| 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
|
(UQAM)
|
| 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¿Â½ner 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
|
(U.Michigan)
|
| 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).
|
