April 23, 2014

July 24-August 4, 2006
Week 2

Workshop on Computational and Combinatorial Commutative Algebra
The Fields Institute, Toronto

Organizing Committee: Ragnar Buchweitz (Toronto), Greg Smith (Queen's), Alexander Yong (Fields/Minnesota)
July 31
August 1
August 2
August 3
Friday afternoon
August 4
9:00-10:00: Mauricio Velasco
11:00-12:00: Cornelia Yuen
2:00-3:00: Uli Walther
3:00-4:00: Achilleas Sinefakopoulos
9:00-10:00: Jessica Sidman
11:00-12:00: Seth Sullivant
2:00-3:00: Graham Leuschke
3:00-4:00: Hugh Thomas
4:00-5:00: Ning Jia
9:00-10:00: Nantel Bergeron
11:00-12:00: Alexander Woo
2:00-3:00: Tony Iarrobino
3:00-4:00: Mike Stillman
4:00-5:00: Hirotachi Abo
9:00-10:00: Adam Van Tuyl
11:00-12:00: Francois Bergeron
2:00-3:00: Mahir Can
3:00-4:00: Diane Maclagan
The sessions will be held in the main conference room at the Fields Institute. An LCD projector and chalkboard will be available.

On Wednesday, August 2, there will be a conference trip to Niagara on the Lake for a wine tasting tour. The cost (includes transportation etc) will be about $40.

Abstracts below are listed alphabetically by speaker (as they become available)

Simple graphs and arrangement of planes in projective fourspace
Hirotachi Abo
(Colorado State U.)
Arrangements of k-planes in projective space can often be associated with special points on parameter spaces such as Hilbert schemes, which represent degenerate cases of the smooth objects being studied. If the k-plane arrangement has the additional property of being locally Cohen-Macaulay (lCM), then liaison theory can be used to produce a smooth variety in such a manner that special properties of the arrangement will be related to special properties of the smooth variety.

In this talk, we expound a method for producing lCM 2-plane arrangements in projective fourspace then apply this to produce a very special 2-plane arrangement related to the Petersen graph. We also discuss a smooth general-type surface linked to this 2-plane arrangement. The exceptional geometry of this surface reflects the combinatorics of the Petersen graph.

This is joint work with Holger Kley and Chris Peterson.

A combinatorial moduli space for polynomials of degree n
Francois Bergeron

The purpose of this talk is to present a new combinatorial classification of the pairs of curves corresponding to the respective zeros of the real and imaginary parts of degree n polynomials over C. A decomposition of resulting moduli space is given in term of forests of non intersecting special trees.
Noncommutative quotients: Invariants and Coinvariants of $\S_n$ in Noncommuting Variables
Nantel Bergeron

(York U.)
We introduce a natural Hopf algebra structure on the space of noncommutative symmetric functions which was recently studied as a vector space by Rosas and Sagan. The bases for this algebra are indexed by set partitions. We consider this algebra as a subspace of noncommutative polynomials and use it to understand the structure of the spaces of harmonics and coinvariants with respect to this collection of noncommutative polynomials and conclude two analogues of Chevalley's theorem in the noncommutative setting.

This is joint with Christophe Reutenauer, Mercedes Rosas and Mike Zabrocki.

The smooth nested Hilbert scheme of points and Catalan numbers
Mahir Can

(U. Penn)
In his inspiring article "t,q-Catalan numbers and the Hilbert scheme," Haiman showed that the Euler characteristic of a certain torus equivariant sheaf on the zero fiber of the Hilbert scheme of points in the plane is the celebrated q,t-Catalan series. Following Haiman's footsteps, we calculate an Atiyah-Bott Lefschetz formula on the smooth nested Hilbert scheme of points and obtain a new class of q,t-series similar to the q,t-Catalan series. We present a conjectural combinatorial model for this new class of series.
Irreducible components of families of level algebras
Tony Iarrobino

(Northeastern U.)
We report on recent progress in describing the possible Hilbert functions H for level Artinian algebras in heights three and four, and in describing the irreducible components of families LevAlg(H). Here LevAlg(H) parametrizes level Artinian algebras having a given Hilbert function H. We report on work joint with M. Boij, and also on work of J.O. Kleppe, F. Zanello, and others.
Duality of antidiagonals and pipe dreams
Ning Jia

(U. Penn)
Associated to every permutation w in S_n is its set of reduced pipe dreams (rc-graphs), each of which is a subset of the n by n grid. Also associated to w is a certain determinantal ideal; the generating minors have antidiagonal terms that can also be considered as subsets of the n by n grid. It is crucial for the geometry of Schubert polynomials that these two collections of subsets of the grid are dual, in a precise sense. This talk is about a direct, elementary combinatorial proof.

This is joint work with Ezra Miller.

Moduli of representations of the McKay quiver
Diane Maclagan

(Rutgers U.)
When G is a finite subgroup of SL(3) the moduli space M_theta of representations of the McKay quiver is a crepant resolution of the quotient singularity C^n/G. I will introduce these objects, and describe joint work with Alastair Craw and Rekha Thomas giving an explicit description of the component of M_theta that is birational to C^n/G for abelian G in GL(n,\mathbb C) for arbitrary n as a (not necessarily normal) toric variety. A special case of the moduli of McKay quiver representations is Nakamura's G-Hilbert scheme, and our explicit description allows us to construct pathological examples of these schemes using Grobner techniques.
Non-commutative desingularization of the generic determinant
Graham Leuschke

(Syracuse U.)
I will describe joint work with Ragnar-Olaf Buchweitz and Michel Van den Bergh showing that the generic determinantal hypersurface has a non-commutative crepant desingularization, which can be explicitly described as a "quiverized Clifford algebra."
Describing secant varieties
Jessica Sidman

(Mt. Holyoke College)
Suppose that X is a variety of dimension n in P^N. The k-th secant variety of X is the closure of the union of all planes spanned by k points of X. Secant varieties arise in classical algebraic geometry and also have interesting connections to other areas, including algebraic statistics. Consider the following question: If we have a concrete description of X, what can we say about its secant varieties? Here, we may wish to understand numerical properties such as dimension and degree, or, we may wish for an explicit description of the ideals of the secant varieties. I will discuss joint work with David Cox on describing the dimension and degree of secant varieties of smooth toric varieties when k=2. I will also discuss some situations in which the defining equations of a secant variety can be computed from the defining equations of X.
On Borel fixed ideals generated in one degree
Achilleas Sinefakopoulos

(Cornell U.)
We construct a (shellable) polyhedral cell complex that supports a minimal free resolution of a Borel fixed ideal, which is minimally generated (in the Borel sense) by just one monomial in $S=\Bbbk[x_1,x_2,...,x_n]$; this includes the case of powers of the homogeneous maximal ideal $(x_1,x_2,...,x_n)$ as a special case. In our most general result we prove that for any Borel fixed ideal $I$ generated in one degree, there exists a polyhedral cell complex that supports a minimal free resolution of $I$.
Applications of monomial ideals and computational algebra in the reverse engineering of biological networks
Mike Stillman

(Cornell U.)
Joint work with Jarrah, Laubenbacher, and Stigler.
Combinatorial secant varieties and symbolic powers
Seth Sullivant

(Harvard U.)
I will discuss the construction of joins, secant varieties, and symbolic powers in the combinatorial context of monomial ideals. For ideals generated by squarefree quadratic monomials, the generators of the secant ideals and symbolic powers record information about coloring properties of the underlying graph and its complementary graph, respectively. As a consequence, perfect graphs play a central role in the theory, and this leads to two commutative algebra versions of the celebrated Strong Perfect Graph Theorem. For general ideals, we use Groeb┐┬Żner degeneration as a tool to reduces questions about secants and symbolic powers to the monomial case. This yields a new approach to the study of determinantal and Pfaffian ideals, and their symbolic powers.
Intersection Theory for Toric Varieties
Hugh Thomas

(U. New Brunswick)
The Chow groups of an algebraic variety encode important geometric information. Unfortunately, they are, in general, extremely large and very difficult to compute. However, for toric varieties, they have a simple description due to Fulton and Sturmfels. According to this description, the Chow group of dimension $i$ can be seen as a certain quotient of the free abelian group on the torus-invariant subvarieties of dimension $i$ (and the generators and relations can be read off from the combinatorial data defining the toric variety). We will discuss this description, and also show how, by making a single global choice, one can define an intersection theory on the level of the free abelian group on the torus-invariant subvarieties, rather than on the quotients which are the Chow groups. This can be computationally more convenient; in particular, in favourable circumstances, it allows one to identify a particular cycle representing a given characteristic class. Applied to the Todd class, this allowed us to answer an old question of Danilov.

Part of this talk concerns joint work with Jamie Pommersheim.

Edge ideals of hypergraphs
Adam van Tuyl

(Lakehead U.)
Given a hypergraph H, we can associate to H a monomial ideal I(H), called the edge ideal of H. One can then ask how the combinatorial structure of the hypergraph H appears within the resolution of its edge ideal I(H). I will describe a family of hypergraphs for which there is a recursive formula to compute all the graded Betti numbers of I(H). This family includes all chordal graphs. I will also describe some bounds on the regularity of I(H) using the combinatorial information about H. I will also introduce a family of hypergraphs for which we have an exact formula for the regularity of I(H). This talk is based upon joint work with Tai Ha (Tulane University)
The total coordinate rings of Del Pezzo surfaces
Mauricio Velasco

(Cornell U.)
In this talk we will describe the total coordinate rings (Cox rings) of the surfaces obtained by blowing up ${\mathbb P}^2$ at 4,5, or 6 general points.

We prove a conjecture of V.V. Batyrev which yields a presentation of these rings as a quotient of a polynomial ring by an ideal generated by quadrics. Finally we will present some links between the Grobner deformations of these ideals and the geometry of the configurations of lines on the surfaces.

This is joint work with Mike Stillman and Damiano Testa.

Slopes of hypergeometric systems along coordinate varieties
Uli Walther

(Purdue U.)
We give a complete combinatorial description of the critical indices (or slopes) of Laurent along coordinate varieties for the $A$-hypergeometric $D$-modules $M_A(\beta)$ introduced by Gel'fand, Graev, and Zelevinski where $A$ is an integer $d\times n$ matrix defining a positive semigroup $\NN A$ and $\beta$ a complex parameter vector.

Generalizing ideas of Adolphson, we introduce the $(A,L)$-umbrella $\Phi_A^L$, a simplicial complex whose facets $\tau\in\Phi_A^L$ correspond to the minimal primes of the graded toric ring. We give an index formula for the multiplicities of such primes. The corresponding components are closures of orbits $O_A^\tau$ of the $d$-torus action.

We show that the $L$-characteristic variety of $M_A(\beta)$ is a union of closures of conormal spaces $C_A^\tau$ to orbits $O_A^\tau$ for all faces $\tau\in\Phi_A^L$. We identify the multiplicity $\mu^{L,\tau}_A$ of $C_A^\tau$ in the $L$-characterisitc cycle of the Euler--Koszul komplex $K_\bullet(S_A;\beta)$ with the intersection multiplicity between the Euler ideal and $S_A$ along $C_A^\tau$, and we provide an explicit index/volume formula. For the multiplicity $\mu_{A,0}^{L,\tau}(\beta)$ of $M_A(\beta)$, we show that $\mu_{A,0}^{L,\tau}(\beta)\ge\mu^{L,\tau}_A$ with equality whenever $\beta$ is generic or $\tau$ is a facet. In the special case where $L=F$ is the order filtration and $\tau=\emptyset$, our inequality reduces to the classical result $\rk(M_A(\beta))\ge\vol(A)$.

For the family of filtrations $L=pF+qV$ indexed by $p/q\in\QQ_{>0}\cup\infty$ where $V$ is the $V$-filtration along a coordinate variety $Y$, our results show that $p/q+1$ is a slope of $M_A(\beta)$ along $Y$ at the origin if and only if the $(A,L')$-umbrella $\Phi_A^{L'}$ jumps at $L'=L$. In particular, this proves a converse to Hotta's Theorem: an $A$-hypergeometric system is regular holonomic precisely when it is homogeneous in the usual sense. Finally, we introduce a natural projectivization of $M_A(\beta)$ to a $\calD$-module $\calM_A(\beta)$ on $(\PP_\CC^1)^n$. We provide generalizations of our prevoius results and in particular give a combinatorial description of the $L$-characteristic variety of $\calM_A(\beta)$.

Partition arrangements and Springer fibers
Alexander Woo

(UC Davis)
We calculate defining ideals for certain S_n invariant subspace arrangements of the braid arrangement and relate them (in part using duality) to the cohomology rings of Springer fibers. This allows us to calculate their graded characters to be particular sums of Hall-Littlewood polynomials. If time permits, I will also describe the connection to the Hilbert scheme of points which originally motivated this project.

This is joint work with Mark Haiman.

Jet schemes of monomial schemes and determinantal varieties
Cornelia Yuen

We will start with a brief introduction to jet schemes, followed by some results on the scheme structure of the jet schemes of certain monomial schemes and determinantal varieties. Some concrete examples will be used for illustration. If time permits, we will talk a little about truncated wedge schemes (a higher dimensional analog of jet schemes).

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