152 research outputs found
Blow-up algebras, determinantal ideals, and Dedekind-Mertens-like formulas
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.Comment: 36 pages, 9 figures. In the updated version, section 7: "Final
remarks and open problems" is new; the introduction was updated accordingly.
References update
Matrix models and stochastic growth in Donaldson-Thomas theory
We show that the partition functions which enumerate Donaldson-Thomas
invariants of local toric Calabi-Yau threefolds without compact divisors can be
expressed in terms of specializations of the Schur measure. We also discuss the
relevance of the Hall-Littlewood and Jack measures in the context of BPS state
counting and study the partition functions at arbitrary points of the Kaehler
moduli space. This rewriting in terms of symmetric functions leads to a unitary
one-matrix model representation for Donaldson-Thomas theory. We describe
explicitly how this result is related to the unitary matrix model description
of Chern-Simons gauge theory. This representation is used to show that the
generating functions for Donaldson-Thomas invariants are related to
tau-functions of the integrable Toda and Toeplitz lattice hierarchies. The
matrix model also leads to an interpretation of Donaldson-Thomas theory in
terms of non-intersecting paths in the lock-step model of vicious walkers. We
further show that these generating functions can be interpreted as
normalization constants of a corner growth/last-passage stochastic model.Comment: 31 pages; v2: comments and references added; v3: presentation
improved, comments added; final version to appear in Journal of Mathematical
Physic
Initial Complex Associated to a Jet Scheme of a Determinantal Variety
We show in this paper that the principal component of the first order jet
scheme over the classical determinantal variety of m x n matrices of rank at
most 1 is arithmetically Cohen-Macaulay, by showing that an associated
Stanley-Reisner simplicial complex is shellable.Comment: To appear in Journal of Pure and Applied Algebr
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