We give conjectures on the possible graded Betti numbers of Cohen�Macaulay modules up to multiplication by positive rational numbers. The idea is that the Betti diagrams should be non-negative linear combinations of pure diagrams. The conjectures are verified in the cases where the structure of resolutions is known, that is: for modules of codimension two, for Gorenstein algebras of codimension three and for complete intersections. The motivation for proposing the conjectures comes from the Multiplicity conjecture of Herzog, Huneke and Srinivasan

Graded betti numbers of Cohen-Macaulay modules and the multiplicity conjecture

Each partition A = (λ 1 , λ 2 ,..., An) determines a so-called Ferrers tableau or, equivalently, a Ferrers bipartite graph. Its edge ideal, dubbed a Ferrers ideal, is a squarefree monomial ideal that is generated by quadrics. We show that such an ideal has a 2-linear minimal free resolution; i.e. it defines a small subscheme. In fact, we prove that this property characterizes Ferrers graphs among bipartite graphs. Furthermore, using a method of Bayer and Sturmfels, we provide an explicit description of the maps in its minimal free resolution. This is obtained by associating a suitable polyhedral cell complex to the ideal/graph. Along the way, we also determine the irredundant primary decomposition of any Ferrers ideal. We conclude our analysis by studying several features of toric rings of Ferrers graphs. In particular we recover/establish formulae for the Hilbert series, the Castelnuovo-Mumford regularity, and the multiplicity of these rings. While most of the previous works in this highly investigated area of research involve path counting arguments, we offer here a new and self-contained approach based on results from Gorenstein liaison theory.

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Monomial and toric ideals associated to Ferrers graphs

Many algebras are expected to have the Weak Lefschetz property though this is often very difficult to establish. We illustrate the subtlety of the problem by studying monomial and some closely related ideals. Our results exemplify the intriguing dependence of the property on the characteristic of the ground field, and on arithmetic properties of the exponent vectors of the monomials.

/pdf/monomial-ideals-almost-complete-intersections-and-the-weak-3h01fm6mze.pdf

Monomial ideals, almost complete intersections and the Weak Lefschetz property

We approach string phenomenology from the perspective of computational algebraic geometry, by providing new and efficient techniques for proving stability and calculating particle spectra in heterotic compactifications. This is done in the context of complete intersection Calabi-Yau manifolds in a single projective space where we classify positive monad bundles. Using a combination of analytic methods and computer algebra we prove stability for all such bundles and compute the complete particle spectrum, including gauge singlets. In particular, we find that the number of anti-generations vanishes for all our bundles and that the spectrum is manifestly moduli-dependent.

/pdf/heterotic-compactification-an-algorithmic-approach-2np60delhs.pdf

Heterotic Compactification, An Algorithmic Approach

We describe a connected component of the space of stability conditions on abelian threefolds, and on Calabi-Yau threefolds obtained as (the crepant resolution of) a finite quotient of an abelian threefold. Our proof includes the following essential steps:

1.

We simultaneously strengthen a conjecture by the first two authors and Toda, and prove that it follows from a more natural and seemingly weaker statement. This conjecture is a Bogomolov-Gieseker type inequality involving the third Chern character of “tilt-stable” two-term complexes on smooth projective threefolds; we extend it from complexes of tilt-slope zero to arbitrary tilt-slope.




2.

We show that this stronger conjecture implies the so-called support property of Bridgeland stability conditions, and the existence of an explicit open subset of the space of stability conditions.




3.

We prove our conjecture for abelian threefolds, thereby reproving and generalizing a result by Maciocia and Piyaratne.






Important in our approach is a more systematic understanding on the behaviour of quadratic inequalities for semistable objects under wall-crossing, closely related to the support property.

/pdf/the-space-of-stability-conditions-on-abelian-threefolds-and-2hg30heyld.pdf

The space of stability conditions on abelian threefolds, and on some Calabi-Yau threefolds

Introduction Preliminaries Gaeta's theorem Divisors on an ACM subscheme of projective spaces Gorenstein ideals and Gorenstein liaison CI-liaison invariants Geometric applications of the CI-liaison invariants Glicci curves on arithmetically Cohen-Macaulay surfaces Unobstructedness and dimension of families of subschemes Dimension of families of determinantal subschemes Bibliography.

Gorenstein liaison, complete intersection liaison invariants and unobstructedness

Monomial ideals, almost complete intersections and the Weak Lefschetz Property

Ideals of general forms and the ubiquity of the Weak Lefschetz property

In [19], A. King states the following conjecture: Any smooth complete toric variety has a tilting bundle whose summands are line bundles. The goal of this paper is to prove King’s conjecture for the following types of smooth complete toric varieties: (i) Any d-dimensional smooth complete toric variety with splitting fan. (ii) Any d-dimensional smooth complete toric variety with Picard number ≤2. (iii) The blow up of any smooth complete minimal toric surface at T-invariants points.

Rosa M. Miró-Roig

Papers

Gorenstein liaison, complete intersection liaison invariants and unobstructedness

Monomial ideals, almost complete intersections and the Weak Lefschetz property

Monomial ideals, almost complete intersections and the Weak Lefschetz Property

Ideals of general forms and the ubiquity of the Weak Lefschetz property

Tilting sheaves on toric varieties