Blow-up algebras, determinantal ideals, and Dedekind-Mertens-like formulas

TL;DR

This paper studies blow-up algebras of specialized Ferrers ideals, revealing their algebraic structure, equations, and properties, and extending classical formulas related to polynomial content.

Contribution

It identifies equations of blow-up algebras for Ferrers ideals, proves they are Cohen-Macaulay and Koszul, and extends Dedekind-Mertens formulas to this context.

Findings

Determinantal ideals generate the blow-up algebra equations.

The rings are normal Cohen-Macaulay and Koszul.

Explicit minimal reductions and Hilbert functions are determined.

Abstract

We investigate Rees algebras and special fiber rings obtained by blowing up specialized Ferrers ideals. This class of monomial ideals includes strongly stable monomial ideals generated in degree two and edge ideals of prominent classes of graphs. We identify the equations of these blow-up algebras. They generate determinantal ideals associated to subregions of a generic symmetric matrix, which may have holes. Exhibiting Gr\"obner bases for these ideals and using methods from Gorenstein liaison theory, we show that these determinantal rings are normal Cohen-Macaulay domains that are Koszul, that the initial ideals correspond to vertex decomposable simplicial complexes, and we determine their Hilbert functions and Castelnuovo-Mumford regularities. As a consequence, we find explicit minimal reductions for all Ferrers and many specialized Ferrers ideals, as well as their reduction numbers. These results can be viewed as extensions of the classical Dedekind-Mertens formula for the content of the product of two polynomials.