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 Gröbner 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.
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The work for this paper was done while the second author was sponsored by the National Security Agency under grant numbers H98230-09-1-0032 and H98230-12-1-0247, and by the Simons Foundation under grants #208869 and #317096. The third author gratefully acknowledges partial support by AFOSR/DARPA grant #FA9550-14-1-0141 during the final phase of this project.
Corso, Alberto; Nagel, Uwe; Petrović, Sonja; and Yuen, Cornelia, "Blow-Up Algebras, Determinantal Ideals, and Dedekind-Mertens-Like Formulas" (2017). Mathematics Faculty Publications. 28.