Motivation & Objective

• To analyze the geometric structure of the exceptional locus E in Proj(A[M]) where A[N] is not finitely generated over A[M].
• To establish dimension and equidimensionality properties of E under conditions on the module N and the ring A.
• To recover and reprove classical results on the height of ideals generated by maximal minors using new geometric techniques.
• To extend the theory of integral dependence of ideals to modules via the conormal geometry framework.
• To apply the results to the dual variety of a projective variety, revealing a counterexample where the main conclusion fails.

Proposed method

• Define A[M] as the A-subalgebra generated by a proper submodule M of a finitely generated torsion-free module N over a Noetherian local domain A.
• Construct C = Proj(A[M]) and define E as the preimage in C of the set W ⊆ Spec(A) where A[N]_p is not finitely generated over A[M]_p.
• Use properties of universally catenary rings and the structure of symmetric algebras to analyze the dimension of E.
• Apply techniques inspired by Gaffney and Massey’s work on Thom’s A_f-condition in equisingularity theory.
• Employ graded overdomains and the Proj construction to relate module-theoretic properties to geometric dimensions.
• Analyze the failure of finite generation via local cohomology and dimension theory in the context of ICIS germs.

Experimental results

Research questions

• RQ1 Under what conditions is the exceptional locus E in Proj(A[M]) of dimension r−1, where r = dim C?
• RQ2 When is the exceptional locus E equidimensional, particularly when A is universally catenary and N is free?
• RQ3 How can the height inequality for maximal minors be recovered and generalized using conormal geometry?
• RQ4 What are the implications of the conormal geometry framework for integral dependence criteria of ideals in modules?
• RQ5 Can the main theorem fail when A[N] is still finitely generated over A[M], and if so, what geometric conditions cause this?

Key findings

• The dimension of the exceptional locus E is r−1 if N is free and A[N] is the symmetric algebra, or if W is nonempty and A is universally catenary.
• The exceptional locus E is equidimensional when N is free and A is universally catenary.
• The main result recovers the classical height inequality for ideals generated by maximal minors.
• An extension of Böger’s criterion for integral dependence of ideals is recovered via the module-theoretic generalization of the conormal geometry framework.
• A counterexample is constructed where the conclusion of the main theorem fails, even though A[N] is finitely generated over A[M], demonstrating the necessity of the assumptions.