Commutative Algebra and Algebraic Geometry
Org: Ragnar-Olaf Buchweitz (Toronto), Graham Leuschke (Syracuse) and Greg Smith (Queen's)
- LUCHEZAR L. AVRAMOV, University of Nebraska
Koszul modules over small graded rings
The title refers to rings defined by more than \binom e2 quadrics in
e indeterminates. It will be shown that generic modules with
certain types of Hilbert series have linear resolutions.
This is joint work with Srikanth Iyengar and Liana Sega.
- SUNIL CHEBOLU, University of Western Ontario
Which finite p-groups are like a finite product of
A ghost in a tensor triangulated category is a map that induces the
zero map on all homotopy groups. We identify the derived categories
of commutative rings and the stable module categories of finite
p-groups that do not have non-trivial ghosts. In particular, we
will solve the riddle posed in the title: which finite p-groups are
like a finite product of fields?
This is joint work with Dan Christensen and Jan Minac.
- KIA DALILI, Dalhousie University, Chase Building, Halifax, NS, B3H 3J5
The reconstruction conjecture for edge ideals
Given a simple graph G on n vertices, let the deck of G be the
collection of unlabeled subgraphs of G obtained by removing one
vertex from G. An invariant of a graph is called reconstructible if
it has the same value for any two graphs with the same deck. Graph
theorists have studied reconstruction of combinatorial invariant of
G as an strategy to prove the isomorphism class of G is
reconstructible. We prove that it is possible to reconstruct several
algebraic properties of the edge ideal from the deck of G. These
properties include Krull dimension, Hilbert function, and all graded
Betti numbers bi,j where j < n.
- HARM DERKSEN, University of Michigan, Ann Arbor
A symmetric function generalization of the Tutte polynomial
I will discuss ideals associated to subspace arrangements. The
Hilbert series of the product of the ideal corresponding to the
subspaces is a combinatorial invariant. Using this, I will construct
a generalization of the Tutte polynomial.
- NEIL EPSTEIN, University of Michigan, Ann Arbor, MI
Pieces of closures
I will discuss recent work on breaking closure operations into
specific, useful parts in various different ways. This applies to
integral closure, tight closure, and other closures, and includes
"special (parts of)" closures and "interiors" of closures. This
is useful, for instance, in analyzing "spreads" (how many elements
does it take to generate an ideal which gives the closure of a given
ideal?) and extending the Briançon-Skoda Theorem.
- SARA FARIDI, Dalhousie University, Halifax, NS, B3J 3J5
On ideals of conjugacy classes of nilpotent matrices
We discuss ideals associated to the closure of conjugacy classes of
nilpotent matrices. These ideals are indexed by partitions of the
size of the matrix. We then restrict to the intersection of the
conjugacy class with diagonal matrices, and use a well-known
generating set by Tanisaki to produce a new and smaller generating set
for these ideals. We also find a minimal generating set in the case
of hook partitions, which enables us to easily compute the minimal
This is joint work with Riccardo Biagioli and Mercedes Rosas.
- NOAM HORWITZ, Cornell University, Ithaca, NY 14853
Linear resolutions of edge ideals
Edge ideals are monomial ideals defined by graphs. We study the
minimal free resolutions of such ideals in the case where the
resolutions are linear. Explicit resolutions are given under the
assumption that the graph associated with the edge ideal satisfies
specific combinatorial conditions. Furthermore, we construct a
regular cell complex supporting the minimal free resolution in such
- COLIN INGALLS, University of New Brunswick, Dept. of Math and Stats
Spaces of Linear Modules on Regular Graded Clifford
The space of regular noncommutative algebras includes regular graded
Clifford algebras, which correspond to base point free linear systems
of quadrics in dimension n in Pn. The schemes of linear modules
for these algebras can be described in terms of this linear system.
We show that the space of line modules on a 4-dimensional algebra is
an Enriques surface called the Reye congruence, and we extend this
result to higher dimensions.
- SRIKANTH B. IYENGAR, University of Nebraska, Department of Mathematics, Lincoln,
Dimension of the stable category of a commutative ring
Rouquier has introduced a notion of dimension for triangulated
categories, and provided estimates for the dimension of stable derived
categories of exterior algebras. In my talk, I will discuss similar
results pertaining to the stable category of a commutative noetherian
- DAVID A. JORGENSEN, University of Texas at Arlington, Arlington, TX 76019, USA
Linear acyclic complexes
In this talk we will investigate the existence of linear acyclic
complexes of finitely generated free modules over a commutative local
ring with radical cube zero.
- ANTONIO LAFACE, Queen's
- DIANE MACLAGAN, Rutgers University, Piscataway, NJ 08854, USA
Equations and degenerations of [`(M)]0,n
The moduli space [`(M)]0,n has as points all stable genus
zero points with n marked points. I will introduce this object, and
describe joint work with Angela Gibney (Penn) that gives explict
equations for [`(M)]0,n in the Cox ring of a related toric
variety. An application of this is an explicit construction of a
degeneration of [`(M)]0,n to a toric variety.
- CLAUDIA MILLER, Syracuse University, Syracuse, NY 13244
A Riemann-Roch formula for the blow-up of a nonsingular
Hilbert polynomials have classically been shown to be determined by
intersections with hyperplanes. We give another approach over regular
local rings via Intersection Theory on the blow-up scheme. The result
comes in the form of a Riemann-Roch formula for the blow-up of a
nonsingular affine scheme.
- IRENA PEEVA, Cornell University, Ithaca, NY 14853, USA
Generalized Green's Theorem
Green proved Green's Theorem on how the Hilbert function changes after
taking a quotient by a generic linear form. He used this result to
provide a new and simple proof of Macaulay's Theorem, which
characterizes the Hilbert functions of graded ideals in a polynomial
ring. Herzog and Popescu extended Green's result to generic forms of
any degree, but under the assumption that the ground field has
characteristic zero. Later, Gasharov found a new proof that works in
all characteristics. We provide a different proof, which works in all
characteristics and which works not only over polynomial rings but
also yields the new result that the theorem holds over
Clements-Lindstrom quotient rings.
This is joint work with Jeff Mermin.
- THUY PHAM, University of Toronto at Scarborough
jdeg of algebraic structures
Let R be a commutative Noetherian ring and A a finitely generated
standard graded R-algebra. We introduce and develop a new degree
jdeg(·) attached to finitely generated graded A-modules.
This construction jdeg(·) coincides with the classical
multiplicity deg(·) when R is an Artinian local ring. It
also acquires a global nature in contrast to other extensions of
deg(·) usually requiring R to be local or graded.
An important application of jdeg(·), which is also the original
motivation of this notion, is to measure the length of the chains of
graded subalgebras between A and its integral closure [`(A)],
constructed by general algorithms. This gives a refinement of recent
results to very general graded algebras.
- RAVI VAKIL, Stanford University, Stanford, CA 94305
Murphy's Law in algebraic geometry: badly-behaved moduli
We consider the question: "How bad can the deformation space of an
object be?" (Alternatively: "What singularities can appear on a
moduli space?") The answer seems to be: "Unless there is some
a priori reason otherwise, the deformation space can be
arbitrarily ugly." Hence many of the most important moduli spaces in
algebraic geometry are arbitrarily singular, justifying a philosophy
More precisely, every singularity of finite type over Z (up
to smooth parameters) appears on the Hilbert scheme of curves in
projective space, and the moduli spaces of: smooth projective
general-type surfaces (or higher-dimensional varieties), plane curves
with nodes and cusps, stable sheaves, isolated threefold
singularities, and more. The objects themselves are not pathological,
and are in fact as nice as can be: the curves are smooth, the surfaces
have very ample canonical bundle, the stable sheaves are torsion of
rank 1, the singularities are normal and Cohen-Macaulay, etc.
Thus one can construct a smooth curve in projective space whose
deformation space has any specified number of components, each with
any specified singularity type, with any specified non-reduced
behaviour along various associated subschemes. Similarly one can give
a surface over Fp that lifts to p7 but not p8. (Of
course the results hold in the holomorphic category as well.)
- ADAM VAN TUYL, Lakehead University, Thunder Bay, ON, P7B 5E1
Some resolutions of double points in P1 × P1
Let Z be a finite set of double points in P1 ×P1 and suppose further that X, the support of Z, is
arithmetically Cohen-Macaulay (ACM). I will present an algorithm,
which depends only upon a combinatorial description of X, for the
bigraded Betti numbers of IZ, the defining ideal of Z.
This is joint work with Elena Guardo of Catania.
- MAURICIO VELASCO, Cornell University, Ithaca, NY
Grobner bases, monomial group actions and the Cox rings of
Del Pezzo surfaces
We introduce the notion of monomial group action and study some of its
consequences for Gröbner basis theory. As an application we prove a
conjecture of V. Batyrev and O. Popov describing the Cox rings of Del
Pezzo surfaces (of degree ³ 3) as quotients of a polynomial ring
by an ideal generated by quadrics.
The results presented in this talk are joint work with Mike Stillman
and Damiano Testa.
- ALEXANDER YONG, University of Minnesota and the Fields Institute
A combinatorial rule for (co)minuscule Schubert calculus
We prove a root system uniform, concise and positive combinatorial
rule for Schubert calculus of minuscule and cominuscule
flag manifolds G/P (the latter are also known as compact
Hermitian symmetric spaces). We connect this geometry to the poset
combinatorics of [Proctor '04], thereby giving a generalization of the
[Schützenberger '77] jeu de taquin formulation of the
Littlewood-Richardson rule, which computes the intersection numbers
of Grassmannian Schubert varieties. Our proof introduces
cominuscule recursions, a general technique to relate the
numbers for different Lie types.
I will also briefly discuss connections of the rule to (geometric)
representation theory, specifically to Kostant's study of Lie algebra
cohomology, and separately, the geometric Satake correspondence of
Ginzburg, Mirkovi\'c-Vilonen et al.
This is based on joint work with Hugh Thomas; see math.AG/0608276.