Daniel Erman
Murphy's Law for Hilbert function strata in the Hilbert scheme of points.
Fri, 05/04/2012 - 15:01 — SomNambulAuthors: Daniel Erman
Subjects: Algebraic Geometry
AbstractAn open question is whether the Hilbert scheme of points of a high
dimensional affine space satisfies Murphy's Law, as formulated by Vakil. In
this short note, we instead consider the loci in the Hilbert scheme
parametrizing punctual schemes with a given Hilbert function, and we show that
these loci satisfy Murphy's Law. We also prove a related result for equivariant
deformations of curve singularities with $\mathbb G_m$-action.Beyond Numerics: The Existence of Pure Filtrations.
Wed, 01/06/2010 - 16:01 — SomNambulAuthors: Daniel Erman, David Eisenbud, Frank-Olaf Schreyer
Subjects: Commutative Algebra
AbstractA recent result of Boij-Soederberg and Eisenbud-Schreyer proves that the
Betti diagram of any graded module decomposes as a positive rational linear
combination of pure diagrams. We consider the follow-up question of whether
this numerical decomposition ever corresponds to an actual filtration of the
minimal free resolution itself. Our main result is an affirmative answer to
this question in many surprising cases. As applications of our technique, we
also obtain new results about the semigroup of Betti diagrams and about very
singular spaces of matrices.A special case of the Buchsbaum-Eisenbud-Horrocks rank conjecture.
Wed, 08/19/2009 - 19:30 — SomNambulAuthors: Daniel Erman
Subjects: Commutative Algebra
AbstractThe Buchsbaum-Eisenbud-Horrocks rank conjecture proposes lower bounds for the
Betti numbers of a graded module M based on the codimension of M. We prove a
special case of this conjecture via Boij-Soederberg theory. More specifically,
we show that the conjecture holds for graded modules where the regularity of M
is small relative to the minimal degree of a first syzygy of M. Our approach
also yields an asymptotic lower bound for the Betti numbers of powers of an
ideal generated in a single degree.Laurent polynomials and Eulerian numbers.
Wed, 08/19/2009 - 19:30 — SomNambulAuthors: Daniel Erman, Gregory G. Smith, Anthony Várilly-Alvarado
Subjects: Combinatorics
AbstractDuistermaat and van der Kallen show that there is no nontrivial complex
Laurent polynomial all of whose powers have a zero constant term. Inspired by
this, Sturmfels posed two questions: Do the constant terms of a generic Laurent
polynomial form a regular sequence? If so, then what is the degree of the
associated zero-dimensional ideal? In this note, we prove that the Eulerian
numbers provide the answer to the second question. The proof involves
reinterpreting the problem in terms of toric geometry.
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