Blow-up algebras, determinantal ideals, and Dedekind–Mertens-like formulas
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 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.
Document Details
- Document Type
- Pub Defense Publication
- Publication Date
- Sep 14, 2016
- Source ID
- 10.1515/forum-2016-0007
Entities
People
- Alberto Corso
- Cornelia Yuen
- Sonja Petrović
- Uwe Nagel
Organizations
- Air Force Office of Scientific Research
- Defense Advanced Research Projects Agency
- Illinois Institute of Technology
- National Security Agency
- Simons Foundation
- State University of New York at Potsdam
- University of Kentucky